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| Boilerpipe Text | Article
20 March 2026
and
School of Mathematics, Statistics and Mechanics, Beijing University of Technology, Beijing 100021, China
*
Author to whom correspondence should be addressed.
Abstract
In this work, we address the nonhomogeneous initial-boundary value problem for the coupled Hirota equation posed on the finite interval
[
0
,
L
]
. To investigate the well-posedness of this problem, we first adopt an appropriate transformation, namely the Laplace transform, which is tailored to the specific characteristics of the problem, and further obtain an explicit solution formula for the linear inhomogeneous coupled system. Subsequently, the local well-posedness of the original nonhomogeneous initial-boundary value problem in
X
s
,
T
×
X
s
,
T
X
s
,
T
=
C
(
0
,
T
;
H
s
(
0
,
1
)
)
∩
L
2
(
0
,
T
;
H
s
+
1
(
0
,
1
)
)
is rigorously established through the combination of this explicit formula, the contraction mapping principle and energy estimates.
MSC:
35Q55; 35Q53; 35G61
1. Introduction
We study the coupled Hirota (cH) equation
i
u
t
+
α
1
∂
x
2
u
+
i
ς
∂
x
3
u
+
i
χ
2
|
u
|
2
u
x
+
|
v
|
2
u
x
+
u
v
¯
v
x
+
δ
|
u
|
2
u
+
|
v
|
2
u
=
0
,
i
v
t
+
α
2
∂
x
2
v
+
i
ς
∂
x
3
v
+
i
χ
|
u
|
2
v
x
+
2
|
v
|
2
v
x
+
u
¯
v
u
x
+
δ
|
u
|
2
v
+
|
v
|
2
v
=
0
,
(cH)
where
α
i
(
i
=
1
,
2
)
,
ς
∈
R
+
,
χ
,
δ
∈
R
are prescribed real constants, and
u
,
v
denote complex-valued functions. Tasgal and Potasek [
1
] first derived the coupled Hirota equation from the single Hirota equation, showing that third-order dispersion and self-steepening can support the simultaneous propagation of two ultrashort pulses while suppressing inelastic Raman scattering effect.
The Hirota equation
i
u
t
+
α
∂
x
2
u
+
i
ς
∂
x
3
u
+
i
χ
∂
x
(
|
u
|
2
u
)
+
δ
|
u
|
2
u
=
0
,
α
ς
χ
δ
≠
0
,
proposed by R. Hirota in 1973 [
2
], stands as a fundamental generalization of the integrable nonlinear Schrödinger (NLS) equation. It provides an accurate description of ultrashort optical pulse propagation in single-mode fibers. To be precise, the NLS equation only accounts for second-order dispersion and the Kerr effect (for picosecond pulses). Extra effects have to be incorporated when the pulse length is on the order of the wavelength or the pulse width reaches the femtosecond regime. In such cases, the NLS equation is generalized to the Hirota equation. The Hirota equation, also known as the Airy–Schrödinger equation, includes the classic NLS equation for
α
=
1
,
ς
=
χ
=
0
, and the derivative NLS equation for
α
=
1
,
ς
=
δ
=
0
.
The Hirota equation is completely integrable for some special parameters [
2
],
i
u
t
+
∂
x
2
u
+
i
ς
∂
x
3
u
+
6
i
ς
|
u
|
2
u
x
+
2
|
u
|
2
u
=
0
,
which possesses infinite conservation laws with respect to the Cauchy problem. Employing the conservation laws, one can readily derive the global well-posedness of the Hirota equation in
H
s
for
s
∈
N
. Evidently, the fact that the Hirota equation is completely integrable does not affect its well-posedness, allowing us to investigate the Hirota equations with more general parameters. Previously, numerous studies have focused on the Hirota equation and the coupled Hirota equation [
3
,
4
,
5
,
6
,
7
,
8
]. The earlier results on higher regularity for the Hirota equation were established by Guo and Tan [
4
] in
H
s
for
s
≥
3
. Following this, Ref. [
3
] generalized the earlier local result to
H
s
for
s
>
3
4
and the global result to
H
s
for
s
=
1
and
s
≥
2
.
In [
5
], Staffilani also proved the low regularity in
H
s
for
s
>
1
4
.
By using more meticulous methods, the results of well-posedness enter the discussion of low-regularity cases (see [
6
,
7
,
8
]). The optimal regularity result for the Hirota equation was achieved in [
7
], which established the local result in
H
s
for
s
>
−
1
4
and global result in
H
s
for
s
>
0
.
In addition, the only existing result concerning the Cauchy problem of (cH) in
H
s
×
H
s
for
s
≥
1
4
was rigorously demonstrated in [
9
].
This manuscript is devoted to investigating the IBVP for (cH) defined on a finite interval:
i
u
t
+
α
1
∂
x
2
u
+
i
ς
∂
x
3
u
+
i
χ
2
|
u
|
2
u
x
+
|
v
|
2
u
x
+
u
v
¯
v
x
+
δ
|
u
|
2
u
+
|
v
|
2
u
=
0
,
i
v
t
+
α
2
∂
x
2
v
+
i
ς
∂
x
3
v
+
i
χ
(
|
u
|
2
v
x
+
2
|
v
|
2
v
x
)
+
u
¯
v
u
x
+
δ
|
u
|
2
v
+
|
v
|
2
v
=
0
,
u
(
x
,
0
)
=
u
0
(
x
)
,
v
(
x
,
0
)
=
v
0
(
x
)
,
x
∈
R
+
,
u
(
0
,
t
)
=
ϑ
1
(
t
)
,
u
(
L
,
t
)
=
ϑ
2
(
t
)
,
u
x
(
L
,
t
)
=
ϑ
3
(
t
)
,
t
∈
R
+
,
v
(
0
,
t
)
=
ϱ
1
(
t
)
,
v
(
L
,
t
)
=
ϱ
2
(
t
)
,
v
x
(
L
,
t
)
=
ϱ
3
(
t
)
,
t
∈
R
+
,
(1)
where
L
is positive nonzero constants, and without loss of generality, let
L
=
1
.
Recall the result of the IBVP concerning
R
+
or a interval
[
0
,
L
]
.
Early research on IBVP primarily focused on high regularity due to the energy method. For the IBVP posed on
R
+
, Colliander, Kenig [
10
], and Holmer [
11
,
12
] extended the well-posedness to the low-regularity case. Their proof relied on the Duhamel boundary operator and Bourgain space
X
s
,
b
, which was proposed by Bourgain in [
13
,
14
]. For the IBVP posed on a finite interval, due to the Duhamel boundary operator being unsuitable for equations exhibiting mixed dispersion effects, Bona et al. [
15
,
16
,
17
] proved the low-regularity existence for the KdV and NLS equation via the Laplace transform. Reference is made to [
16
,
18
,
19
,
20
,
21
,
22
,
23
,
24
,
25
,
26
,
27
,
28
,
29
,
30
,
31
] for results involving the IBVP for other dispersive equations.
Our primary goal herein is to analyze the IBVP of the cH equation within a bounded interval framework. In doing so, we resolve the core issues of existence and uniqueness, and further demonstrate the continuous dependence on the prescribed data under suitable regularity constraints. The Laplace transform is specially selected in this work to address the core challenges brought by the mixed dispersion inherent in the coupled Hirota equation. The third-order spatial derivative term complicates the standard Bourgain space methods for addressing the associated IBVP, making them far more intricate than the approaches applied to the second-order NLS equation. It is evident from the proof that existence can be obtained on the initial data
(
u
0
(
x
)
,
v
0
(
x
)
)
∈
H
s
(
0
,
1
)
:
=
H
s
(
0
,
1
)
×
H
s
(
0
,
1
)
and boundary data
ϑ
1
(
t
)
,
ϑ
2
(
t
)
,
ϑ
3
(
t
)
,
ϱ
1
(
t
)
,
ϱ
2
(
t
)
,
ϱ
3
(
t
)
∈
Z
s
:
=
H
s
+
1
3
(
R
+
)
×
H
s
+
1
3
(
R
+
)
×
H
s
3
(
R
+
)
.
In order to address the well-posedness of IBVP, the given data must satisfy the following compatibility condition and definition:
u
0
,
ι
(
0
)
=
ϑ
1
(
ι
)
(
0
)
,
u
0
,
ι
(
1
)
=
ϑ
2
(
ι
)
(
0
)
,
u
0
,
ι
′
(
1
)
=
ϑ
3
(
ι
)
(
0
)
,
v
0
,
ι
(
0
)
=
ϱ
1
(
ι
)
(
0
)
,
v
0
,
ι
(
1
)
=
ϱ
2
(
ι
)
(
0
)
,
v
0
,
ι
′
(
1
)
=
ϱ
3
(
ι
)
(
0
)
,
where
u
0
,
ι
(
0
)
=
u
0
(
x
)
,
v
0
,
ι
(
0
)
=
v
0
(
x
)
if
ι
=
0
and for
ι
≥
1
,
u
0
,
ι
(
x
)
=
i
α
1
u
0
,
ι
−
1
″
−
ς
u
0
,
ι
−
1
‴
+
i
δ
∑
j
=
0
ι
−
1
(
|
u
0
,
j
|
2
+
|
v
0
,
j
|
2
)
u
0
,
ι
−
1
−
j
−
χ
∑
j
=
0
ι
−
1
{
(
2
|
u
0
,
j
|
2
+
|
v
0
,
j
|
2
)
u
0
,
ι
−
1
−
j
′
+
u
0
,
j
v
¯
0
,
ι
−
1
−
j
v
0
,
ι
−
1
−
j
′
}
,
v
0
,
ι
(
x
)
=
i
α
2
v
0
,
ι
−
1
″
−
ς
v
0
,
ι
−
1
‴
+
i
δ
∑
j
=
0
ι
−
1
(
|
u
0
,
j
|
2
+
|
v
0
,
j
|
2
)
v
0
,
ι
−
1
−
j
−
χ
∑
j
=
0
ι
−
1
{
(
|
u
0
,
j
|
2
+
2
|
v
0
,
j
|
2
)
v
0
,
ι
−
1
−
j
′
+
u
¯
0
,
j
u
0
,
j
′
v
0
,
ι
−
1
−
j
}
.
Definition
1.
For arbitrary
s
∈
[
0
,
+
∞
)
and
T
∈
(
0
,
+
∞
)
, the compatibility condition for
u
0
(
x
)
,
v
0
(
x
)
,
ϑ
→
(
t
)
,
ϱ
→
(
t
)
=
u
0
(
x
)
,
v
0
(
x
)
,
ϑ
1
(
t
)
,
ϑ
2
(
t
)
,
ϑ
3
(
t
)
,
ϱ
1
(
t
)
,
ϱ
2
(
t
)
,
ϱ
3
(
t
)
∈
H
s
(
0
,
1
)
×
Z
s
×
Z
s
is as follows:
1.
If
s
−
3
s
3
≤
1
2
,
u
0
,
ι
(
0
)
=
ϑ
1
(
ι
)
(
0
)
,
u
0
,
ι
(
1
)
=
ϑ
2
(
ι
)
(
0
)
,
v
0
,
ι
(
0
)
=
ϱ
1
(
ι
)
(
0
)
,
v
0
,
ι
(
1
)
=
ϱ
2
(
ι
)
(
0
)
,
(2)
is valid for
ι
=
0
,
1
,
⋯
,
s
3
−
1
.
2.
If
1
2
<
s
−
3
s
3
<
3
2
, (
2
) is valid for
ι
=
0
,
1
,
⋯
,
s
3
.
3.
If
s
−
3
s
3
>
3
2
,
u
0
,
ι
(
0
)
=
ϑ
1
(
ι
)
(
0
)
,
u
0
,
ι
(
1
)
=
ϑ
2
(
ι
)
(
0
)
,
u
0
,
ι
′
(
1
)
=
ϑ
3
(
ι
)
(
0
)
v
0
,
ι
(
0
)
=
ϱ
1
(
ι
)
(
0
)
,
v
0
,
ι
(
1
)
=
ϱ
2
(
ι
)
(
0
)
,
v
0
,
ι
′
(
1
)
=
ϱ
3
(
ι
)
(
0
)
.
is valid for
ι
=
0
,
1
,
⋯
,
s
3
−
1
.
4.
We agree that (
2
) is vacuous if
s
3
−
1
<
0
.
It is obvious that higher-order time derivatives cannot be directly observed from boundary value, and they must be computed recursively through the equation itself along with its initial conditions. In light of the previously introduced compatibility conditions, we now formulate a theorem, which constitutes the central focus of our investigation.
Theorem
1.
For
s
∈
[
0
,
3
2
)
and
T
>
0
, we assume that
(
U
0
,
ϱ
→
)
:
=
u
0
(
x
)
,
v
0
(
x
)
,
ϑ
→
(
t
)
,
ϱ
→
(
t
)
=
u
0
(
x
)
,
v
0
(
x
)
,
ϑ
1
(
t
)
,
ϑ
2
(
t
)
,
ϑ
3
(
t
)
,
ϱ
1
(
t
)
,
ϱ
2
(
t
)
,
ϱ
3
(
t
)
∈
H
s
(
0
,
1
)
×
Z
s
×
Z
s
,
which satisfies Definition 1. Then, IBVP (
1
) admits a unique solution in
X
s
,
T
×
X
s
,
T
,
where
X
s
,
T
=
C
0
,
T
;
H
s
(
0
,
1
)
∩
L
2
0
,
T
;
H
s
+
1
(
0
,
1
)
and
T
:
=
T
u
0
H
s
,
v
0
H
s
,
ϑ
→
Z
s
,
ϱ
→
Z
s
.
In addition, Lipschitz continuity also holds for the given initial and boundary values.
Corollary
1.
For
s
≥
3
2
and
T
>
0
, assume that
u
0
(
x
)
,
v
0
(
x
)
,
ϑ
→
(
t
)
,
ϱ
→
(
t
)
∈
H
s
(
0
,
1
)
×
Z
s
×
Z
s
, which satisfies Definition 1. Then there exists a unique solution of (
1
) for sufficiently small T.
Compared with the scalar Hirota equation, the coupled Hirota (cH) system introduces interdependent nonlinear cross-coupling terms (i.e.,
u
2
v
x
,
u
v
¯
v
x
,
v
2
u
x
,
u
¯
v
u
x
) and asymmetric linear dispersion terms with distinct coefficients
α
1
,
α
2
for the two components
u
and
v
.
These coupling characteristics give rise to fundamental mathematical challenges, such as cross-dependent nonlinearities in energy estimates and compatibility conditions for the coupled system, which do not arise in the scalar case.
Theorem 1 and Corollary 1 establish the local low regularity in the energy space
H
s
(
0
,
1
)
×
H
s
(
0
,
1
)
. To clarify this definition of regularity, it is necessary to specify the precise meaning of
(
u
,
v
)
as a solution to (
1
). This is particularly crucial for low regularity cases, where the solution must satisfy the equation in a suitable sense. For IBVP with a bounded interval, the concept of a mild solution is typically defined to ensure that the solution satisfies the nonlinear term along with the given data.
In earlier research of IBVP, the standard approach to handling nonhomogeneous boundary data involved transforming the dependent variable to achieve homogeneous boundary conditions. The specific approach involved solving a combination of the boundary value and the boundary conditions, thereby making the boundary conditions equal to 0, but at this point, the new variable satisfied a new form of equation. While this approach permits converting the IBVP into an initial value problem for investigation, it requires the boundary data to possess higher regularity compared to more recent methods. To achieve low-regularization of the well-posedness of IBVP, the Duhamel boundary forcing operator serves as a powerful tool. However, due to its specific construction form, it is not applicable to equations exhibiting mixed dispersion effects (e.g., Hirota equation). Proof of the main results adapt the strategy proposed by Bona et al. [
16
], who developed a strategy involving the Laplace transform to construct a solution formula that incorporates both the linear equation and the boundary operator. Specifically, the prior estimates encompass both linear and nonlinear estimates. The linear estimate is derived based on the energy method, while the nonlinear estimate is obtained by Lemma 4 and the interpolation theorem.
Below is the structure of the rest of this work.
Section 2
establishes the foundational framework: it standardizes notation, formalizes function space definitions, and deriving a solution representation that decomposes the solution into linear and nonlinear evolution. The analytical backbone of this work is presented in
Section 3
, where we develop the critical estimates required to implement a contraction mapping argument. Finally,
Section 4
is dedicated to rigorously proving the main result (Theorem 1) and its associated corollary (Corollary 1).
2. Preliminaries
2.1. Notations
We use the following notation for function spaces:
C
t
0
H
x
s
=
C
0
,
T
;
H
s
(
0
,
1
)
L
t
p
H
x
s
=
L
p
0
,
T
;
H
s
(
0
,
1
)
,
p
∈
[
1
,
∞
)
X
s
,
T
=
C
0
,
T
;
H
s
(
0
,
1
)
∩
L
2
0
,
T
;
H
s
+
1
(
0
,
1
)
,
and
Z
s
=
Z
s
:
=
H
s
+
1
3
(
R
+
)
×
H
s
+
1
3
(
R
+
)
×
H
s
3
(
R
+
)
,
where we write the inhomogeneous
L
2
-based space
H
s
=
H
s
(
R
)
as
∥
g
∥
H
s
(
R
)
=
⟨
ξ
⟩
s
g
^
(
ξ
)
L
ξ
2
.
We denote the space
H
0
s
as
H
0
s
(
φ
)
=
the
closure
of
D
(
φ
)
in
H
s
(
φ
)
,
Let
φ
be an open set in
R
1
, and let both m and n be continuous functions. Define the space
D
(
φ
)
as the set of all functions
ϕ
, such that
ϕ
is infinitely differentiable on
φ
and has compact support contained in
φ
.
Finally, throughout this paper, unless otherwise stated, we abbreviate
∥
·
∥
H
s
as
∥
·
∥
H
x
s
(
0
,
1
)
for simplicity. In addition, the notation
a
≲
(
≳
)
b
is used to denote that
a
≤
(
≥
)
C
b
, where
C
is a positive constant.
2.2. Solution Formula
With the aim of constructing a solution formula for Equation (
1
) using the Laplace transform. In order to establish the solution to problem (
1
), we examine the following nonhomogeneous initial-boundary value problem:
i
u
t
+
α
1
∂
x
2
u
+
i
ς
∂
x
3
u
+
Q
1
(
u
,
v
)
=
0
,
i
v
t
+
α
2
∂
x
2
v
+
i
ς
∂
x
3
v
+
Q
2
(
u
,
v
)
=
0
,
u
(
x
,
0
)
=
u
0
(
x
)
,
v
(
x
,
0
)
=
v
0
(
x
)
,
u
(
0
,
t
)
=
ϑ
1
(
t
)
,
u
(
1
,
t
)
=
ϑ
2
(
t
)
,
u
x
(
1
,
t
)
=
ϑ
3
(
t
)
,
t
∈
R
+
,
v
(
0
,
t
)
=
ϱ
1
(
t
)
,
v
(
1
,
t
)
=
ϱ
2
(
t
)
,
v
x
(
1
,
t
)
=
ϱ
3
(
t
)
,
t
∈
R
+
,
(3)
where
Q
1
(
u
)
=
i
χ
2
|
u
|
2
u
x
+
|
v
|
2
u
x
+
u
v
¯
v
x
+
δ
|
u
|
2
u
+
|
v
|
2
u
,
Q
2
(
u
,
v
)
=
i
χ
(
|
u
|
2
v
x
+
2
|
v
|
2
v
x
)
+
u
¯
v
u
x
+
δ
|
u
|
2
v
+
|
v
|
2
v
and
(
u
0
,
v
0
,
ϑ
→
,
ϱ
→
)
=
(
u
0
,
v
0
,
ϑ
1
,
ϑ
2
,
ϑ
3
,
ϱ
1
,
ϱ
2
,
ϱ
3
)
∈
H
s
×
Z
s
×
Z
s
that satisfies Definition 1. We first decompose the IBVP (
3
) into two distinct problems: a nonlinear system which satisfies initial data
i
u
t
+
α
1
∂
x
2
u
+
i
ς
∂
x
3
u
+
Q
1
(
u
,
v
)
=
0
,
i
v
t
+
α
2
∂
x
2
v
+
i
ς
∂
x
3
v
+
Q
2
(
u
,
v
)
=
0
,
u
(
x
,
0
)
=
u
0
(
x
)
,
v
(
x
,
0
)
=
v
0
(
x
)
,
u
(
0
,
t
)
=
0
,
u
(
1
,
t
)
=
0
,
u
x
(
1
,
t
)
=
0
,
t
∈
R
+
,
v
(
0
,
t
)
=
0
,
v
(
1
,
t
)
=
0
,
v
x
(
1
,
t
)
=
0
,
t
∈
R
+
,
(4)
and a linear system which satisfies nonhomogeneous boundary conditions
i
u
t
+
α
1
∂
x
2
u
+
i
ς
∂
x
3
u
=
0
,
i
v
t
+
α
2
∂
x
2
v
+
i
ς
∂
x
3
v
=
0
,
u
(
x
,
0
)
=
0
,
v
(
x
,
0
)
=
0
,
u
(
0
,
t
)
=
ϑ
1
(
t
)
,
u
(
1
,
t
)
=
ϑ
2
(
t
)
,
u
x
(
1
,
t
)
=
ϑ
3
(
t
)
,
v
(
0
,
t
)
=
ϱ
1
(
t
)
,
v
(
1
,
t
)
=
ϱ
2
(
t
)
,
v
x
(
1
,
t
)
=
ϱ
3
(
t
)
.
(5)
Guided by the solution construction strategy from [
16
], we thus express the solution as
(
Ψ
1
(
u
,
v
)
,
Ψ
2
(
u
,
v
)
)
=
(
u
(
x
,
t
)
,
v
(
x
,
t
)
)
=
(
ϕ
1
,
ϕ
2
)
+
(
ψ
1
,
ψ
2
)
,
(6)
where
Ψ
1
(
u
,
v
)
and
Ψ
2
(
u
,
v
)
are the integral operators, the initial solution operators
ϕ
1
,
ϕ
2
are the solutions of (
4
) with initial value, and the boundary solution operators
ψ
1
,
ψ
2
are the solutions of the linear problem (
5
) with nonhomogeneous boundary conditions. In fact, to derive these two boundary operators for (
5
), it suffices to consider only one equation in the system, since the dispersion effects are identical for both equations and, consequently, the boundary operators take the same form. Therefore, we focus on the following equation:
i
Υ
t
+
α
Υ
x
x
+
i
ς
Υ
x
x
x
=
0
,
Υ
(
x
,
0
)
=
0
,
Υ
(
0
,
t
)
=
h
1
(
t
)
,
Υ
(
1
,
t
)
=
h
2
(
t
)
,
Υ
x
(
1
,
t
)
=
h
3
(
t
)
,
(7)
where
α
=
α
1
or
α
2
,
Υ
=
u
or
v
,
h
j
(
t
)
=
ϑ
j
(
t
)
or
ϱ
j
(
t
)
for
j
=
1
,
2
,
3
.
Equation (
7
) can be reformulated as a family of BVP by Laplace transform in
t
:
i
δ
Υ
˜
(
x
,
δ
)
+
α
Υ
˜
x
x
+
i
ς
Υ
˜
x
x
x
=
0
Υ
˜
(
0
,
δ
)
=
h
˜
1
(
δ
)
,
Υ
˜
(
1
,
δ
)
=
h
˜
2
(
δ
)
,
Υ
˜
(
1
,
δ
)
=
h
˜
3
(
δ
)
,
(8)
where
Υ
˜
(
x
,
δ
)
=
∫
0
+
∞
e
−
δ
t
Υ
(
x
,
t
)
d
t
. The characteristic equation of (8) is
δ
−
i
α
φ
2
+
ς
φ
3
=
0
.
It is clear that
x
3
=
i
has three roots:
cos
1
6
π
+
i
sin
1
6
π
,
cos
5
6
π
+
i
sin
5
6
π
,
cos
9
6
π
+
i
sin
9
6
π
,
Letting
δ
=
(
i
ξ
)
3
=
−
i
ξ
3
(
ξ
≥
0
)
, by means of the perturbation analysis, the roots of the characteristic equation for
ξ
→
+
∞
can be derived as
φ
1
(
ξ
)
=
−
i
ξ
+
∘
(
1
ξ
)
,
φ
2
(
ξ
)
=
ξ
(
cos
1
6
π
+
i
sin
1
6
π
)
+
∘
(
1
ξ
)
,
φ
3
(
ξ
)
=
ξ
(
cos
5
6
π
+
i
sin
5
6
π
)
+
∘
(
1
ξ
)
.
The solution
Υ
˜
(
x
,
δ
)
of (8) takes the form
u
˜
(
x
,
δ
)
=
∑
j
=
1
3
c
j
(
δ
)
e
φ
j
(
δ
)
x
,
where
c
j
satisfies the following linear equations:
c
1
+
c
2
+
c
3
=
h
˜
1
(
δ
)
,
c
1
e
h
1
(
δ
)
+
c
2
e
φ
2
(
δ
)
+
c
3
e
φ
3
(
δ
)
=
h
˜
2
(
δ
)
,
c
1
φ
1
(
δ
)
e
φ
1
(
δ
)
+
c
2
φ
2
(
δ
)
e
φ
2
(
δ
)
+
c
3
φ
3
(
δ
)
e
φ
3
(
δ
)
=
h
˜
3
(
δ
)
.
(9)
Applying Cramer’s rule, we obtain
c
j
=
D
j
(
δ
)
D
(
δ
)
,
where
D
(
s
)
represents the determinant of the coefficient matrix, with
D
j
(
s
)
given by substituting the
j
-th column with
(
h
˜
1
(
s
)
,
h
˜
2
(
s
)
,
h
˜
3
(
s
)
)
.
Υ
admits the following representation after taking the Mellin transform with a given
χ
>
0
:
Υ
(
x
,
t
)
=
∑
j
=
1
3
1
2
π
i
∫
χ
−
i
∞
χ
+
i
∞
e
δ
t
Υ
˜
(
x
,
δ
)
d
δ
=
∑
j
=
1
3
1
2
π
i
∫
χ
−
i
∞
χ
+
i
∞
e
δ
t
D
j
(
δ
)
D
(
δ
)
e
φ
j
x
d
δ
.
(10)
The solution
Υ
of (7) can be expressed a sum:
Υ
(
x
,
t
)
=
S
0
,
h
→
(
0
)
=
Υ
1
(
x
,
t
)
+
Υ
2
(
x
,
t
)
+
Υ
3
(
x
,
t
)
,
where
Υ
m
is a solution to (7) with
φ
j
=
0
for
j
≠
m
. Invoking (10), we have
Υ
m
(
x
,
t
)
=
∑
j
=
1
3
1
2
π
i
∫
χ
−
i
∞
χ
+
i
∞
e
δ
t
D
j
m
(
δ
)
D
(
δ
)
e
φ
j
(
δ
)
x
h
˜
m
(
δ
)
d
δ
:
=
S
m
(
t
)
φ
m
,
where
D
j
m
(
s
)
is derived from
D
j
(
s
)
by setting
h
j
˜
(
δ
)
=
1
and
h
k
˜
(
δ
)
=
0
for
k
≠
m
(
k
,
m
=
1
,
2
,
3
)
. We fix
χ
=
0
, in which case the expression of
Υ
m
can be denoted as
Υ
m
(
x
,
t
)
=
∑
j
=
1
3
1
2
π
i
∫
0
+
i
∞
e
δ
t
D
j
m
+
(
δ
)
D
+
(
δ
)
e
φ
j
+
(
δ
)
x
h
˜
m
+
(
δ
)
d
δ
+
∑
j
=
1
3
1
2
π
i
∫
−
i
∞
0
e
−
δ
t
D
j
m
−
(
δ
)
D
−
(
δ
)
e
φ
j
−
(
δ
)
x
h
˜
m
−
(
δ
)
d
δ
:
=
I
m
+
I
I
m
.
(11)
Therefore,
I
m
and
I
I
m
can be written as
I
m
=
∑
j
=
1
3
1
2
π
∫
0
+
∞
e
i
ξ
3
t
D
j
m
+
(
ξ
)
D
+
(
ξ
)
e
φ
j
+
(
ξ
)
x
h
˜
m
+
(
ξ
)
(
3
ξ
2
)
d
ξ
,
I
I
m
=
∑
j
=
1
3
1
2
π
∫
0
∞
e
−
i
ξ
3
t
D
j
m
−
(
ξ
)
D
−
(
ξ
)
e
φ
j
−
(
ξ
)
x
h
˜
m
−
(
ξ
)
(
3
ξ
2
)
d
ξ
,
where
h
˜
±
(
ξ
)
=
g
˜
(
±
i
ξ
3
)
. The quantities carrying the superscript
+
, such as
h
˜
m
+
(
ξ
)
,
D
j
m
+
(
ξ
)
,
D
+
(
ξ
)
, and
φ
j
+
(
ξ
)
, are complex conjugates of their counterparts with the superscript −. For notational simplicity, we henceforth suppress the superscript + and write
h
˜
m
+
(
ξ
)
,
D
+
(
ξ
)
,
D
j
m
+
(
ξ
)
, and
φ
j
+
(
ξ
)
in place of
h
˜
m
(
ξ
)
,
D
(
ξ
)
,
D
j
m
(
ξ
)
, and
φ
j
(
ξ
)
, respectively.
Remark
1.
Indeed, since the nonlinear terms involve both u and v, the solution operators corresponding to u and v should depend on both initial data. That said, when introducing the notation, u is associated only with
u
0
and v only with
v
0
. This is because the final nonlinear estimates are incorporated as a whole within the linear estimates, and the detailed calculation process can be found in Lemma 4.
The solution of (
1
) is established in the space
X
s
,
T
×
X
s
,
T
by using the classical fixed-point principle and a priori estimates. Furthermore, this smooth solution is shown to satisfy Equation (
1
) in the classical sense. To establish the contraction mapping property, we decompose the boundary operator
S
0
,
h
→
(
0
)
into three distinct components and derive corresponding estimates for each component. We now apply the preceding lemmas to furnish the boundary operator
Υ
1
. The estimates of
Υ
2
and
Υ
3
are demonstrated in a manner analogous to that of
Υ
1
.
Lemma
1
([
32
])
.
We assume that φ is a bounded open set in
R
n
with a
n
−
1
dsmooth boundary. Let
θ
∈
0
,
1
,
s
1
,
s
2
≥
0
and
s
1
,
s
2
=
i
n
t
e
g
e
r
+
1
2
.
1.
If
s
1
≥
s
2
≥
0
and
(
1
−
θ
)
s
1
+
θ
s
2
≠
i
n
t
e
g
e
r
+
1
2
,
[
H
0
s
1
(
φ
)
,
H
0
s
2
(
φ
)
]
θ
=
H
0
(
1
−
θ
)
s
1
+
θ
s
2
(
φ
)
(12)
2.
If
s
2
≥
s
1
≥
0
and
(
1
−
θ
)
s
1
+
θ
s
2
≠
i
n
t
e
g
e
r
+
1
2
,
we have
[
H
0
−
s
1
(
φ
)
,
H
0
−
s
2
(
φ
)
]
θ
=
H
0
−
(
1
−
θ
)
s
1
−
θ
s
2
(
φ
)
(13)
where
H
−
s
(
φ
)
=
(
H
s
(
φ
)
)
′
.
3.
If
(
1
−
θ
)
s
1
+
θ
s
2
≠
N
+
1
2
,
i
n
t
e
g
e
r
N
≥
0
we have
[
H
0
s
1
(
φ
)
,
H
0
−
s
2
(
φ
)
]
θ
=
H
0
(
1
−
θ
)
s
1
−
θ
s
2
(
φ
)
if
(
1
−
θ
)
s
1
−
θ
s
2
≥
0
H
(
1
−
θ
)
s
1
−
θ
s
2
(
φ
)
if
(
1
−
θ
)
s
1
−
θ
s
2
≤
0
(14)
Lemma
2
([
33
], Theorem 2)
.
We assume that
V
0
⊂
V
1
and
W
0
⊂
W
1
are Banach spaces. The mapping
L
:
V
i
→
W
i
(
i
=
0
,
1
)
satisfies
∥
L
u
−
L
v
∥
W
1
≤
m
∥
u
∥
V
1
,
∥
v
∥
V
1
∥
u
−
v
∥
V
1
c
,
∀
u
,
v
∈
V
1
,
∥
T
u
∥
W
0
≤
n
∥
u
∥
V
1
∥
u
∥
V
0
d
,
∀
u
∈
V
0
.
For any
0
<
θ
<
1
and
1
≤
p
≤
+
∞
, we obtain
L
:
V
0
,
V
1
θ
,
p
→
W
0
,
W
1
κ
,
q
.
Moreover,
∥
L
u
∥
W
0
,
W
1
κ
,
q
≤
C
f
∥
u
∥
V
1
∥
u
∥
V
0
,
V
1
θ
,
p
(
1
−
κ
)
d
+
κ
c
,
where
f
(
t
)
=
n
(
2
t
)
1
−
κ
m
(
t
,
2
t
)
κ
,
1
−
κ
κ
=
1
−
θ
θ
·
c
d
,
q
=
max
1
,
p
(
1
−
κ
)
d
+
κ
c
=
max
1
,
1
−
θ
d
+
θ
c
p
,
m
(
·
)
and
n
(
·
)
are continuous functions.
Throughout this paper, in Lemma 2, the parameters are specified as
c
=
d
=
1
,
θ
=
κ
,
and
p
=
q
=
2
. In the process of estimating the boundary operator, we still need to introduce a technical lemma from [
16
].
Lemma
3
([
16
])
.
Let
g
∈
L
2
(
0
,
∞
)
, and define the operator
L
f
by
L
g
(
x
)
=
∫
0
∞
e
χ
(
μ
)
x
g
(
μ
)
d
μ
,
where
χ
(
μ
)
:
(
0
,
∞
)
→
C
is a continuous function satisfying the following assumptions:
1.
For some constant
σ
>
0
and sufficient small
ε
>
0
, it holds that
sup
0
<
μ
<
σ
|
R
e
χ
(
μ
)
|
μ
≥
ε
.
2.
For a complex number
α
+
i
ς
, it holds that
lim
μ
→
∞
χ
(
μ
)
μ
=
α
+
i
ς
.
Under these conditions, for all
g
∈
L
2
(
0
,
+
∞
)
, we have
∥
L
g
∥
L
2
(
0
,
1
)
≤
C
e
R
e
χ
(
·
)
g
(
·
)
L
2
R
+
+
∥
g
(
·
)
∥
L
2
R
+
.
3. A Prior Estimates
In this section, the prior bounds for the solution operator will be derived. The solution operator is constructed from the solution components
S
u
0
,
0
→
(
Q
1
)
,
S
v
0
,
0
→
(
Q
2
)
, and
S
0
,
h
→
(
0
)
, given in
Section 2
. To initiate our analysis, we utilize energy techniques to obtain bounds for the solution operator related to the initial data, a step that in turn enables the derivation of the required nonlinear bounds.
3.1. Linear Estimates
We begin by establishing the linear estimates associated with the nonlinear problem, subject to nonhomogeneous initial and homogeneous boundary conditions. Since proofs of many analogous properties have been presented in other literature (see [
31
]), we provide a concise proof of the relevant lemma here to enrich our demonstration.
Lemma
4.
For any
s
≥
0
,
u
0
,
v
0
∈
H
s
(
0
,
1
)
. One can find a positive constant
C
(
T
)
>
0
depending solely on T, such that
ϕ
1
X
s
,
T
≤
C
(
T
)
u
0
H
s
+
∥
Q
1
∥
L
t
2
H
s
−
2
,
if
0
≤
s
≤
2
,
C
(
T
)
u
0
H
s
+
∥
Q
1
∥
L
t
2
H
0
s
−
2
,
if
s
>
2
,
s
≠
integer
+
5
2
,
(15)
and
ϕ
2
X
s
,
T
≤
C
(
T
)
v
0
H
s
+
∥
Q
2
∥
L
t
2
H
s
−
2
,
if
0
≤
s
≤
2
,
C
(
T
)
v
0
H
s
+
∥
Q
2
∥
L
t
2
H
0
s
−
2
,
if
s
>
2
,
s
≠
integer
+
5
2
.
(16)
Proof.
For the case
s
=
0
, since the dispersive effects of the two equations in the system differ only by a constant, their estimation procedures are analogous. Therefore, it suffices to consider a single equation for our analysis. Suppose that
Υ
(
x
,
t
)
solves the homogeneous boundary value problem
i
∂
t
Υ
+
α
∂
x
x
Υ
+
i
ς
∂
x
x
x
Υ
=
Q
(
x
,
t
)
Υ
(
x
,
0
)
=
Υ
0
(
x
)
Υ
(
0
,
t
)
=
0
,
Υ
(
1
,
t
)
=
0
,
Υ
x
(
1
,
t
)
=
0
.
(17)
where
Q
=
Q
1
or
Q
2
. We multiply (17) by
(
1
+
x
)
Υ
¯
, and integrate over the space–time domain
[
0
,
1
]
×
(
0
,
t
)
to arrive at
1
2
∫
0
1
(
1
+
x
)
Υ
2
d
x
+
ς
2
∫
0
t
Υ
x
2
|
x
=
0
d
s
+
3
ς
2
∫
0
t
∫
0
1
Υ
x
2
d
x
d
s
=
1
2
∫
0
1
(
1
+
x
)
Υ
0
2
d
x
+
ℑ
∫
0
t
∫
0
1
Q
(
Υ
)
(
1
+
x
)
Υ
¯
d
x
d
s
≤
Υ
0
L
2
2
+
ℑ
∫
0
t
∫
0
1
Q
(
1
+
x
)
Υ
¯
d
x
d
s
+
ℑ
α
∫
0
t
∫
0
1
Υ
x
Υ
¯
d
x
d
s
,
(18)
where the last term of (
18
) can be controlled by integration by parts together with the Cauchy–Schwarz inequality. Meanwhile, the second term on the last line of (
18
) provides the essential bridge between the linear and nonlinear estimates. Its bound is based on the observation that
∂
x
−
1
u
(
x
)
=
∫
0
x
u
(
y
)
d
y
. We therefore get
∫
0
t
∫
0
1
Q
1
(
Υ
)
(
1
+
x
)
Υ
¯
d
x
d
s
=
∫
0
t
∫
0
1
∂
x
−
2
∂
x
2
Q
1
(
Υ
)
(
1
+
x
)
Υ
¯
d
x
d
s
,
≤
ϵ
∫
0
t
∫
0
1
(
∂
x
Υ
)
2
d
x
d
s
+
1
ϵ
Q
1
L
2
(
0
,
T
;
H
−
2
(
0
,
1
)
)
2
,
where we clearly point out that the
H
−
2
space is specifically chosen in this work to balance the spatial derivative loss induced by the nonlinear terms in the coupled Hirota equation and the dispersive smoothing effect of the third-order spatial derivative operator. By means of the aforementioned calculations, we have
∥
Υ
∥
X
0
,
T
≲
∥
Υ
0
∥
L
2
(
0
,
1
)
)
+
∥
Q
1
∥
L
2
(
0
,
T
;
H
−
2
(
0
,
1
)
)
,
(19)
where
X
0
,
T
=
C
(
0
,
T
;
L
2
(
0
,
1
)
)
∩
L
2
(
0
,
T
;
H
1
(
0
,
1
)
)
. In the following, we allow
s
∈
(
0
,
+
∞
)
. For
s
=
3
, we derive from (17) that
∂
x
3
Υ
(
x
,
t
)
∣
x
=
0
,
1
=
∂
x
4
Υ
(
x
,
t
)
∣
x
=
1
=
0
.
By applying
∂
x
3
to both sides of (17), we obtain that
Υ
X
3
,
T
≲
C
(
T
)
Υ
0
H
x
3
+
Q
1
L
2
(
0
,
T
;
H
0
1
)
.
(20)
By Lemma 1, we get that
∥
Υ
∥
X
3
(
1
−
θ
)
,
T
≲
Υ
0
H
3
(
1
−
θ
)
+
∥
Q
1
∥
L
2
0
,
T
;
H
0
1
−
3
θ
where
θ
∈
0
,
1
3
,
θ
≠
1
6
, and
∥
Υ
∥
X
3
(
1
−
θ
)
,
T
≲
Υ
0
H
3
(
1
−
θ
)
+
∥
Q
1
∥
L
2
0
,
T
;
H
1
−
3
θ
for
θ
∈
1
3
,
1
. It follows that
S
Υ
0
,
0
→
(
Q
1
)
X
s
,
T
≲
Υ
0
H
s
+
∥
Q
1
∥
L
t
2
H
s
−
2
,
if
0
≤
s
≤
2
,
Υ
0
H
s
+
∥
Q
1
∥
L
t
2
H
0
s
−
2
,
if
2
<
s
≤
3
,
s
≠
5
2
.
(21)
We next consider the case
s
≥
3
and
k
∈
Z
+
, from which we obtain
Υ
[
X
3
(
k
+
1
)
,
T
,
X
3
k
,
T
]
θ
≲
Υ
0
[
L
x
2
H
x
3
(
k
+
1
)
,
L
x
2
H
x
3
k
]
θ
+
Q
1
[
L
x
2
H
0
3
(
k
+
1
)
−
2
,
L
x
2
H
0
3
k
−
2
]
θ
.
By using Lemma 2, for
θ
∈
(
0
,
1
)
, we obtain
Υ
X
3
(
k
+
1
)
−
3
θ
,
T
≲
Υ
0
H
x
3
(
k
+
1
)
−
3
θ
+
Q
1
L
2
(
0
,
T
;
H
0
3
(
k
+
1
)
−
3
θ
−
2
)
,
where
θ
∈
(
0
,
1
)
, and
θ
does not belong to the set
{
k
+
1
6
−
integer
3
}
,
for some
k
∈
Z
+
.
We deduce from setting
s
=
3
(
k
+
1
)
−
3
θ
that
Υ
X
s
,
T
≲
Υ
0
H
x
s
+
Q
1
L
2
(
0
,
T
;
H
0
s
−
2
)
,
(22)
where
s
≠
integer
+
5
2
. Consequently, by combining relations (21) and (22), we obtain (
15
). Following a similar line of reasoning, we may also derive (
16
). □
The estimates for the boundary operators below are identical to those established in [
31
]; therefore, their proofs are omitted here for brevity. The boundary operator estimates for the coupled Hirota system (cH system) depend entirely on the linear terms of the equations (i.e.,
i
u
t
+
α
1
∂
x
2
u
+
i
ς
∂
x
3
u
=
0
and
i
v
t
+
α
2
∂
x
2
v
+
i
ς
∂
x
3
u
=
0
) rather than the nonlinear coupling terms. For the two components
u
and
v
, their linear parts share the fundamental third-order dispersion term
i
ς
∂
x
3
, differing only in the second-order dispersion coefficients
α
1
and
α
2
, which do not affect the boundary operator. We outline the key steps: (1) decompose the boundary operator
ψ
1
,
ψ
2
into three components by analogy with the scalar case; (2) derive the integral representations for each component using Laplace transforms and inverse Laplace transforms; (3) prove the
X
s
,
T
norm estimates for each component using the Lemma 3; (4) derive a unified estimate for the coupled system by summing the bounds of the three components, where the constant coefficients
α
1
,
α
2
introduce only finite constant bounds in the estimate.
Lemma
5.
For any
s
≥
0
. Then, for any
ϑ
→
=
(
ϑ
1
,
ϑ
2
,
ϑ
3
)
,
ϱ
→
=
(
ϱ
1
,
ϱ
2
,
ϱ
3
)
satisfy
ϑ
→
,
ϱ
→
∈
Z
s
, we have
ψ
1
X
s
,
T
≤
C
(
T
)
ϑ
→
Z
s
and
ψ
2
X
s
,
T
≤
C
(
T
)
ϱ
→
Z
s
.
(23)
3.2. Nonlinear Estimates
We now turn to the nonlinear estimates for (
1
) involving the nonlinear term
Q
1
(
u
)
=
i
χ
2
|
u
|
2
+
|
v
|
2
u
x
+
u
v
¯
v
x
+
δ
|
u
|
2
+
|
v
|
2
u
,
Q
2
(
u
,
v
)
=
i
χ
(
|
u
|
2
+
2
|
v
|
2
)
v
x
+
u
¯
v
u
x
+
δ
|
u
|
2
+
|
v
|
2
v
and the parameters
χ
,
δ
. Note that
L
t
2
H
0
s
−
2
=
L
t
2
H
s
−
2
for
[
2
,
5
2
)
. It suffices to estimate the nonlinear terms in
L
t
2
H
x
s
−
2
for
[
2
,
5
2
)
, and this also remains the case for the terms established in this section.
Lemma
6.
For arbitrary
u
,
v
∈
X
s
,
T
for
s
∈
[
2
,
5
2
)
and
T
∈
(
0
,
+
∞
)
, we obtain
Q
1
(
u
,
v
)
L
t
2
H
s
−
2
≲
C
(
T
)
u
X
s
,
T
3
+
u
X
s
,
T
v
X
s
,
T
2
,
(24)
and
Q
2
(
u
,
v
)
L
t
2
H
s
−
2
≲
C
(
T
)
v
X
s
,
T
3
+
u
X
s
,
T
2
v
X
s
,
T
,
(25)
where
C
(
T
)
→
0
as
T
→
0
.
Proof.
We first focus on the case
s
=
2
. The Sobolev embedding theorem
H
k
+
3
2
+
ϵ
(
0
,
1
)
↪
C
k
(
0
,
k
)
,
k
∈
Z
+
,
ϵ
>
0
is crucial for our analysis; it transforms the
H
s
Sobolev norm of solutions into a unified
L
∞
norm. This move aims to constrain the nonlinear product terms arising from the dot product between the solution and its derivatives, such as
|
u
|
2
u
x
and
|
v
|
2
u
x
. By the definition of
X
2
,
T
=
C
0
,
T
;
H
2
(
0
,
1
)
∩
L
2
0
,
T
;
H
3
(
0
,
1
)
, all dot-product nonlinear terms are thereby explicitly defined and possess boundedness. Using the Sobolev embedding theorem and the definition of
X
2
,
T
, we have
Q
1
(
u
,
v
)
L
t
2
L
x
2
2
=
u
2
u
x
L
t
2
L
x
2
2
+
u
x
v
2
L
t
2
L
x
2
2
+
u
v
¯
v
x
L
t
2
L
x
2
2
+
u
2
u
L
t
2
L
x
2
2
+
v
2
u
L
t
2
L
x
2
2
=
∫
0
T
u
2
u
x
L
2
2
d
t
+
∫
0
T
u
x
v
2
L
2
2
d
t
+
∫
0
T
u
v
¯
v
x
L
2
2
d
t
+
∫
0
T
u
2
u
L
2
2
d
t
+
∫
0
T
v
2
u
L
2
2
d
t
≲
∫
0
T
sup
x
∈
[
0
,
1
]
u
4
u
x
L
2
2
d
t
+
∫
0
T
sup
x
∈
[
0
,
1
]
v
4
u
x
L
2
2
d
t
+
∫
0
T
sup
x
∈
[
0
,
1
]
u
2
v
¯
v
x
L
2
2
d
t
+
∫
0
T
sup
x
∈
[
0
,
1
]
u
4
u
L
2
2
d
t
+
∫
0
T
sup
x
∈
[
0
,
1
]
v
4
u
L
2
2
d
t
≲
∫
0
T
sup
x
∈
[
0
,
1
]
u
4
(
u
x
L
2
2
+
u
L
2
2
)
+
sup
x
∈
[
0
,
1
]
v
4
(
u
x
L
2
2
+
u
L
2
2
)
d
t
+
∫
0
T
sup
x
∈
[
0
,
1
]
u
2
(
v
L
2
4
+
v
x
L
2
4
)
d
t
≲
u
X
0
,
T
6
+
u
X
0
,
T
2
v
X
0
,
T
4
.
(26)
In the case
s
=
3
, one can find
u
2
u
x
L
t
2
H
x
1
+
u
x
v
2
L
t
2
H
x
1
+
u
v
¯
v
x
L
t
2
H
x
1
+
u
2
u
L
t
2
H
x
1
+
v
2
u
L
t
2
H
x
1
=
∂
x
(
u
2
)
u
x
L
t
2
L
x
2
2
+
u
2
u
x
x
L
t
2
L
x
2
2
+
u
x
x
v
2
L
t
2
L
x
2
2
+
u
x
∂
x
(
v
2
)
L
t
2
L
x
2
2
+
u
x
v
¯
v
x
+
u
v
¯
x
v
x
+
u
v
¯
v
x
x
L
t
2
L
x
2
2
+
∂
x
(
u
2
)
u
L
t
2
L
x
2
2
+
u
2
u
x
L
t
2
L
x
2
2
+
v
2
u
x
L
t
2
L
x
2
2
+
∂
x
(
v
2
)
u
L
t
2
L
x
2
2
≲
u
x
2
u
x
L
t
2
L
x
2
2
+
u
x
2
v
¯
L
t
2
L
x
2
2
+
u
2
u
x
x
L
t
2
L
x
2
2
+
u
x
x
v
2
L
t
2
L
x
2
2
+
u
x
v
¯
v
x
L
t
2
L
x
2
2
+
u
x
v
¯
x
v
L
t
2
L
x
2
2
+
u
v
x
2
L
t
2
L
x
2
2
+
u
v
¯
v
x
x
L
t
2
L
x
2
2
+
u
2
u
x
L
t
2
L
x
2
2
+
u
2
u
¯
x
L
t
2
L
x
2
2
+
u
x
v
2
L
t
2
L
x
2
2
+
u
v
x
v
¯
L
t
2
L
x
2
2
+
u
v
v
¯
x
L
t
2
L
x
2
2
≲
∫
0
T
u
H
1
6
d
t
+
∫
0
T
u
H
2
2
u
H
1
4
d
t
+
∫
0
T
u
H
2
2
v
H
1
4
d
t
+
∫
0
T
u
H
1
2
v
H
1
4
d
t
+
∫
0
T
u
H
1
2
v
H
2
4
d
t
≲
∫
0
T
u
H
1
6
d
t
+
u
X
2
,
T
2
u
X
1
,
T
4
+
u
X
1
,
T
2
v
X
1
,
T
4
+
u
X
1
,
T
2
v
X
2
,
T
4
d
t
.
(27)
Employing the Gagliardo–Nirenberg inequality, we can deduce that
u
H
1
≤
u
L
2
+
u
L
2
1
2
u
x
x
L
2
1
2
,
(28)
This inequality interpolates between the low-order
L
2
norm and the high-order
H
2
norm, serving as a crucial step in defining the nonlinear term within the
L
t
2
H
x
s
−
2
space. It enables the
H
1
norm to be controlled by the solution space
X
s
,
T
. Equation (
28
) gives that
∫
0
T
u
H
1
6
d
t
≲
∫
0
T
(
u
L
2
+
u
L
2
1
2
u
x
x
L
2
1
2
)
6
d
t
≲
∫
0
T
u
L
2
6
d
t
+
∫
0
T
u
L
2
3
u
x
x
L
2
3
d
t
≲
C
(
T
)
(
u
X
0
,
T
6
+
u
X
0
,
T
3
u
X
1
,
T
3
)
.
(29)
We deduce from (26) and (29) that
u
2
u
x
L
t
2
H
x
1
+
u
x
v
2
L
t
2
H
x
1
+
u
v
¯
v
x
L
t
2
H
x
1
+
u
2
u
L
t
2
H
x
1
+
v
2
u
L
t
2
H
x
1
≲
(
u
X
0
,
T
6
+
u
X
0
,
T
3
u
X
1
,
T
3
)
+
u
X
2
,
T
2
u
X
1
,
T
2
+
u
X
2
,
T
2
v
X
1
,
T
4
+
u
X
1
,
T
2
v
X
1
,
T
4
+
u
X
1
,
T
2
v
X
2
,
T
4
≲
(
u
X
3
,
T
6
+
u
X
3
,
T
2
u
X
3
,
T
4
)
.
(30)
where
C
(
T
)
→
0
as
T
→
0
.
This shows that, in the case
s
=
3
, relation (
24
) holds true. By Tatar’s interpolation theorem, we obtain that (
24
) holds for
s
∈
[
2
,
5
2
)
. Indeed, the estimation procedures for
Q
1
and
Q
2
are similar, so the estimate for
Q
2
follows naturally by the same argument. □
We now prove the nonlinear estimates result for
s
<
2
.
Lemma
7.
For arbitrary
u
,
v
∈
X
s
,
T
with
0
≤
s
<
2
and
T
∈
(
0
,
+
∞
)
, we obtain
S
0
,
0
→
(
Q
1
)
X
s
,
T
≲
u
X
s
,
T
3
+
u
X
s
,
T
v
X
s
,
T
2
,
(31)
and
S
0
,
0
→
(
Q
2
)
X
s
,
T
≤
v
X
s
,
T
3
+
u
X
s
,
T
2
v
X
s
,
T
.
(32)
Proof.
We begin with the case
s
=
0
.
Let
Λ
=
S
0
,
0
→
(
Q
1
)
. Then,
Λ
satisfies
i
Λ
t
+
α
1
∂
x
2
Λ
+
i
ς
∂
x
3
Λ
=
Q
1
(
u
,
v
)
,
Λ
(
x
,
0
)
=
0
,
Λ
(
0
,
t
)
=
0
,
Λ
(
L
,
t
)
=
0
,
Λ
x
(
L
,
t
)
=
0
,
t
∈
R
+
.
(33)
An application of integration by parts, after multiplying (
33
) by
(
1
+
x
)
Λ
¯
, and taking the imaginary part, leads to
1
2
d
d
t
∫
0
1
Λ
2
(
1
+
x
)
d
x
+
ς
2
Λ
x
2
|
x
=
0
+
3
2
ς
∫
0
1
Λ
x
2
d
x
=
ℑ
∫
0
1
(
1
+
x
)
Λ
¯
Q
1
d
x
+
ℑ
α
∫
0
1
Λ
Λ
¯
d
x
where
∫
0
1
(
1
+
x
)
Λ
¯
Q
1
d
x
≲
Λ
L
∞
u
L
2
3
+
Λ
L
2
u
L
∞
2
u
H
1
+
Λ
L
2
v
L
∞
2
u
H
1
+
v
L
∞
∫
0
1
u
Λ
v
x
d
x
+
Λ
L
2
v
L
∞
2
u
L
2
.
An argument analogous to that in Lemma 4, combined with integration in
t
, yields
Λ
L
2
2
+
Λ
x
L
t
2
L
x
2
2
≲
∫
0
T
Λ
L
∞
u
L
2
3
+
Λ
L
2
u
L
∞
2
u
H
1
+
Λ
L
2
v
L
∞
u
H
1
+
v
L
∞
u
L
∞
Λ
L
2
v
x
L
2
+
v
L
∞
2
u
L
2
Λ
L
2
d
t
:
=
Q
1
+
Q
2
+
Q
3
+
Q
4
+
Q
5
.
(34)
We deduce from Lemma 2 and the Young’s inequality that
Q
1
≤
∫
0
1
Λ
L
∞
3
d
t
1
3
∫
0
1
u
L
2
9
2
d
t
2
3
≤
T
2
3
Λ
L
t
2
L
x
∞
u
L
t
∞
L
x
2
3
≤
T
2
3
Λ
X
0
,
T
u
X
0
,
T
3
,
(35)
and
Q
2
≤
∫
0
T
Λ
L
2
6
d
t
1
6
∫
0
T
u
L
∞
6
d
t
1
3
∫
0
T
u
H
1
2
d
t
1
2
≤
T
1
6
Λ
X
0
,
T
u
X
0
,
T
3
.
(36)
Sobolev embedding theorem and Young’s inequality are employed here to separate the norms of
Λ
and
u
, which is crucial for the fixed-point argument. We need to constrain the norm of the nonlinear operator using the norm of the solution itself. The estimates for the remaining three terms follow arguments similar to those used for
Q
1
and
Q
2
, from which we directly obtain
Q
3
≤
T
1
6
Λ
X
0
,
T
u
X
0
,
T
v
X
0
,
T
2
,
(37)
Q
4
≤
T
1
2
Λ
X
0
,
T
v
X
0
,
T
2
u
X
0
,
T
,
(38)
Q
5
≤
T
1
6
v
X
0
,
T
2
Λ
X
0
,
T
u
X
0
,
T
.
(39)
It follows from (
34
)–(39) that
Λ
L
2
2
+
Λ
x
L
t
2
L
x
2
2
≲
Λ
X
0
,
T
(
u
X
0
,
T
3
+
u
X
0
,
T
v
X
0
,
T
2
)
,
where
0
≤
t
≤
T
. We deduce that
Λ
X
0
,
T
2
≲
Λ
X
0
,
T
(
u
X
0
,
T
3
+
u
X
0
,
T
v
X
0
,
T
2
)
,
which leads to
Λ
X
0
,
T
≲
u
X
0
,
T
3
+
u
X
0
,
T
v
X
0
,
T
2
.
Furthermore, in analogy with Lemma 6, for
s
=
2
, we obtain
Λ
X
2
,
T
≲
u
X
2
,
T
3
+
u
X
2
,
T
v
X
2
,
T
2
.
We deduce from Lemmas 2 and 3 that
Λ
X
2
θ
,
T
≲
u
X
2
θ
,
T
3
+
u
X
2
θ
,
T
v
X
2
θ
,
T
2
,
θ
∈
(
0
,
1
)
,
which completes the proof of (
31
).
Indeed, the estimation procedures for
S
0
,
0
→
(
Q
1
)
and
S
0
,
0
→
(
Q
2
)
are similar, both relying on the same energy estimates. Thus, the estimate for
S
0
,
0
→
(
Q
2
)
follows naturally by the same argument, and we therefore omit its proof here. □
4. Proof of Main Results
4.1. Proof of Theorem 1
In the current section, we carry out the proof of local well-posedness associated with system (
6
), with the aid of the estimates from
Section 3
. The preceding estimates reveal a fundamental link between the regularity of the solution operator and that of the initial and boundary conditions. The solution operator for the IBVP (
1
) admits the representation
(
u
(
x
,
t
)
,
v
(
x
,
t
)
)
=
(
Ψ
1
(
u
,
v
)
,
Ψ
2
(
u
,
v
)
)
=
(
ϕ
1
,
ϕ
2
)
+
(
ψ
1
,
ψ
2
)
,
where
Q
1
(
u
)
=
i
χ
{
2
|
u
|
2
u
x
+
|
v
|
2
u
x
+
u
v
¯
v
x
}
+
δ
|
u
|
2
u
+
|
v
|
2
u
and
Q
2
(
u
,
v
)
=
i
χ
{
(
|
u
|
2
v
x
+
2
|
v
|
2
v
x
)
+
u
¯
v
u
x
}
+
δ
|
u
|
2
v
+
|
v
|
2
v
. We define the Banach space
B
s
,
M
as the closed ball
B
s
,
R
=
{
(
u
,
v
)
∈
X
s
,
T
∗
×
X
s
,
T
∗
:
u
X
s
,
T
∗
≤
R
,
v
X
s
,
T
∗
≤
R
}
such that
Ψ
1
:
B
s
,
R
→
B
s
,
R
and
Ψ
2
:
B
s
,
R
→
B
s
,
R
,
where
R
is a fixed constant,
T
∗
∈
(
0
,
T
)
and
X
s
,
T
∗
=
C
0
,
T
∗
;
H
s
(
0
,
1
)
∩
L
2
0
,
T
∗
;
H
s
+
1
(
0
,
1
)
.
We deduce from combining Lemmas 4–7 that
Ψ
1
X
s
,
T
∗
≤
C
(
T
∗
)
u
0
H
s
(
0
,
1
)
+
ϑ
→
Z
s
+
C
(
T
∗
)
u
X
s
,
T
∗
3
+
C
(
T
∗
)
u
X
s
,
T
∗
v
X
s
,
T
∗
2
≤
C
(
T
∗
)
u
0
H
s
(
0
,
1
)
+
ϑ
→
Z
s
+
C
(
T
∗
)
R
3
+
C
(
T
∗
)
R
3
≤
R
2
+
R
8
≤
R
,
and
Ψ
2
X
s
,
T
∗
≤
C
(
T
∗
)
v
0
H
s
(
0
,
1
)
+
ϱ
→
Z
s
+
C
(
T
∗
)
v
X
s
,
T
∗
3
+
C
(
T
∗
)
u
X
s
,
T
∗
2
v
X
s
,
T
∗
≤
C
(
T
∗
)
v
0
H
s
(
0
,
1
)
+
ϱ
→
Z
s
+
C
(
T
∗
)
R
3
+
C
(
T
∗
)
R
3
≤
R
2
+
R
8
≤
R
,
where
T
∗
∈
(
0
,
T
)
is chosen small enough to ensure that
C
(
T
∗
)
(
u
0
H
s
(
0
,
1
)
+
v
0
H
s
(
0
,
1
)
+
ϑ
→
Z
s
+
ϱ
→
Z
s
)
≤
R
2
and
C
(
T
∗
)
R
2
≤
1
16
. For any
u
1
,
u
2
,
v
1
,
v
2
∈
B
s
,
R
, by using the definitions of
Ψ
1
,
Ψ
2
, it follows that
Ψ
1
−
Ψ
2
=
S
u
0
,
ϑ
→
(
Q
1
(
u
1
)
)
−
S
u
0
,
ϑ
→
(
Q
1
(
u
2
)
)
=
S
0
,
0
→
(
(
u
1
−
u
2
)
,
u
1
,
∂
x
u
1
)
+
S
0
,
0
→
(
u
2
,
(
u
1
−
u
2
)
,
∂
x
u
1
)
+
S
0
,
0
→
(
u
2
,
(
u
1
−
u
2
)
x
,
u
2
)
+
S
0
,
0
→
(
(
v
1
−
v
2
)
,
v
1
,
∂
x
u
1
)
+
S
0
,
0
→
(
v
2
,
(
v
1
−
v
2
)
,
∂
x
u
1
)
+
S
0
,
0
→
(
v
2
,
v
2
,
(
u
1
−
u
2
)
x
)
+
S
0
,
0
→
(
(
u
1
−
u
2
)
,
v
1
,
∂
x
v
1
)
+
S
0
,
0
→
(
u
2
,
(
v
1
−
v
2
)
,
∂
x
v
1
)
+
S
0
,
0
→
(
u
2
,
v
2
,
(
v
1
−
v
2
)
x
)
+
S
0
,
0
→
(
(
u
1
−
u
2
)
,
u
1
,
u
1
)
+
S
0
,
0
→
(
u
2
,
(
u
1
−
u
2
)
,
u
1
)
+
S
0
,
0
→
(
u
2
,
u
2
,
(
u
1
−
u
2
)
)
+
S
0
,
0
→
(
(
v
1
−
v
2
)
,
v
1
,
u
1
)
+
S
0
,
0
→
(
v
2
,
(
v
1
−
v
2
)
,
u
1
)
+
S
0
,
0
→
(
v
2
,
v
2
,
(
u
1
−
u
2
)
)
.
(40)
Then, we infer from gathering (
40
) and Lemmas 6 and 7 that
∥
Ψ
1
−
Ψ
2
∥
X
s
,
T
∗
=
∥
S
u
0
,
ϑ
→
(
Q
1
(
u
1
,
v
1
)
)
−
S
u
0
,
ϑ
→
(
Q
1
(
u
2
,
v
2
)
)
∥
X
s
,
T
∗
=
S
u
0
,
ϑ
→
(
u
1
,
u
1
,
u
1
)
−
S
u
0
,
ϑ
→
(
u
2
,
u
2
,
u
2
)
X
s
,
T
∗
≲
S
0
,
0
→
(
(
u
1
−
u
2
)
,
u
1
,
∂
x
u
1
)
X
s
,
T
∗
+
S
0
,
0
→
(
u
2
,
(
u
1
−
u
2
)
,
∂
x
u
1
)
X
s
,
T
∗
+
(
S
0
,
0
→
(
u
2
,
(
u
1
−
u
2
)
x
,
u
2
)
X
s
,
T
∗
+
S
0
,
0
→
(
(
v
1
−
v
2
)
,
v
1
,
∂
x
u
1
)
X
s
,
T
∗
+
S
0
,
0
→
(
v
2
,
(
v
1
−
v
2
)
,
∂
x
u
1
)
X
s
,
T
∗
+
S
0
,
0
→
(
v
2
,
v
2
,
(
u
1
−
u
2
)
x
)
X
s
,
T
∗
+
S
0
,
0
→
(
(
u
1
−
u
2
)
,
v
1
,
∂
x
v
1
)
X
s
,
T
∗
+
S
0
,
0
→
(
u
2
,
(
v
1
−
v
2
)
,
∂
x
v
1
)
X
s
,
T
∗
+
S
0
,
0
→
(
u
2
,
v
2
,
(
v
1
−
v
2
)
x
)
X
s
,
T
∗
+
S
0
,
0
→
(
(
u
1
−
u
2
)
,
u
1
,
u
1
)
X
s
,
T
∗
+
S
0
,
0
→
(
u
2
,
(
u
1
−
u
2
)
,
u
1
)
X
s
,
T
∗
+
S
0
,
0
→
(
u
2
,
u
2
,
(
u
1
−
u
2
)
)
X
s
,
T
∗
+
S
0
,
0
→
(
(
v
1
−
v
2
)
,
v
1
,
u
1
)
X
s
,
T
∗
+
S
0
,
0
→
(
v
2
,
(
v
1
−
v
2
)
,
u
1
)
X
s
,
T
∗
+
S
0
,
0
→
(
v
2
,
v
2
,
(
u
1
−
u
2
)
)
X
s
,
T
∗
≤
C
(
T
∗
)
(
u
1
X
s
,
T
∗
2
+
u
2
X
s
,
T
∗
2
+
u
1
X
s
,
T
∗
u
2
X
s
,
T
∗
+
v
1
X
s
,
T
∗
2
+
v
2
X
s
,
T
∗
2
)
∥
u
1
−
u
2
∥
X
s
,
T
∗
+
(
u
1
X
s
,
T
∗
v
1
X
s
,
T
∗
+
u
2
X
s
,
T
∗
v
2
X
s
,
T
∗
+
u
1
X
s
,
T
∗
v
2
X
s
,
T
∗
+
u
2
X
s
,
T
∗
v
1
X
s
,
T
∗
+
u
1
X
s
,
T
∗
v
2
X
s
,
T
∗
+
v
1
X
s
,
T
∗
u
1
X
s
,
T
∗
)
∥
v
1
−
v
2
∥
X
s
,
T
∗
≤
5
M
2
C
(
T
∗
)
∥
u
1
−
u
2
∥
X
s
,
T
∗
+
6
M
2
C
(
T
∗
)
∥
v
1
−
v
2
∥
X
s
,
T
∗
≤
5
16
∥
u
1
−
u
2
∥
X
s
,
T
∗
+
3
8
∥
v
1
−
v
2
∥
X
s
,
T
∗
,
where
C
(
T
∗
)
M
2
≤
1
16
. Let
(
u
,
v
)
be the unique solution to the IBVP (
1
) corresponding to the data
(
u
0
,
v
0
,
ϑ
→
,
ϱ
→
)
, whose existence is guaranteed by the fixed-point principle. Consider now sequences
u
0
n
→
u
0
,
v
0
n
→
v
0
in
H
s
(
0
,
1
)
and
ϑ
→
n
→
ϑ
→
,
ϱ
→
n
→
ϱ
→
in
Z
s
,
and denote by the
u
n
,
v
n
associated solutions with data
(
u
0
n
,
v
0
n
,
ϑ
→
n
,
ϱ
→
n
)
. Invoking the estimates from Lemmas 4–7, we deduce that
(
u
n
,
v
n
)
→
(
u
,
v
)
in
X
s
,
T
∗
×
X
s
,
T
∗
as
n
→
∞
.
Gathering the contraction mapping framework with the a priori estimates we rigorously established earlier naturally gives rise to the continuous dependence of the solutions. More precisely, it follows from the local well-posedness theory that there exist two pairs of solutions,
(
u
,
v
)
=
(
Ψ
1
(
u
0
,
ϑ
→
)
,
Ψ
2
(
v
0
,
ϱ
→
)
)
∈
B
s
,
M
×
B
s
,
M
and
(
u
˜
,
v
˜
)
=
(
Ψ
1
(
u
˜
0
,
ϑ
→
˜
)
,
Ψ
2
(
v
˜
0
,
ϱ
→
˜
)
)
∈
B
s
,
M
×
B
s
,
M
, to the IBVP (
1
) corresponding to the data
(
(
u
0
,
v
0
)
,
ϑ
→
,
ϱ
→
)
and
(
(
u
˜
0
,
v
˜
0
)
,
ϑ
→
˜
,
ϱ
→
˜
)
in
H
s
(
0
,
1
)
×
Z
s
×
Z
s
that satisfy the
s
-compatibility conditions, respectively. Furthermore, the difference between the two solution pairs satisfies the corresponding initial and boundary values. Specifically, the initial and boundary values of the difference are simply the differences in the respective data. A direct computation gives that
(
Ψ
1
(
u
0
,
ϑ
→
)
,
Ψ
2
(
v
0
,
ϱ
→
)
)
−
(
Ψ
1
(
u
˜
0
,
ϑ
→
˜
)
,
Ψ
2
(
v
˜
0
,
ϱ
→
˜
)
)
X
s
,
T
∗
×
X
s
,
T
∗
=
(
u
,
v
)
−
(
u
˜
,
v
˜
)
X
s
,
T
∗
×
X
s
,
T
∗
≲
(
S
u
0
,
ϑ
→
(
u
)
,
S
v
0
,
ϱ
→
(
v
)
)
−
(
S
u
˜
0
,
ϑ
→
˜
(
u
)
,
S
v
˜
0
,
ϑ
→
˜
(
v
)
)
X
s
,
T
∗
×
X
s
,
T
∗
≲
S
u
0
−
u
˜
0
,
ϑ
→
−
ϑ
→
˜
(
(
u
1
−
u
2
)
,
u
1
,
∂
x
u
1
)
X
s
,
T
∗
+
S
u
0
−
u
˜
0
,
ϑ
→
−
ϑ
→
˜
(
u
2
,
(
u
1
−
u
2
)
,
∂
x
u
1
)
X
s
,
T
∗
+
(
S
u
0
−
u
˜
0
,
ϑ
→
−
ϑ
→
˜
(
u
2
,
(
u
1
−
u
2
)
x
,
u
2
)
X
s
,
T
∗
+
S
u
0
−
u
˜
0
,
ϑ
→
−
ϑ
→
˜
(
(
v
1
−
v
2
)
,
v
1
,
∂
x
u
1
)
X
s
,
T
∗
+
S
u
0
−
u
˜
0
,
ϑ
→
−
ϑ
→
˜
(
v
2
,
(
v
1
−
v
2
)
,
∂
x
u
1
)
X
s
,
T
∗
+
S
u
0
−
u
˜
0
,
ϑ
→
−
ϑ
→
˜
(
v
2
,
v
2
,
(
u
1
−
u
2
)
x
)
X
s
,
T
∗
+
S
u
0
−
u
˜
0
,
ϑ
→
−
ϑ
→
˜
(
(
u
1
−
u
2
)
,
v
1
,
∂
x
v
1
)
X
s
,
T
∗
+
S
u
0
−
u
˜
0
,
ϑ
→
−
ϑ
→
˜
(
u
2
,
(
v
1
−
v
2
)
,
∂
x
v
1
)
X
s
,
T
∗
+
S
u
0
−
u
˜
0
,
ϑ
→
−
ϑ
→
˜
(
u
2
,
v
2
,
(
v
1
−
v
2
)
x
)
X
s
,
T
∗
+
S
u
0
−
u
˜
0
,
ϑ
→
−
ϑ
→
˜
(
(
u
1
−
u
2
)
,
u
1
,
u
1
)
X
s
,
T
∗
+
S
u
0
−
u
˜
0
,
ϑ
→
−
ϑ
→
˜
(
u
2
,
(
u
1
−
u
2
)
,
u
1
)
X
s
,
T
∗
+
S
u
0
−
u
˜
0
,
ϑ
→
−
ϑ
→
˜
(
u
2
,
u
2
,
(
u
1
−
u
2
)
)
X
s
,
T
∗
+
S
u
0
−
u
˜
0
,
ϑ
→
−
ϑ
→
˜
(
(
v
1
−
v
2
)
,
v
1
,
u
1
)
X
s
,
T
∗
+
S
u
0
−
u
˜
0
,
ϑ
→
−
ϑ
→
˜
(
v
2
,
(
v
1
−
v
2
)
,
u
1
)
X
s
,
T
∗
+
S
u
0
−
u
˜
0
,
ϑ
→
−
ϑ
→
˜
(
v
2
,
v
2
,
(
u
1
−
u
2
)
)
X
s
,
T
∗
≤
C
(
T
∗
)
u
0
−
u
˜
0
H
s
+
v
0
−
v
˜
0
H
s
+
ϑ
→
−
ϑ
→
˜
Z
s
+
ϱ
→
−
ϱ
→
˜
Z
s
+
1
2
(
u
,
v
)
−
(
u
˜
,
v
˜
)
X
s
,
T
∗
×
X
s
,
T
∗
,
where the last term depends on
C
(
T
∗
)
M
2
≤
1
16
,
C
(
T
∗
)
M
2
≤
1
16
for small
T
∗
.
The proof of the Lipschitz continuous for (
1
) is now complete.
4.2. Proof of Corollary 1
This section is concerned with the proof of Corollary 1. In the course of proving Theorem 1, the nonlinear estimates we use impose a limitation: they only work for
s
∈
[
0
,
5
2
)
, which motivates our method for the case
s
>
5
2
. We deduce from Lemmas 4 and 5 that
(
S
u
0
,
ϑ
→
(
0
)
,
S
v
0
,
ϱ
→
(
0
)
)
X
s
,
T
≲
u
0
H
s
+
v
0
H
s
+
ϑ
→
Z
s
+
ϱ
→
Z
s
,
(41)
where
s
≥
0
and the solution corresponding to the problem (
1
) admits
(
u
(
x
,
t
)
,
v
(
x
,
t
)
)
=
(
S
u
0
,
ϑ
→
(
0
)
,
S
v
0
,
ϱ
→
(
0
)
)
+
(
S
0
,
0
→
(
Q
1
)
,
S
0
,
0
→
(
Q
2
)
)
.
(42)
where
Q
1
(
u
)
=
i
χ
{
2
|
u
|
2
+
|
v
|
2
u
x
+
u
v
¯
v
x
}
+
δ
|
u
|
2
+
|
v
|
2
u
and
Q
2
(
u
,
v
)
=
i
χ
{
(
|
u
|
2
+
2
|
v
|
2
)
v
x
+
u
¯
v
u
x
}
+
δ
|
u
|
2
+
|
v
|
2
v
. In the case of homogeneous initial-boundary data, we define
ω
1
=
∂
t
S
0
,
0
→
(
Q
1
)
,
ω
2
=
∂
t
S
0
,
0
→
(
Q
2
)
, which in turn yields that
i
∂
t
ω
1
+
α
1
∂
x
2
ω
1
+
i
ς
∂
x
3
ω
1
+
L
u
,
v
(
ω
1
,
ω
2
)
=
0
,
i
∂
t
ω
2
+
α
2
∂
x
2
ω
2
+
i
ς
∂
x
3
ω
2
+
L
˜
u
,
v
(
ω
1
,
ω
2
)
=
0
,
ω
1
(
x
,
0
)
=
0
,
ω
2
(
x
,
0
)
=
0
,
ω
1
(
0
,
t
)
=
0
,
ω
1
(
1
,
t
)
=
0
,
ψ
1
x
(
1
,
t
)
=
0
,
ω
2
(
0
,
t
)
=
0
,
ω
2
(
1
,
t
)
=
0
,
ψ
2
x
(
1
,
t
)
=
0
,
(43)
where the operators
L
u
,
v
(
ω
1
,
ω
2
)
and
L
˜
u
,
v
(
ω
1
,
ω
2
)
are nonlinear terms related to
ω
1
,
ω
2
.
Since the linearization operators
L
u
,
v
(
ω
1
,
ω
2
)
and
L
˜
u
,
v
(
ω
1
,
ω
2
)
are linear with respect to
ω
1
,
ω
2
, the standard linear energy estimates can be directly applied. It follows from Lemma 7 that
ω
1
X
s
,
T
≤
(
u
X
s
,
T
2
+
v
X
s
,
T
2
)
∂
t
u
X
s
,
T
,
(44)
and
ω
2
X
s
,
T
≤
(
v
X
s
,
T
2
+
u
X
s
,
T
2
)
∂
t
v
X
s
,
T
,
(45)
for
s
∈
[
0
,
1
)
. Furthermore, it relies on the key observation that
∂
t
u
X
s
,
T
=
∂
t
S
u
0
,
ϑ
→
(
0
)
+
∂
t
S
0
,
0
→
(
Q
1
)
X
s
,
T
≲
∂
t
S
u
0
,
ϑ
→
(
0
)
X
s
,
T
+
ω
1
X
s
,
T
,
(46)
∂
t
v
X
s
,
T
=
∂
t
S
v
0
,
ϱ
→
(
0
)
+
∂
t
S
0
,
0
→
(
Q
2
)
X
s
,
T
≲
∂
t
S
v
0
,
ϱ
→
(
0
)
X
s
,
T
+
ω
2
X
s
,
T
.
(47)
After setting
s
=
0
in (
44
) and (
45
), we substitute this result into (
46
) and (
47
), respectively, thereby obtaining that
ω
1
X
0
,
T
≲
(
u
X
0
,
T
2
+
v
X
0
,
T
2
)
(
∂
t
S
u
0
,
ϑ
→
(
0
)
X
0
,
T
+
ω
1
X
0
,
T
)
,
ω
2
X
0
,
T
≲
(
v
X
0
,
T
2
+
u
X
0
,
T
2
)
(
∂
t
S
u
0
,
ϱ
→
(
0
)
X
0
,
T
+
ω
2
X
0
,
T
)
.
Theorem 1 yields the key estimate
u
X
0
,
T
+
v
X
0
,
T
≲
u
0
L
2
+
v
0
L
2
+
ϑ
→
Z
0
+
ϱ
→
Z
0
.
For
T
∗
∈
(
0
,
T
)
to be chosen, it must be sufficiently small to ensure that
C
(
T
∗
)
(
u
X
0
,
T
2
+
v
X
0
,
T
2
)
≤
1
8
. Under this condition, we deduce
ω
1
X
0
,
T
∗
≲
∂
t
S
u
0
,
ϑ
→
(
0
)
X
0
,
T
∗
,
(48)
ω
2
X
0
,
T
∗
≲
∂
t
S
u
0
,
ϑ
→
(
0
)
X
0
,
T
∗
.
(49)
Take the
X
0
,
T
∗
norm on both sides of (
43
), and apply the triangle inequality for the following norms:
∥
∂
t
u
∥
X
0
,
T
∗
≲
α
1
∥
∂
x
2
u
∥
X
0
,
T
∗
+
ς
∥
∂
x
3
u
∥
X
0
,
T
∗
+
∥
Q
1
(
u
,
v
)
∥
X
0
,
T
∗
,
(50)
∥
∂
t
v
∥
X
0
,
T
∗
≲
α
2
∥
∂
x
2
v
∥
X
0
,
T
∗
+
ς
∥
∂
x
3
v
∥
X
0
,
T
∗
+
∥
Q
2
(
u
,
v
)
∥
X
0
,
T
∗
.
(51)
It follows from gathering (
41
), (
42
), and (
48
)–(51) that
∂
t
u
X
0
,
T
∗
≲
∂
t
S
u
0
,
ϑ
→
(
0
)
X
0
,
T
∗
+
ω
1
X
0
,
T
∗
≲
∂
t
S
u
0
,
ϑ
→
(
0
)
X
0
,
T
∗
≲
∂
t
u
L
2
(
0
)
+
∂
t
ϑ
→
Z
0
≲
u
0
H
3
+
ϑ
→
Z
3
,
(52)
and
∂
t
v
X
0
,
T
∗
≲
v
0
H
3
+
ϱ
→
Z
3
.
(53)
Moreover, by using (
1
), we have
ς
∂
x
3
u
=
−
u
t
+
i
α
1
∂
x
2
u
+
i
Q
1
(
u
,
v
)
,
ς
∂
x
3
v
=
−
v
t
+
i
α
2
∂
x
2
v
+
i
Q
2
(
u
,
v
)
,
which, when combined with Young’s inequality and the Gagliardo–Nirenberg inequality, can be used to derive that
u
X
3
,
T
∗
≲
∂
t
u
X
0
,
T
∗
+
∂
x
3
u
X
0
,
T
∗
+
∂
x
2
u
X
0
,
T
∗
≲
u
0
H
3
+
ϑ
→
Z
3
,
and
v
X
3
,
T
∗
≲
v
0
H
3
+
ϱ
→
Z
3
.
Then, the solution mappings
Ψ
1
,
Ψ
2
for (1) satisfy
Ψ
1
,
Ψ
2
:
H
3
×
Z
3
→
X
3
,
T
∗
,
Ψ
1
X
3
,
T
∗
≲
u
0
H
3
+
ϑ
→
Z
3
,
Ψ
2
X
3
,
T
∗
≲
v
0
H
3
+
ϱ
→
Z
3
,
and for the case
s
=
0
, we further have that
Ψ
1
,
Ψ
2
:
L
2
×
Z
0
→
X
0
,
T
∗
,
Ψ
1
(
u
0
,
ϑ
→
)
−
Ψ
1
(
u
0
,
ϑ
→
˜
)
X
0
,
T
∗
≲
u
0
−
u
˜
0
L
2
+
ϑ
→
−
ϑ
→
˜
Z
0
,
Ψ
2
(
v
0
,
ϱ
→
)
−
Ψ
2
(
v
0
,
ϱ
→
˜
)
X
0
,
T
∗
≲
v
0
−
v
˜
0
L
2
+
ϱ
→
−
ϱ
→
˜
Z
0
.
Moreover, we deduce from Lemma 2 that
Ψ
1
,
Ψ
2
:
H
3
(
1
−
θ
)
×
Z
3
(
1
−
θ
)
→
X
3
(
1
−
θ
)
,
T
∗
,
Ψ
1
(
u
0
,
ϑ
→
)
X
3
(
1
−
θ
)
,
T
∗
≲
u
0
H
3
(
1
−
θ
)
+
ϑ
→
Z
3
(
1
−
θ
)
,
Ψ
2
(
v
0
,
ϱ
→
)
X
3
(
1
−
θ
)
,
T
∗
≲
v
0
H
3
(
1
−
θ
)
+
ϱ
→
Z
3
(
1
−
θ
)
,
(54)
where
3
(
1
−
θ
)
>
5
2
,
θ
∈
(
0
,
1
6
)
. We deduce from Lemma 2 and setting
s
=
3
k
(
1
−
θ
)
,
k
=
2
,
3
,
…
,
for
θ
∈
(
0
,
1
)
that
Ψ
1
,
Ψ
2
:
H
s
×
Z
s
→
X
s
,
T
∗
,
Ψ
1
(
u
0
,
ϑ
→
)
X
s
,
T
∗
≲
u
0
H
s
+
ϑ
→
Z
s
,
Ψ
2
(
v
0
,
ϱ
→
)
X
s
,
T
∗
≲
v
0
H
s
+
ϱ
→
Z
s
.
(55)
This finishes the proof of Corollary 1, gathering (
54
) and (55).
5. Conclusions
We consider the well-posedness of (
1
) on a bounded interval. By utilizing the Laplace transform, which is well suited to the mixed dispersive structure of the equation, we derive a solution operator formula for the linear nonhomogeneous coupled system. Combining the fixed-point principle with energy estimates, we establish the local well-posedness of the problem in
H
s
(
0
,
1
)
when
s
∈
[
0
,
3
/
2
)
is rigorously proven. Furthermore, a corollary extending the local result to the case
s
≥
3
/
2
is proved. It is clarified that when the initial values
(
u
0
,
v
0
)
∈
H
s
(
0
,
1
)
×
H
s
(
0
,
1
)
and the boundary values belong to space
Z
s
and satisfy compatibility conditions, (
1
) admits a unique solution and the associated solution map is Lipschitz continuous in
X
s
,
T
×
X
s
,
T
. During the research process, by decomposing the solution operator into an initial value operator and boundary operator, linear and nonlinear estimates were established, respectively. Tools such as Cauchy–Schwarz inequality, Gagliardo–Nirenberg inequality, and interpolation theorem are utilized to complete a priori estimates. This approach addresses the balance between the smoothing effect of dispersion brought by third-order derivatives and the loss of spatial derivatives in nonlinear terms, and also compensates for the limitations of the Duhamel forcing operator method in dealing with the mixed dispersion problem of this equation. The research results provide a reference framework for the study of dispersion equations involving third-order derivatives. Within this context, modern mesh-free methods demonstrate significant potential for solving high-order boundary value problems. For instance, Karageorghis, Noorizadegan, and Chen have recently employed Radial Basis Function (RBF) methods, such as the fictitious center RBF method, which have been successfully applied to high-order boundary value problems, as demonstrated in [
34
]. Future research could explore the application and adaptation of such mesh-free techniques to numerically solve the IBVP of (cH), for which our analytical results on solution stability and regularity provide a crucial theoretical underpinning.
Author Contributions
S.W. and H.W. contributed to the draft of the manuscript. All authors have read and agreed to the published version of the manuscript.
Funding
This research received no external funding.
Data Availability Statement
The data presented in this study are available on request from the corresponding author.
Conflicts of Interest
The authors declare no conflicts of interest.
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[Abstract](https://www.mdpi.com/2075-1680/15/3/230#Abstract)[Introduction](https://www.mdpi.com/2075-1680/15/3/230#Introduction)[Preliminaries](https://www.mdpi.com/2075-1680/15/3/230#Preliminaries)[A Prior Estimates](https://www.mdpi.com/2075-1680/15/3/230#A_Prior_Estimates)[Proof of Main Results](https://www.mdpi.com/2075-1680/15/3/230#Proof_of_Main_Results)[Conclusions](https://www.mdpi.com/2075-1680/15/3/230#Conclusions)[Author Contributions](https://www.mdpi.com/2075-1680/15/3/230#Author_Contributions)[Funding](https://www.mdpi.com/2075-1680/15/3/230#Funding)[Data Availability Statement](https://www.mdpi.com/2075-1680/15/3/230#Data_Availability_Statement)[Conflicts of Interest](https://www.mdpi.com/2075-1680/15/3/230#Conflicts_of_Interest)[References](https://www.mdpi.com/2075-1680/15/3/230#References)[Article Metrics](https://www.mdpi.com/2075-1680/15/3/230#Article_Metrics)
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- 
20 March 2026
# Well-Posedness of the Nonhomogeneous Initial-Boundary Value Problem for the Coupled Hirota Equation
Shu Wang
and
Huifeng Wang\*
School of Mathematics, Statistics and Mechanics, Beijing University of Technology, Beijing 100021, China
\*
Author to whom correspondence should be addressed.
[Axioms](https://www.mdpi.com/journal/axioms)**2026**, *15*(3), 230;<https://doi.org/10.3390/axioms15030230>
[Version Notes](https://www.mdpi.com/2075-1680/15/3/230/notes)
[Order Reprints](https://www.mdpi.com/2075-1680/15/3/230/reprints)
## Abstract
In this work, we address the nonhomogeneous initial-boundary value problem for the coupled Hirota equation posed on the finite interval
\[
0
,
L
\]
. To investigate the well-posedness of this problem, we first adopt an appropriate transformation, namely the Laplace transform, which is tailored to the specific characteristics of the problem, and further obtain an explicit solution formula for the linear inhomogeneous coupled system. Subsequently, the local well-posedness of the original nonhomogeneous initial-boundary value problem in
X
s
,
T
×
X
s
,
T
X
s
,
T
\=
C
(
0
,
T
;
H
s
(
0
,
1
)
)
∩
L
2
(
0
,
T
;
H
s
\+
1
(
0
,
1
)
)
is rigorously established through the combination of this explicit formula, the contraction mapping principle and energy estimates.
Keywords:
[coupled Hirota equation](https://www.mdpi.com/search?q=coupled+Hirota+equation); [nonhomogeneous boundary value problem](https://www.mdpi.com/search?q=nonhomogeneous+boundary+value+problem); [local well-posedness](https://www.mdpi.com/search?q=local+well-posedness)
MSC:
35Q55; 35Q53; 35G61
## 1\. Introduction
We study the coupled Hirota (cH) equation
i
u
t
\+
α
1
∂
x
2
u
\+
i
ς
∂
x
3
u
\+
i
χ
2
\|
u
\|
2
u
x
\+
\|
v
\|
2
u
x
\+
u
v
¯
v
x
\+
δ
\|
u
\|
2
u
\+
\|
v
\|
2
u
\=
0
,
i
v
t
\+
α
2
∂
x
2
v
\+
i
ς
∂
x
3
v
\+
i
χ
\|
u
\|
2
v
x
\+
2
\|
v
\|
2
v
x
\+
u
¯
v
u
x
\+
δ
\|
u
\|
2
v
\+
\|
v
\|
2
v
\=
0
,
(cH)
where
α
i
(
i
\=
1
,
2
)
,
ς
∈
R
\+
,
χ
,
δ
∈
R
are prescribed real constants, and
u
,
v
denote complex-valued functions. Tasgal and Potasek \[1\] first derived the coupled Hirota equation from the single Hirota equation, showing that third-order dispersion and self-steepening can support the simultaneous propagation of two ultrashort pulses while suppressing inelastic Raman scattering effect.
The Hirota equation
i
u
t
\+
α
∂
x
2
u
\+
i
ς
∂
x
3
u
\+
i
χ
∂
x
(
\|
u
\|
2
u
)
\+
δ
\|
u
\|
2
u
\=
0
,
α
ς
χ
δ
≠
0
,
proposed by R. Hirota in 1973 \[2\], stands as a fundamental generalization of the integrable nonlinear Schrödinger (NLS) equation. It provides an accurate description of ultrashort optical pulse propagation in single-mode fibers. To be precise, the NLS equation only accounts for second-order dispersion and the Kerr effect (for picosecond pulses). Extra effects have to be incorporated when the pulse length is on the order of the wavelength or the pulse width reaches the femtosecond regime. In such cases, the NLS equation is generalized to the Hirota equation. The Hirota equation, also known as the Airy–Schrödinger equation, includes the classic NLS equation for
α
\=
1
,
ς
\=
χ
\=
0
, and the derivative NLS equation for
α
\=
1
,
ς
\=
δ
\=
0
.
The Hirota equation is completely integrable for some special parameters \[2\],
i
u
t
\+
∂
x
2
u
\+
i
ς
∂
x
3
u
\+
6
i
ς
\|
u
\|
2
u
x
\+
2
\|
u
\|
2
u
\=
0
,
which possesses infinite conservation laws with respect to the Cauchy problem. Employing the conservation laws, one can readily derive the global well-posedness of the Hirota equation in
H
s
for
s
∈
N
. Evidently, the fact that the Hirota equation is completely integrable does not affect its well-posedness, allowing us to investigate the Hirota equations with more general parameters. Previously, numerous studies have focused on the Hirota equation and the coupled Hirota equation \[3,4,5,6,7,8\]. The earlier results on higher regularity for the Hirota equation were established by Guo and Tan \[4\] in
H
s
for
s
≥
3
. Following this, Ref. \[3\] generalized the earlier local result to
H
s
for
s
\>
3
4
and the global result to
H
s
for
s
\=
1
and
s
≥
2
.
In \[5\], Staffilani also proved the low regularity in
H
s
for
s
\>
1
4
.
By using more meticulous methods, the results of well-posedness enter the discussion of low-regularity cases (see \[6,7,8\]). The optimal regularity result for the Hirota equation was achieved in \[7\], which established the local result in
H
s
for
s
\>
−
1
4
and global result in
H
s
for
s
\>
0
.
In addition, the only existing result concerning the Cauchy problem of (cH) in
H
s
×
H
s
for
s
≥
1
4
was rigorously demonstrated in \[9\].
This manuscript is devoted to investigating the IBVP for (cH) defined on a finite interval:
i
u
t
\+
α
1
∂
x
2
u
\+
i
ς
∂
x
3
u
\+
i
χ
2
\|
u
\|
2
u
x
\+
\|
v
\|
2
u
x
\+
u
v
¯
v
x
\+
δ
\|
u
\|
2
u
\+
\|
v
\|
2
u
\=
0
,
i
v
t
\+
α
2
∂
x
2
v
\+
i
ς
∂
x
3
v
\+
i
χ
(
\|
u
\|
2
v
x
\+
2
\|
v
\|
2
v
x
)
\+
u
¯
v
u
x
\+
δ
\|
u
\|
2
v
\+
\|
v
\|
2
v
\=
0
,
u
(
x
,
0
)
\=
u
0
(
x
)
,
v
(
x
,
0
)
\=
v
0
(
x
)
,
x
∈
R
\+
,
u
(
0
,
t
)
\=
ϑ
1
(
t
)
,
u
(
L
,
t
)
\=
ϑ
2
(
t
)
,
u
x
(
L
,
t
)
\=
ϑ
3
(
t
)
,
t
∈
R
\+
,
v
(
0
,
t
)
\=
ϱ
1
(
t
)
,
v
(
L
,
t
)
\=
ϱ
2
(
t
)
,
v
x
(
L
,
t
)
\=
ϱ
3
(
t
)
,
t
∈
R
\+
,
(1)
where L is positive nonzero constants, and without loss of generality, let
L
\=
1
.
Recall the result of the IBVP concerning
R
\+
or a interval
\[
0
,
L
\]
.
Early research on IBVP primarily focused on high regularity due to the energy method. For the IBVP posed on
R
\+
, Colliander, Kenig \[10\], and Holmer \[11,12\] extended the well-posedness to the low-regularity case. Their proof relied on the Duhamel boundary operator and Bourgain space
X
s
,
b
, which was proposed by Bourgain in \[13,14\]. For the IBVP posed on a finite interval, due to the Duhamel boundary operator being unsuitable for equations exhibiting mixed dispersion effects, Bona et al. \[15,16,17\] proved the low-regularity existence for the KdV and NLS equation via the Laplace transform. Reference is made to \[16,18,19,20,21,22,23,24,25,26,27,28,29,30,31\] for results involving the IBVP for other dispersive equations.
Our primary goal herein is to analyze the IBVP of the cH equation within a bounded interval framework. In doing so, we resolve the core issues of existence and uniqueness, and further demonstrate the continuous dependence on the prescribed data under suitable regularity constraints. The Laplace transform is specially selected in this work to address the core challenges brought by the mixed dispersion inherent in the coupled Hirota equation. The third-order spatial derivative term complicates the standard Bourgain space methods for addressing the associated IBVP, making them far more intricate than the approaches applied to the second-order NLS equation. It is evident from the proof that existence can be obtained on the initial data
(
u
0
(
x
)
,
v
0
(
x
)
)
∈
H
s
(
0
,
1
)
:
\=
H
s
(
0
,
1
)
×
H
s
(
0
,
1
)
and boundary data
ϑ
1
(
t
)
,
ϑ
2
(
t
)
,
ϑ
3
(
t
)
,
ϱ
1
(
t
)
,
ϱ
2
(
t
)
,
ϱ
3
(
t
)
∈
Z
s
:
\=
H
s
\+
1
3
(
R
\+
)
×
H
s
\+
1
3
(
R
\+
)
×
H
s
3
(
R
\+
)
.
In order to address the well-posedness of IBVP, the given data must satisfy the following compatibility condition and definition:
u
0
,
ι
(
0
)
\=
ϑ
1
(
ι
)
(
0
)
,
u
0
,
ι
(
1
)
\=
ϑ
2
(
ι
)
(
0
)
,
u
0
,
ι
′
(
1
)
\=
ϑ
3
(
ι
)
(
0
)
,
v
0
,
ι
(
0
)
\=
ϱ
1
(
ι
)
(
0
)
,
v
0
,
ι
(
1
)
\=
ϱ
2
(
ι
)
(
0
)
,
v
0
,
ι
′
(
1
)
\=
ϱ
3
(
ι
)
(
0
)
,
where
u
0
,
ι
(
0
)
\=
u
0
(
x
)
,
v
0
,
ι
(
0
)
\=
v
0
(
x
)
if
ι
\=
0
and for
ι
≥
1
,
u
0
,
ι
(
x
)
\=
i
α
1
u
0
,
ι
−
1
″
−
ς
u
0
,
ι
−
1
‴
\+
i
δ
∑
j
\=
0
ι
−
1
(
\|
u
0
,
j
\|
2
\+
\|
v
0
,
j
\|
2
)
u
0
,
ι
−
1
−
j
−
χ
∑
j
\=
0
ι
−
1
{
(
2
\|
u
0
,
j
\|
2
\+
\|
v
0
,
j
\|
2
)
u
0
,
ι
−
1
−
j
′
\+
u
0
,
j
v
¯
0
,
ι
−
1
−
j
v
0
,
ι
−
1
−
j
′
}
,
v
0
,
ι
(
x
)
\=
i
α
2
v
0
,
ι
−
1
″
−
ς
v
0
,
ι
−
1
‴
\+
i
δ
∑
j
\=
0
ι
−
1
(
\|
u
0
,
j
\|
2
\+
\|
v
0
,
j
\|
2
)
v
0
,
ι
−
1
−
j
−
χ
∑
j
\=
0
ι
−
1
{
(
\|
u
0
,
j
\|
2
\+
2
\|
v
0
,
j
\|
2
)
v
0
,
ι
−
1
−
j
′
\+
u
¯
0
,
j
u
0
,
j
′
v
0
,
ι
−
1
−
j
}
.
**Definition** **1\.**
For arbitrary
s
∈
\[
0
,
\+
∞
)
and
T
∈
(
0
,
\+
∞
)
, the compatibility condition for
u
0
(
x
)
,
v
0
(
x
)
,
ϑ
→
(
t
)
,
ϱ
→
(
t
)
\=
u
0
(
x
)
,
v
0
(
x
)
,
ϑ
1
(
t
)
,
ϑ
2
(
t
)
,
ϑ
3
(
t
)
,
ϱ
1
(
t
)
,
ϱ
2
(
t
)
,
ϱ
3
(
t
)
∈
H
s
(
0
,
1
)
×
Z
s
×
Z
s
is as follows:
1\.
If
s
−
3
s
3
≤
1
2
,
u
0
,
ι
(
0
)
\=
ϑ
1
(
ι
)
(
0
)
,
u
0
,
ι
(
1
)
\=
ϑ
2
(
ι
)
(
0
)
,
v
0
,
ι
(
0
)
\=
ϱ
1
(
ι
)
(
0
)
,
v
0
,
ι
(
1
)
\=
ϱ
2
(
ι
)
(
0
)
,
(2)
is valid for
ι
\=
0
,
1
,
⋯
,
s
3
−
1
.
2\.
If
1
2
\<
s
−
3
s
3
\<
3
2
, ([2](https://www.mdpi.com/2075-1680/15/3/230#FD2-axioms-15-00230)) is valid for
ι
\=
0
,
1
,
⋯
,
s
3
.
3\.
If
s
−
3
s
3
\>
3
2
,
u
0
,
ι
(
0
)
\=
ϑ
1
(
ι
)
(
0
)
,
u
0
,
ι
(
1
)
\=
ϑ
2
(
ι
)
(
0
)
,
u
0
,
ι
′
(
1
)
\=
ϑ
3
(
ι
)
(
0
)
v
0
,
ι
(
0
)
\=
ϱ
1
(
ι
)
(
0
)
,
v
0
,
ι
(
1
)
\=
ϱ
2
(
ι
)
(
0
)
,
v
0
,
ι
′
(
1
)
\=
ϱ
3
(
ι
)
(
0
)
.
is valid for
ι
\=
0
,
1
,
⋯
,
s
3
−
1
.
4\.
We agree that ([2](https://www.mdpi.com/2075-1680/15/3/230#FD2-axioms-15-00230)) is vacuous if
s
3
−
1
\<
0
.
It is obvious that higher-order time derivatives cannot be directly observed from boundary value, and they must be computed recursively through the equation itself along with its initial conditions. In light of the previously introduced compatibility conditions, we now formulate a theorem, which constitutes the central focus of our investigation.
**Theorem** **1\.**
For
s
∈
\[
0
,
3
2
)
and
T
\>
0
, we assume that
(
U
0
,
ϱ
→
)
:
\=
u
0
(
x
)
,
v
0
(
x
)
,
ϑ
→
(
t
)
,
ϱ
→
(
t
)
\=
u
0
(
x
)
,
v
0
(
x
)
,
ϑ
1
(
t
)
,
ϑ
2
(
t
)
,
ϑ
3
(
t
)
,
ϱ
1
(
t
)
,
ϱ
2
(
t
)
,
ϱ
3
(
t
)
∈
H
s
(
0
,
1
)
×
Z
s
×
Z
s
,
which satisfies Definition 1. Then, IBVP ([1](https://www.mdpi.com/2075-1680/15/3/230#FD1-axioms-15-00230)) admits a unique solution in
X
s
,
T
×
X
s
,
T
,
where
X
s
,
T
\=
C
0
,
T
;
H
s
(
0
,
1
)
∩
L
2
0
,
T
;
H
s
\+
1
(
0
,
1
)
and
T
:
\=
T
u
0
H
s
,
v
0
H
s
,
ϑ
→
Z
s
,
ϱ
→
Z
s
.
In addition, Lipschitz continuity also holds for the given initial and boundary values.
**Corollary** **1\.**
For
s
≥
3
2
and
T
\>
0
, assume that
u
0
(
x
)
,
v
0
(
x
)
,
ϑ
→
(
t
)
,
ϱ
→
(
t
)
∈
H
s
(
0
,
1
)
×
Z
s
×
Z
s
, which satisfies Definition 1. Then there exists a unique solution of ([1](https://www.mdpi.com/2075-1680/15/3/230#FD1-axioms-15-00230)) for sufficiently small T.
Compared with the scalar Hirota equation, the coupled Hirota (cH) system introduces interdependent nonlinear cross-coupling terms (i.e.,
u
2
v
x
,
u
v
¯
v
x
,
v
2
u
x
,
u
¯
v
u
x
) and asymmetric linear dispersion terms with distinct coefficients
α
1
,
α
2
for the two components u and
v
.
These coupling characteristics give rise to fundamental mathematical challenges, such as cross-dependent nonlinearities in energy estimates and compatibility conditions for the coupled system, which do not arise in the scalar case.
Theorem 1 and Corollary 1 establish the local low regularity in the energy space
H
s
(
0
,
1
)
×
H
s
(
0
,
1
)
. To clarify this definition of regularity, it is necessary to specify the precise meaning of
(
u
,
v
)
as a solution to ([1](https://www.mdpi.com/2075-1680/15/3/230#FD1-axioms-15-00230)). This is particularly crucial for low regularity cases, where the solution must satisfy the equation in a suitable sense. For IBVP with a bounded interval, the concept of a mild solution is typically defined to ensure that the solution satisfies the nonlinear term along with the given data.
In earlier research of IBVP, the standard approach to handling nonhomogeneous boundary data involved transforming the dependent variable to achieve homogeneous boundary conditions. The specific approach involved solving a combination of the boundary value and the boundary conditions, thereby making the boundary conditions equal to 0, but at this point, the new variable satisfied a new form of equation. While this approach permits converting the IBVP into an initial value problem for investigation, it requires the boundary data to possess higher regularity compared to more recent methods. To achieve low-regularization of the well-posedness of IBVP, the Duhamel boundary forcing operator serves as a powerful tool. However, due to its specific construction form, it is not applicable to equations exhibiting mixed dispersion effects (e.g., Hirota equation). Proof of the main results adapt the strategy proposed by Bona et al. \[16\], who developed a strategy involving the Laplace transform to construct a solution formula that incorporates both the linear equation and the boundary operator. Specifically, the prior estimates encompass both linear and nonlinear estimates. The linear estimate is derived based on the energy method, while the nonlinear estimate is obtained by Lemma 4 and the interpolation theorem.
Below is the structure of the rest of this work. [Section 2](https://www.mdpi.com/2075-1680/15/3/230#sec2-axioms-15-00230) establishes the foundational framework: it standardizes notation, formalizes function space definitions, and deriving a solution representation that decomposes the solution into linear and nonlinear evolution. The analytical backbone of this work is presented in [Section 3](https://www.mdpi.com/2075-1680/15/3/230#sec3-axioms-15-00230), where we develop the critical estimates required to implement a contraction mapping argument. Finally, [Section 4](https://www.mdpi.com/2075-1680/15/3/230#sec4-axioms-15-00230) is dedicated to rigorously proving the main result (Theorem 1) and its associated corollary (Corollary 1).
## 2\. Preliminaries
### 2\.1. Notations
We use the following notation for function spaces:
C
t
0
H
x
s
\=
C
0
,
T
;
H
s
(
0
,
1
)
L
t
p
H
x
s
\=
L
p
0
,
T
;
H
s
(
0
,
1
)
,
p
∈
\[
1
,
∞
)
X
s
,
T
\=
C
0
,
T
;
H
s
(
0
,
1
)
∩
L
2
0
,
T
;
H
s
\+
1
(
0
,
1
)
,
and
Z
s
\=
Z
s
:
\=
H
s
\+
1
3
(
R
\+
)
×
H
s
\+
1
3
(
R
\+
)
×
H
s
3
(
R
\+
)
,
where we write the inhomogeneous
L
2
\-based space
H
s
\=
H
s
(
R
)
as
∥
g
∥
H
s
(
R
)
\=
⟨
ξ
⟩
s
g
^
(
ξ
)
L
ξ
2
.
We denote the space
H
0
s
as
H
0
s
(
φ
)
\=
the
closure
of
D
(
φ
)
in
H
s
(
φ
)
,
Let
φ
be an open set in
R
1
, and let both m and n be continuous functions. Define the space
D
(
φ
)
as the set of all functions
ϕ
, such that
ϕ
is infinitely differentiable on
φ
and has compact support contained in
φ
.
Finally, throughout this paper, unless otherwise stated, we abbreviate
∥
·
∥
H
s
as
∥
·
∥
H
x
s
(
0
,
1
)
for simplicity. In addition, the notation
a
≲
(
≳
)
b
is used to denote that
a
≤
(
≥
)
C
b
, where C is a positive constant.
### 2\.2. Solution Formula
With the aim of constructing a solution formula for Equation ([1](https://www.mdpi.com/2075-1680/15/3/230#FD1-axioms-15-00230)) using the Laplace transform. In order to establish the solution to problem ([1](https://www.mdpi.com/2075-1680/15/3/230#FD1-axioms-15-00230)), we examine the following nonhomogeneous initial-boundary value problem:
i
u
t
\+
α
1
∂
x
2
u
\+
i
ς
∂
x
3
u
\+
Q
1
(
u
,
v
)
\=
0
,
i
v
t
\+
α
2
∂
x
2
v
\+
i
ς
∂
x
3
v
\+
Q
2
(
u
,
v
)
\=
0
,
u
(
x
,
0
)
\=
u
0
(
x
)
,
v
(
x
,
0
)
\=
v
0
(
x
)
,
u
(
0
,
t
)
\=
ϑ
1
(
t
)
,
u
(
1
,
t
)
\=
ϑ
2
(
t
)
,
u
x
(
1
,
t
)
\=
ϑ
3
(
t
)
,
t
∈
R
\+
,
v
(
0
,
t
)
\=
ϱ
1
(
t
)
,
v
(
1
,
t
)
\=
ϱ
2
(
t
)
,
v
x
(
1
,
t
)
\=
ϱ
3
(
t
)
,
t
∈
R
\+
,
(3)
where
Q
1
(
u
)
\=
i
χ
2
\|
u
\|
2
u
x
\+
\|
v
\|
2
u
x
\+
u
v
¯
v
x
\+
δ
\|
u
\|
2
u
\+
\|
v
\|
2
u
,
Q
2
(
u
,
v
)
\=
i
χ
(
\|
u
\|
2
v
x
\+
2
\|
v
\|
2
v
x
)
\+
u
¯
v
u
x
\+
δ
\|
u
\|
2
v
\+
\|
v
\|
2
v
and
(
u
0
,
v
0
,
ϑ
→
,
ϱ
→
)
\=
(
u
0
,
v
0
,
ϑ
1
,
ϑ
2
,
ϑ
3
,
ϱ
1
,
ϱ
2
,
ϱ
3
)
∈
H
s
×
Z
s
×
Z
s
that satisfies Definition 1. We first decompose the IBVP ([3](https://www.mdpi.com/2075-1680/15/3/230#FD3-axioms-15-00230)) into two distinct problems: a nonlinear system which satisfies initial data
i
u
t
\+
α
1
∂
x
2
u
\+
i
ς
∂
x
3
u
\+
Q
1
(
u
,
v
)
\=
0
,
i
v
t
\+
α
2
∂
x
2
v
\+
i
ς
∂
x
3
v
\+
Q
2
(
u
,
v
)
\=
0
,
u
(
x
,
0
)
\=
u
0
(
x
)
,
v
(
x
,
0
)
\=
v
0
(
x
)
,
u
(
0
,
t
)
\=
0
,
u
(
1
,
t
)
\=
0
,
u
x
(
1
,
t
)
\=
0
,
t
∈
R
\+
,
v
(
0
,
t
)
\=
0
,
v
(
1
,
t
)
\=
0
,
v
x
(
1
,
t
)
\=
0
,
t
∈
R
\+
,
(4)
and a linear system which satisfies nonhomogeneous boundary conditions
i
u
t
\+
α
1
∂
x
2
u
\+
i
ς
∂
x
3
u
\=
0
,
i
v
t
\+
α
2
∂
x
2
v
\+
i
ς
∂
x
3
v
\=
0
,
u
(
x
,
0
)
\=
0
,
v
(
x
,
0
)
\=
0
,
u
(
0
,
t
)
\=
ϑ
1
(
t
)
,
u
(
1
,
t
)
\=
ϑ
2
(
t
)
,
u
x
(
1
,
t
)
\=
ϑ
3
(
t
)
,
v
(
0
,
t
)
\=
ϱ
1
(
t
)
,
v
(
1
,
t
)
\=
ϱ
2
(
t
)
,
v
x
(
1
,
t
)
\=
ϱ
3
(
t
)
.
(5)
Guided by the solution construction strategy from \[16\], we thus express the solution as
(
Ψ
1
(
u
,
v
)
,
Ψ
2
(
u
,
v
)
)
\=
(
u
(
x
,
t
)
,
v
(
x
,
t
)
)
\=
(
ϕ
1
,
ϕ
2
)
\+
(
ψ
1
,
ψ
2
)
,
(6)
where
Ψ
1
(
u
,
v
)
and
Ψ
2
(
u
,
v
)
are the integral operators, the initial solution operators
ϕ
1
,
ϕ
2
are the solutions of ([4](https://www.mdpi.com/2075-1680/15/3/230#FD4-axioms-15-00230)) with initial value, and the boundary solution operators
ψ
1
,
ψ
2
are the solutions of the linear problem ([5](https://www.mdpi.com/2075-1680/15/3/230#FD5-axioms-15-00230)) with nonhomogeneous boundary conditions. In fact, to derive these two boundary operators for ([5](https://www.mdpi.com/2075-1680/15/3/230#FD5-axioms-15-00230)), it suffices to consider only one equation in the system, since the dispersion effects are identical for both equations and, consequently, the boundary operators take the same form. Therefore, we focus on the following equation:
i
Υ
t
\+
α
Υ
x
x
\+
i
ς
Υ
x
x
x
\=
0
,
Υ
(
x
,
0
)
\=
0
,
Υ
(
0
,
t
)
\=
h
1
(
t
)
,
Υ
(
1
,
t
)
\=
h
2
(
t
)
,
Υ
x
(
1
,
t
)
\=
h
3
(
t
)
,
(7)
where
α
\=
α
1
or
α
2
,
Υ
\=
u
or
v
,
h
j
(
t
)
\=
ϑ
j
(
t
)
or
ϱ
j
(
t
)
for
j
\=
1
,
2
,
3
.
Equation ([7](https://www.mdpi.com/2075-1680/15/3/230#FD7-axioms-15-00230)) can be reformulated as a family of BVP by Laplace transform in t:
i
δ
Υ
˜
(
x
,
δ
)
\+
α
Υ
˜
x
x
\+
i
ς
Υ
˜
x
x
x
\=
0
Υ
˜
(
0
,
δ
)
\=
h
˜
1
(
δ
)
,
Υ
˜
(
1
,
δ
)
\=
h
˜
2
(
δ
)
,
Υ
˜
(
1
,
δ
)
\=
h
˜
3
(
δ
)
,
(8)
where
Υ
˜
(
x
,
δ
)
\=
∫
0
\+
∞
e
−
δ
t
Υ
(
x
,
t
)
d
t
. The characteristic equation of (8) is
δ
−
i
α
φ
2
\+
ς
φ
3
\=
0
.
It is clear that
x
3
\=
i
has three roots:
cos
1
6
π
\+
i
sin
1
6
π
,
cos
5
6
π
\+
i
sin
5
6
π
,
cos
9
6
π
\+
i
sin
9
6
π
,
Letting
δ
\=
(
i
ξ
)
3
\=
−
i
ξ
3
(
ξ
≥
0
)
, by means of the perturbation analysis, the roots of the characteristic equation for
ξ
→
\+
∞
can be derived as
φ
1
(
ξ
)
\=
−
i
ξ
\+
∘
(
1
ξ
)
,
φ
2
(
ξ
)
\=
ξ
(
cos
1
6
π
\+
i
sin
1
6
π
)
\+
∘
(
1
ξ
)
,
φ
3
(
ξ
)
\=
ξ
(
cos
5
6
π
\+
i
sin
5
6
π
)
\+
∘
(
1
ξ
)
.
The solution
Υ
˜
(
x
,
δ
)
of (8) takes the form
u
˜
(
x
,
δ
)
\=
∑
j
\=
1
3
c
j
(
δ
)
e
φ
j
(
δ
)
x
,
where
c
j
satisfies the following linear equations:
c
1
\+
c
2
\+
c
3
\=
h
˜
1
(
δ
)
,
c
1
e
h
1
(
δ
)
\+
c
2
e
φ
2
(
δ
)
\+
c
3
e
φ
3
(
δ
)
\=
h
˜
2
(
δ
)
,
c
1
φ
1
(
δ
)
e
φ
1
(
δ
)
\+
c
2
φ
2
(
δ
)
e
φ
2
(
δ
)
\+
c
3
φ
3
(
δ
)
e
φ
3
(
δ
)
\=
h
˜
3
(
δ
)
.
(9)
Applying Cramer’s rule, we obtain
c
j
\=
D
j
(
δ
)
D
(
δ
)
,
where
D
(
s
)
represents the determinant of the coefficient matrix, with
D
j
(
s
)
given by substituting the j\-th column with
(
h
˜
1
(
s
)
,
h
˜
2
(
s
)
,
h
˜
3
(
s
)
)
.
Υ
admits the following representation after taking the Mellin transform with a given
χ
\>
0
:
Υ
(
x
,
t
)
\=
∑
j
\=
1
3
1
2
π
i
∫
χ
−
i
∞
χ
\+
i
∞
e
δ
t
Υ
˜
(
x
,
δ
)
d
δ
\=
∑
j
\=
1
3
1
2
π
i
∫
χ
−
i
∞
χ
\+
i
∞
e
δ
t
D
j
(
δ
)
D
(
δ
)
e
φ
j
x
d
δ
.
(10)
The solution
Υ
of (7) can be expressed a sum:
Υ
(
x
,
t
)
\=
S
0
,
h
→
(
0
)
\=
Υ
1
(
x
,
t
)
\+
Υ
2
(
x
,
t
)
\+
Υ
3
(
x
,
t
)
,
where
Υ
m
is a solution to (7) with
φ
j
\=
0
for
j
≠
m
. Invoking (10), we have
Υ
m
(
x
,
t
)
\=
∑
j
\=
1
3
1
2
π
i
∫
χ
−
i
∞
χ
\+
i
∞
e
δ
t
D
j
m
(
δ
)
D
(
δ
)
e
φ
j
(
δ
)
x
h
˜
m
(
δ
)
d
δ
:
\=
S
m
(
t
)
φ
m
,
where
D
j
m
(
s
)
is derived from
D
j
(
s
)
by setting
h
j
˜
(
δ
)
\=
1
and
h
k
˜
(
δ
)
\=
0
for
k
≠
m
(
k
,
m
\=
1
,
2
,
3
)
. We fix
χ
\=
0
, in which case the expression of
Υ
m
can be denoted as
Υ
m
(
x
,
t
)
\=
∑
j
\=
1
3
1
2
π
i
∫
0
\+
i
∞
e
δ
t
D
j
m
\+
(
δ
)
D
\+
(
δ
)
e
φ
j
\+
(
δ
)
x
h
˜
m
\+
(
δ
)
d
δ
\+
∑
j
\=
1
3
1
2
π
i
∫
−
i
∞
0
e
−
δ
t
D
j
m
−
(
δ
)
D
−
(
δ
)
e
φ
j
−
(
δ
)
x
h
˜
m
−
(
δ
)
d
δ
:
\=
I
m
\+
I
I
m
.
(11)
Therefore,
I
m
and
I
I
m
can be written as
I
m
\=
∑
j
\=
1
3
1
2
π
∫
0
\+
∞
e
i
ξ
3
t
D
j
m
\+
(
ξ
)
D
\+
(
ξ
)
e
φ
j
\+
(
ξ
)
x
h
˜
m
\+
(
ξ
)
(
3
ξ
2
)
d
ξ
,
I
I
m
\=
∑
j
\=
1
3
1
2
π
∫
0
∞
e
−
i
ξ
3
t
D
j
m
−
(
ξ
)
D
−
(
ξ
)
e
φ
j
−
(
ξ
)
x
h
˜
m
−
(
ξ
)
(
3
ξ
2
)
d
ξ
,
where
h
˜
±
(
ξ
)
\=
g
˜
(
±
i
ξ
3
)
. The quantities carrying the superscript
\+
, such as
h
˜
m
\+
(
ξ
)
,
D
j
m
\+
(
ξ
)
,
D
\+
(
ξ
)
, and
φ
j
\+
(
ξ
)
, are complex conjugates of their counterparts with the superscript −. For notational simplicity, we henceforth suppress the superscript + and write
h
˜
m
\+
(
ξ
)
,
D
\+
(
ξ
)
,
D
j
m
\+
(
ξ
)
, and
φ
j
\+
(
ξ
)
in place of
h
˜
m
(
ξ
)
,
D
(
ξ
)
,
D
j
m
(
ξ
)
, and
φ
j
(
ξ
)
, respectively.
**Remark** **1\.**
Indeed, since the nonlinear terms involve both u and v, the solution operators corresponding to u and v should depend on both initial data. That said, when introducing the notation, u is associated only with
u
0
and v only with
v
0
. This is because the final nonlinear estimates are incorporated as a whole within the linear estimates, and the detailed calculation process can be found in Lemma 4.
The solution of ([1](https://www.mdpi.com/2075-1680/15/3/230#FD1-axioms-15-00230)) is established in the space
X
s
,
T
×
X
s
,
T
by using the classical fixed-point principle and a priori estimates. Furthermore, this smooth solution is shown to satisfy Equation ([1](https://www.mdpi.com/2075-1680/15/3/230#FD1-axioms-15-00230)) in the classical sense. To establish the contraction mapping property, we decompose the boundary operator
S
0
,
h
→
(
0
)
into three distinct components and derive corresponding estimates for each component. We now apply the preceding lemmas to furnish the boundary operator
Υ
1
. The estimates of
Υ
2
and
Υ
3
are demonstrated in a manner analogous to that of
Υ
1
.
**Lemma** **1**
(\[32\])**.**
We assume that φ is a bounded open set in
R
n
with a
n
−
1
dsmooth boundary. Let
θ
∈
0
,
1
,
s
1
,
s
2
≥
0
and
s
1
,
s
2
\=
i
n
t
e
g
e
r
\+
1
2
.
1\.
If
s
1
≥
s
2
≥
0
and
(
1
−
θ
)
s
1
\+
θ
s
2
≠
i
n
t
e
g
e
r
\+
1
2
,
\[
H
0
s
1
(
φ
)
,
H
0
s
2
(
φ
)
\]
θ
\=
H
0
(
1
−
θ
)
s
1
\+
θ
s
2
(
φ
)
(12)
2\.
If
s
2
≥
s
1
≥
0
and
(
1
−
θ
)
s
1
\+
θ
s
2
≠
i
n
t
e
g
e
r
\+
1
2
,
we have
\[
H
0
−
s
1
(
φ
)
,
H
0
−
s
2
(
φ
)
\]
θ
\=
H
0
−
(
1
−
θ
)
s
1
−
θ
s
2
(
φ
)
(13)
where
H
−
s
(
φ
)
\=
(
H
s
(
φ
)
)
′
.
3\.
If
(
1
−
θ
)
s
1
\+
θ
s
2
≠
N
\+
1
2
,
i
n
t
e
g
e
r
N
≥
0
we have
\[
H
0
s
1
(
φ
)
,
H
0
−
s
2
(
φ
)
\]
θ
\=
H
0
(
1
−
θ
)
s
1
−
θ
s
2
(
φ
)
if
(
1
−
θ
)
s
1
−
θ
s
2
≥
0
H
(
1
−
θ
)
s
1
−
θ
s
2
(
φ
)
if
(
1
−
θ
)
s
1
−
θ
s
2
≤
0
(14)
**Lemma** **2**
(\[33\], Theorem 2)**.**
We assume that
V
0
⊂
V
1
and
W
0
⊂
W
1
are Banach spaces. The mapping
L
:
V
i
→
W
i
(
i
\=
0
,
1
)
satisfies
∥
L
u
−
L
v
∥
W
1
≤
m
∥
u
∥
V
1
,
∥
v
∥
V
1
∥
u
−
v
∥
V
1
c
,
∀
u
,
v
∈
V
1
,
∥
T
u
∥
W
0
≤
n
∥
u
∥
V
1
∥
u
∥
V
0
d
,
∀
u
∈
V
0
.
For any
0
\<
θ
\<
1
and
1
≤
p
≤
\+
∞
, we obtain
L
:
V
0
,
V
1
θ
,
p
→
W
0
,
W
1
κ
,
q
.
Moreover,
∥
L
u
∥
W
0
,
W
1
κ
,
q
≤
C
f
∥
u
∥
V
1
∥
u
∥
V
0
,
V
1
θ
,
p
(
1
−
κ
)
d
\+
κ
c
,
where
f
(
t
)
\=
n
(
2
t
)
1
−
κ
m
(
t
,
2
t
)
κ
,
1
−
κ
κ
\=
1
−
θ
θ
·
c
d
,
q
\=
max
1
,
p
(
1
−
κ
)
d
\+
κ
c
\=
max
1
,
1
−
θ
d
\+
θ
c
p
,
m
(
·
)
and
n
(
·
)
are continuous functions.
Throughout this paper, in Lemma 2, the parameters are specified as
c
\=
d
\=
1
,
θ
\=
κ
,
and
p
\=
q
\=
2
. In the process of estimating the boundary operator, we still need to introduce a technical lemma from \[16\].
**Lemma** **3**
(\[16\])**.**
Let
g
∈
L
2
(
0
,
∞
)
, and define the operator
L
f
by
L
g
(
x
)
\=
∫
0
∞
e
χ
(
μ
)
x
g
(
μ
)
d
μ
,
where
χ
(
μ
)
:
(
0
,
∞
)
→
C
is a continuous function satisfying the following assumptions:
1\.
For some constant
σ
\>
0
and sufficient small
ε
\>
0
, it holds that
sup
0
\<
μ
\<
σ
\|
R
e
χ
(
μ
)
\|
μ
≥
ε
.
2\.
For a complex number
α
\+
i
ς
, it holds that
lim
μ
→
∞
χ
(
μ
)
μ
\=
α
\+
i
ς
.
Under these conditions, for all
g
∈
L
2
(
0
,
\+
∞
)
, we have
∥
L
g
∥
L
2
(
0
,
1
)
≤
C
e
R
e
χ
(
·
)
g
(
·
)
L
2
R
\+
\+
∥
g
(
·
)
∥
L
2
R
\+
.
## 3\. A Prior Estimates
In this section, the prior bounds for the solution operator will be derived. The solution operator is constructed from the solution components
S
u
0
,
0
→
(
Q
1
)
,
S
v
0
,
0
→
(
Q
2
)
, and
S
0
,
h
→
(
0
)
, given in [Section 2](https://www.mdpi.com/2075-1680/15/3/230#sec2-axioms-15-00230). To initiate our analysis, we utilize energy techniques to obtain bounds for the solution operator related to the initial data, a step that in turn enables the derivation of the required nonlinear bounds.
### 3\.1. Linear Estimates
We begin by establishing the linear estimates associated with the nonlinear problem, subject to nonhomogeneous initial and homogeneous boundary conditions. Since proofs of many analogous properties have been presented in other literature (see \[31\]), we provide a concise proof of the relevant lemma here to enrich our demonstration.
**Lemma** **4\.**
For any
s
≥
0
,
u
0
,
v
0
∈
H
s
(
0
,
1
)
. One can find a positive constant
C
(
T
)
\>
0
depending solely on T, such that
ϕ
1
X
s
,
T
≤
C
(
T
)
u
0
H
s
\+
∥
Q
1
∥
L
t
2
H
s
−
2
,
if
0
≤
s
≤
2
,
C
(
T
)
u
0
H
s
\+
∥
Q
1
∥
L
t
2
H
0
s
−
2
,
if
s
\>
2
,
s
≠
integer
\+
5
2
,
(15)
and
ϕ
2
X
s
,
T
≤
C
(
T
)
v
0
H
s
\+
∥
Q
2
∥
L
t
2
H
s
−
2
,
if
0
≤
s
≤
2
,
C
(
T
)
v
0
H
s
\+
∥
Q
2
∥
L
t
2
H
0
s
−
2
,
if
s
\>
2
,
s
≠
integer
\+
5
2
.
(16)
**Proof.**
For the case
s
\=
0
, since the dispersive effects of the two equations in the system differ only by a constant, their estimation procedures are analogous. Therefore, it suffices to consider a single equation for our analysis. Suppose that
Υ
(
x
,
t
)
solves the homogeneous boundary value problem
i
∂
t
Υ
\+
α
∂
x
x
Υ
\+
i
ς
∂
x
x
x
Υ
\=
Q
(
x
,
t
)
Υ
(
x
,
0
)
\=
Υ
0
(
x
)
Υ
(
0
,
t
)
\=
0
,
Υ
(
1
,
t
)
\=
0
,
Υ
x
(
1
,
t
)
\=
0
.
(17)
where
Q
\=
Q
1
or
Q
2
. We multiply (17) by
(
1
\+
x
)
Υ
¯
, and integrate over the space–time domain
\[
0
,
1
\]
×
(
0
,
t
)
to arrive at
1
2
∫
0
1
(
1
\+
x
)
Υ
2
d
x
\+
ς
2
∫
0
t
Υ
x
2
\|
x
\=
0
d
s
\+
3
ς
2
∫
0
t
∫
0
1
Υ
x
2
d
x
d
s
\=
1
2
∫
0
1
(
1
\+
x
)
Υ
0
2
d
x
\+
ℑ
∫
0
t
∫
0
1
Q
(
Υ
)
(
1
\+
x
)
Υ
¯
d
x
d
s
≤
Υ
0
L
2
2
\+
ℑ
∫
0
t
∫
0
1
Q
(
1
\+
x
)
Υ
¯
d
x
d
s
\+
ℑ
α
∫
0
t
∫
0
1
Υ
x
Υ
¯
d
x
d
s
,
(18)
where the last term of ([18](https://www.mdpi.com/2075-1680/15/3/230#FD18-axioms-15-00230)) can be controlled by integration by parts together with the Cauchy–Schwarz inequality. Meanwhile, the second term on the last line of ([18](https://www.mdpi.com/2075-1680/15/3/230#FD18-axioms-15-00230)) provides the essential bridge between the linear and nonlinear estimates. Its bound is based on the observation that
∂
x
−
1
u
(
x
)
\=
∫
0
x
u
(
y
)
d
y
. We therefore get
∫
0
t
∫
0
1
Q
1
(
Υ
)
(
1
\+
x
)
Υ
¯
d
x
d
s
\=
∫
0
t
∫
0
1
∂
x
−
2
∂
x
2
Q
1
(
Υ
)
(
1
\+
x
)
Υ
¯
d
x
d
s
,
≤
ϵ
∫
0
t
∫
0
1
(
∂
x
Υ
)
2
d
x
d
s
\+
1
ϵ
Q
1
L
2
(
0
,
T
;
H
−
2
(
0
,
1
)
)
2
,
where we clearly point out that the
H
−
2
space is specifically chosen in this work to balance the spatial derivative loss induced by the nonlinear terms in the coupled Hirota equation and the dispersive smoothing effect of the third-order spatial derivative operator. By means of the aforementioned calculations, we have
∥
Υ
∥
X
0
,
T
≲
∥
Υ
0
∥
L
2
(
0
,
1
)
)
\+
∥
Q
1
∥
L
2
(
0
,
T
;
H
−
2
(
0
,
1
)
)
,
(19)
where
X
0
,
T
\=
C
(
0
,
T
;
L
2
(
0
,
1
)
)
∩
L
2
(
0
,
T
;
H
1
(
0
,
1
)
)
. In the following, we allow
s
∈
(
0
,
\+
∞
)
. For
s
\=
3
, we derive from (17) that
∂
x
3
Υ
(
x
,
t
)
∣
x
\=
0
,
1
\=
∂
x
4
Υ
(
x
,
t
)
∣
x
\=
1
\=
0
.
By applying
∂
x
3
to both sides of (17), we obtain that
Υ
X
3
,
T
≲
C
(
T
)
Υ
0
H
x
3
\+
Q
1
L
2
(
0
,
T
;
H
0
1
)
.
(20)
By Lemma 1, we get that
∥
Υ
∥
X
3
(
1
−
θ
)
,
T
≲
Υ
0
H
3
(
1
−
θ
)
\+
∥
Q
1
∥
L
2
0
,
T
;
H
0
1
−
3
θ
where
θ
∈
0
,
1
3
,
θ
≠
1
6
, and
∥
Υ
∥
X
3
(
1
−
θ
)
,
T
≲
Υ
0
H
3
(
1
−
θ
)
\+
∥
Q
1
∥
L
2
0
,
T
;
H
1
−
3
θ
for
θ
∈
1
3
,
1
. It follows that
S
Υ
0
,
0
→
(
Q
1
)
X
s
,
T
≲
Υ
0
H
s
\+
∥
Q
1
∥
L
t
2
H
s
−
2
,
if
0
≤
s
≤
2
,
Υ
0
H
s
\+
∥
Q
1
∥
L
t
2
H
0
s
−
2
,
if
2
\<
s
≤
3
,
s
≠
5
2
.
(21)
We next consider the case
s
≥
3
and
k
∈
Z
\+
, from which we obtain
Υ
\[
X
3
(
k
\+
1
)
,
T
,
X
3
k
,
T
\]
θ
≲
Υ
0
\[
L
x
2
H
x
3
(
k
\+
1
)
,
L
x
2
H
x
3
k
\]
θ
\+
Q
1
\[
L
x
2
H
0
3
(
k
\+
1
)
−
2
,
L
x
2
H
0
3
k
−
2
\]
θ
.
By using Lemma 2, for
θ
∈
(
0
,
1
)
, we obtain
Υ
X
3
(
k
\+
1
)
−
3
θ
,
T
≲
Υ
0
H
x
3
(
k
\+
1
)
−
3
θ
\+
Q
1
L
2
(
0
,
T
;
H
0
3
(
k
\+
1
)
−
3
θ
−
2
)
,
where
θ
∈
(
0
,
1
)
, and
θ
does not belong to the set
{
k
\+
1
6
−
integer
3
}
,
for some
k
∈
Z
\+
.
We deduce from setting
s
\=
3
(
k
\+
1
)
−
3
θ
that
Υ
X
s
,
T
≲
Υ
0
H
x
s
\+
Q
1
L
2
(
0
,
T
;
H
0
s
−
2
)
,
(22)
where
s
≠
integer
\+
5
2
. Consequently, by combining relations (21) and (22), we obtain ([15](https://www.mdpi.com/2075-1680/15/3/230#FD15-axioms-15-00230)). Following a similar line of reasoning, we may also derive ([16](https://www.mdpi.com/2075-1680/15/3/230#FD16-axioms-15-00230)). □
The estimates for the boundary operators below are identical to those established in \[31\]; therefore, their proofs are omitted here for brevity. The boundary operator estimates for the coupled Hirota system (cH system) depend entirely on the linear terms of the equations (i.e.,
i
u
t
\+
α
1
∂
x
2
u
\+
i
ς
∂
x
3
u
\=
0
and
i
v
t
\+
α
2
∂
x
2
v
\+
i
ς
∂
x
3
u
\=
0
) rather than the nonlinear coupling terms. For the two components u and v, their linear parts share the fundamental third-order dispersion term
i
ς
∂
x
3
, differing only in the second-order dispersion coefficients
α
1
and
α
2
, which do not affect the boundary operator. We outline the key steps: (1) decompose the boundary operator
ψ
1
,
ψ
2
into three components by analogy with the scalar case; (2) derive the integral representations for each component using Laplace transforms and inverse Laplace transforms; (3) prove the
X
s
,
T
norm estimates for each component using the Lemma 3; (4) derive a unified estimate for the coupled system by summing the bounds of the three components, where the constant coefficients
α
1
,
α
2
introduce only finite constant bounds in the estimate.
**Lemma** **5\.**
For any
s
≥
0
. Then, for any
ϑ
→
\=
(
ϑ
1
,
ϑ
2
,
ϑ
3
)
,
ϱ
→
\=
(
ϱ
1
,
ϱ
2
,
ϱ
3
)
satisfy
ϑ
→
,
ϱ
→
∈
Z
s
, we have
ψ
1
X
s
,
T
≤
C
(
T
)
ϑ
→
Z
s
and
ψ
2
X
s
,
T
≤
C
(
T
)
ϱ
→
Z
s
.
(23)
### 3\.2. Nonlinear Estimates
We now turn to the nonlinear estimates for ([1](https://www.mdpi.com/2075-1680/15/3/230#FD1-axioms-15-00230)) involving the nonlinear term
Q
1
(
u
)
\=
i
χ
2
\|
u
\|
2
\+
\|
v
\|
2
u
x
\+
u
v
¯
v
x
\+
δ
\|
u
\|
2
\+
\|
v
\|
2
u
,
Q
2
(
u
,
v
)
\=
i
χ
(
\|
u
\|
2
\+
2
\|
v
\|
2
)
v
x
\+
u
¯
v
u
x
\+
δ
\|
u
\|
2
\+
\|
v
\|
2
v
and the parameters
χ
,
δ
. Note that
L
t
2
H
0
s
−
2
\=
L
t
2
H
s
−
2
for
\[
2
,
5
2
)
. It suffices to estimate the nonlinear terms in
L
t
2
H
x
s
−
2
for
\[
2
,
5
2
)
, and this also remains the case for the terms established in this section.
**Lemma** **6\.**
For arbitrary
u
,
v
∈
X
s
,
T
for
s
∈
\[
2
,
5
2
)
and
T
∈
(
0
,
\+
∞
)
, we obtain
Q
1
(
u
,
v
)
L
t
2
H
s
−
2
≲
C
(
T
)
u
X
s
,
T
3
\+
u
X
s
,
T
v
X
s
,
T
2
,
(24)
and
Q
2
(
u
,
v
)
L
t
2
H
s
−
2
≲
C
(
T
)
v
X
s
,
T
3
\+
u
X
s
,
T
2
v
X
s
,
T
,
(25)
where
C
(
T
)
→
0
as
T
→
0
.
**Proof.**
We first focus on the case
s
\=
2
. The Sobolev embedding theorem
H
k
\+
3
2
\+
ϵ
(
0
,
1
)
↪
C
k
(
0
,
k
)
,
k
∈
Z
\+
,
ϵ
\>
0
is crucial for our analysis; it transforms the
H
s
Sobolev norm of solutions into a unified
L
∞
norm. This move aims to constrain the nonlinear product terms arising from the dot product between the solution and its derivatives, such as
\|
u
\|
2
u
x
and
\|
v
\|
2
u
x
. By the definition of
X
2
,
T
\=
C
0
,
T
;
H
2
(
0
,
1
)
∩
L
2
0
,
T
;
H
3
(
0
,
1
)
, all dot-product nonlinear terms are thereby explicitly defined and possess boundedness. Using the Sobolev embedding theorem and the definition of
X
2
,
T
, we have
Q
1
(
u
,
v
)
L
t
2
L
x
2
2
\=
u
2
u
x
L
t
2
L
x
2
2
\+
u
x
v
2
L
t
2
L
x
2
2
\+
u
v
¯
v
x
L
t
2
L
x
2
2
\+
u
2
u
L
t
2
L
x
2
2
\+
v
2
u
L
t
2
L
x
2
2
\=
∫
0
T
u
2
u
x
L
2
2
d
t
\+
∫
0
T
u
x
v
2
L
2
2
d
t
\+
∫
0
T
u
v
¯
v
x
L
2
2
d
t
\+
∫
0
T
u
2
u
L
2
2
d
t
\+
∫
0
T
v
2
u
L
2
2
d
t
≲
∫
0
T
sup
x
∈
\[
0
,
1
\]
u
4
u
x
L
2
2
d
t
\+
∫
0
T
sup
x
∈
\[
0
,
1
\]
v
4
u
x
L
2
2
d
t
\+
∫
0
T
sup
x
∈
\[
0
,
1
\]
u
2
v
¯
v
x
L
2
2
d
t
\+
∫
0
T
sup
x
∈
\[
0
,
1
\]
u
4
u
L
2
2
d
t
\+
∫
0
T
sup
x
∈
\[
0
,
1
\]
v
4
u
L
2
2
d
t
≲
∫
0
T
sup
x
∈
\[
0
,
1
\]
u
4
(
u
x
L
2
2
\+
u
L
2
2
)
\+
sup
x
∈
\[
0
,
1
\]
v
4
(
u
x
L
2
2
\+
u
L
2
2
)
d
t
\+
∫
0
T
sup
x
∈
\[
0
,
1
\]
u
2
(
v
L
2
4
\+
v
x
L
2
4
)
d
t
≲
u
X
0
,
T
6
\+
u
X
0
,
T
2
v
X
0
,
T
4
.
(26)
In the case
s
\=
3
, one can find
u
2
u
x
L
t
2
H
x
1
\+
u
x
v
2
L
t
2
H
x
1
\+
u
v
¯
v
x
L
t
2
H
x
1
\+
u
2
u
L
t
2
H
x
1
\+
v
2
u
L
t
2
H
x
1
\=
∂
x
(
u
2
)
u
x
L
t
2
L
x
2
2
\+
u
2
u
x
x
L
t
2
L
x
2
2
\+
u
x
x
v
2
L
t
2
L
x
2
2
\+
u
x
∂
x
(
v
2
)
L
t
2
L
x
2
2
\+
u
x
v
¯
v
x
\+
u
v
¯
x
v
x
\+
u
v
¯
v
x
x
L
t
2
L
x
2
2
\+
∂
x
(
u
2
)
u
L
t
2
L
x
2
2
\+
u
2
u
x
L
t
2
L
x
2
2
\+
v
2
u
x
L
t
2
L
x
2
2
\+
∂
x
(
v
2
)
u
L
t
2
L
x
2
2
≲
u
x
2
u
x
L
t
2
L
x
2
2
\+
u
x
2
v
¯
L
t
2
L
x
2
2
\+
u
2
u
x
x
L
t
2
L
x
2
2
\+
u
x
x
v
2
L
t
2
L
x
2
2
\+
u
x
v
¯
v
x
L
t
2
L
x
2
2
\+
u
x
v
¯
x
v
L
t
2
L
x
2
2
\+
u
v
x
2
L
t
2
L
x
2
2
\+
u
v
¯
v
x
x
L
t
2
L
x
2
2
\+
u
2
u
x
L
t
2
L
x
2
2
\+
u
2
u
¯
x
L
t
2
L
x
2
2
\+
u
x
v
2
L
t
2
L
x
2
2
\+
u
v
x
v
¯
L
t
2
L
x
2
2
\+
u
v
v
¯
x
L
t
2
L
x
2
2
≲
∫
0
T
u
H
1
6
d
t
\+
∫
0
T
u
H
2
2
u
H
1
4
d
t
\+
∫
0
T
u
H
2
2
v
H
1
4
d
t
\+
∫
0
T
u
H
1
2
v
H
1
4
d
t
\+
∫
0
T
u
H
1
2
v
H
2
4
d
t
≲
∫
0
T
u
H
1
6
d
t
\+
u
X
2
,
T
2
u
X
1
,
T
4
\+
u
X
1
,
T
2
v
X
1
,
T
4
\+
u
X
1
,
T
2
v
X
2
,
T
4
d
t
.
(27)
Employing the Gagliardo–Nirenberg inequality, we can deduce that
u
H
1
≤
u
L
2
\+
u
L
2
1
2
u
x
x
L
2
1
2
,
(28)
This inequality interpolates between the low-order
L
2
norm and the high-order
H
2
norm, serving as a crucial step in defining the nonlinear term within the
L
t
2
H
x
s
−
2
space. It enables the
H
1
norm to be controlled by the solution space
X
s
,
T
. Equation ([28](https://www.mdpi.com/2075-1680/15/3/230#FD28-axioms-15-00230)) gives that
∫
0
T
u
H
1
6
d
t
≲
∫
0
T
(
u
L
2
\+
u
L
2
1
2
u
x
x
L
2
1
2
)
6
d
t
≲
∫
0
T
u
L
2
6
d
t
\+
∫
0
T
u
L
2
3
u
x
x
L
2
3
d
t
≲
C
(
T
)
(
u
X
0
,
T
6
\+
u
X
0
,
T
3
u
X
1
,
T
3
)
.
(29)
We deduce from (26) and (29) that
u
2
u
x
L
t
2
H
x
1
\+
u
x
v
2
L
t
2
H
x
1
\+
u
v
¯
v
x
L
t
2
H
x
1
\+
u
2
u
L
t
2
H
x
1
\+
v
2
u
L
t
2
H
x
1
≲
(
u
X
0
,
T
6
\+
u
X
0
,
T
3
u
X
1
,
T
3
)
\+
u
X
2
,
T
2
u
X
1
,
T
2
\+
u
X
2
,
T
2
v
X
1
,
T
4
\+
u
X
1
,
T
2
v
X
1
,
T
4
\+
u
X
1
,
T
2
v
X
2
,
T
4
≲
(
u
X
3
,
T
6
\+
u
X
3
,
T
2
u
X
3
,
T
4
)
.
(30)
where
C
(
T
)
→
0
as
T
→
0
.
This shows that, in the case
s
\=
3
, relation ([24](https://www.mdpi.com/2075-1680/15/3/230#FD24-axioms-15-00230)) holds true. By Tatar’s interpolation theorem, we obtain that ([24](https://www.mdpi.com/2075-1680/15/3/230#FD24-axioms-15-00230)) holds for
s
∈
\[
2
,
5
2
)
. Indeed, the estimation procedures for
Q
1
and
Q
2
are similar, so the estimate for
Q
2
follows naturally by the same argument. □
We now prove the nonlinear estimates result for
s
\<
2
.
**Lemma** **7\.**
For arbitrary
u
,
v
∈
X
s
,
T
with
0
≤
s
\<
2
and
T
∈
(
0
,
\+
∞
)
, we obtain
S
0
,
0
→
(
Q
1
)
X
s
,
T
≲
u
X
s
,
T
3
\+
u
X
s
,
T
v
X
s
,
T
2
,
(31)
and
S
0
,
0
→
(
Q
2
)
X
s
,
T
≤
v
X
s
,
T
3
\+
u
X
s
,
T
2
v
X
s
,
T
.
(32)
**Proof.**
We begin with the case
s
\=
0
.
Let
Λ
\=
S
0
,
0
→
(
Q
1
)
. Then,
Λ
satisfies
i
Λ
t
\+
α
1
∂
x
2
Λ
\+
i
ς
∂
x
3
Λ
\=
Q
1
(
u
,
v
)
,
Λ
(
x
,
0
)
\=
0
,
Λ
(
0
,
t
)
\=
0
,
Λ
(
L
,
t
)
\=
0
,
Λ
x
(
L
,
t
)
\=
0
,
t
∈
R
\+
.
(33)
An application of integration by parts, after multiplying ([33](https://www.mdpi.com/2075-1680/15/3/230#FD33-axioms-15-00230)) by
(
1
\+
x
)
Λ
¯
, and taking the imaginary part, leads to
1
2
d
d
t
∫
0
1
Λ
2
(
1
\+
x
)
d
x
\+
ς
2
Λ
x
2
\|
x
\=
0
\+
3
2
ς
∫
0
1
Λ
x
2
d
x
\=
ℑ
∫
0
1
(
1
\+
x
)
Λ
¯
Q
1
d
x
\+
ℑ
α
∫
0
1
Λ
Λ
¯
d
x
where
∫
0
1
(
1
\+
x
)
Λ
¯
Q
1
d
x
≲
Λ
L
∞
u
L
2
3
\+
Λ
L
2
u
L
∞
2
u
H
1
\+
Λ
L
2
v
L
∞
2
u
H
1
\+
v
L
∞
∫
0
1
u
Λ
v
x
d
x
\+
Λ
L
2
v
L
∞
2
u
L
2
.
An argument analogous to that in Lemma 4, combined with integration in
t
, yields
Λ
L
2
2
\+
Λ
x
L
t
2
L
x
2
2
≲
∫
0
T
Λ
L
∞
u
L
2
3
\+
Λ
L
2
u
L
∞
2
u
H
1
\+
Λ
L
2
v
L
∞
u
H
1
\+
v
L
∞
u
L
∞
Λ
L
2
v
x
L
2
\+
v
L
∞
2
u
L
2
Λ
L
2
d
t
:
\=
Q
1
\+
Q
2
\+
Q
3
\+
Q
4
\+
Q
5
.
(34)
We deduce from Lemma 2 and the Young’s inequality that
Q
1
≤
∫
0
1
Λ
L
∞
3
d
t
1
3
∫
0
1
u
L
2
9
2
d
t
2
3
≤
T
2
3
Λ
L
t
2
L
x
∞
u
L
t
∞
L
x
2
3
≤
T
2
3
Λ
X
0
,
T
u
X
0
,
T
3
,
(35)
and
Q
2
≤
∫
0
T
Λ
L
2
6
d
t
1
6
∫
0
T
u
L
∞
6
d
t
1
3
∫
0
T
u
H
1
2
d
t
1
2
≤
T
1
6
Λ
X
0
,
T
u
X
0
,
T
3
.
(36)
Sobolev embedding theorem and Young’s inequality are employed here to separate the norms of
Λ
and u, which is crucial for the fixed-point argument. We need to constrain the norm of the nonlinear operator using the norm of the solution itself. The estimates for the remaining three terms follow arguments similar to those used for
Q
1
and
Q
2
, from which we directly obtain
Q
3
≤
T
1
6
Λ
X
0
,
T
u
X
0
,
T
v
X
0
,
T
2
,
(37)
Q
4
≤
T
1
2
Λ
X
0
,
T
v
X
0
,
T
2
u
X
0
,
T
,
(38)
Q
5
≤
T
1
6
v
X
0
,
T
2
Λ
X
0
,
T
u
X
0
,
T
.
(39)
It follows from ([34](https://www.mdpi.com/2075-1680/15/3/230#FD34-axioms-15-00230))–(39) that
Λ
L
2
2
\+
Λ
x
L
t
2
L
x
2
2
≲
Λ
X
0
,
T
(
u
X
0
,
T
3
\+
u
X
0
,
T
v
X
0
,
T
2
)
,
where
0
≤
t
≤
T
. We deduce that
Λ
X
0
,
T
2
≲
Λ
X
0
,
T
(
u
X
0
,
T
3
\+
u
X
0
,
T
v
X
0
,
T
2
)
,
which leads to
Λ
X
0
,
T
≲
u
X
0
,
T
3
\+
u
X
0
,
T
v
X
0
,
T
2
.
Furthermore, in analogy with Lemma 6, for
s
\=
2
, we obtain
Λ
X
2
,
T
≲
u
X
2
,
T
3
\+
u
X
2
,
T
v
X
2
,
T
2
.
We deduce from Lemmas 2 and 3 that
Λ
X
2
θ
,
T
≲
u
X
2
θ
,
T
3
\+
u
X
2
θ
,
T
v
X
2
θ
,
T
2
,
θ
∈
(
0
,
1
)
,
which completes the proof of ([31](https://www.mdpi.com/2075-1680/15/3/230#FD31-axioms-15-00230)).
Indeed, the estimation procedures for
S
0
,
0
→
(
Q
1
)
and
S
0
,
0
→
(
Q
2
)
are similar, both relying on the same energy estimates. Thus, the estimate for
S
0
,
0
→
(
Q
2
)
follows naturally by the same argument, and we therefore omit its proof here. □
## 4\. Proof of Main Results
### 4\.1. Proof of Theorem 1
In the current section, we carry out the proof of local well-posedness associated with system ([6](https://www.mdpi.com/2075-1680/15/3/230#FD6-axioms-15-00230)), with the aid of the estimates from [Section 3](https://www.mdpi.com/2075-1680/15/3/230#sec3-axioms-15-00230). The preceding estimates reveal a fundamental link between the regularity of the solution operator and that of the initial and boundary conditions. The solution operator for the IBVP ([1](https://www.mdpi.com/2075-1680/15/3/230#FD1-axioms-15-00230)) admits the representation
(
u
(
x
,
t
)
,
v
(
x
,
t
)
)
\=
(
Ψ
1
(
u
,
v
)
,
Ψ
2
(
u
,
v
)
)
\=
(
ϕ
1
,
ϕ
2
)
\+
(
ψ
1
,
ψ
2
)
,
where
Q
1
(
u
)
\=
i
χ
{
2
\|
u
\|
2
u
x
\+
\|
v
\|
2
u
x
\+
u
v
¯
v
x
}
\+
δ
\|
u
\|
2
u
\+
\|
v
\|
2
u
and
Q
2
(
u
,
v
)
\=
i
χ
{
(
\|
u
\|
2
v
x
\+
2
\|
v
\|
2
v
x
)
\+
u
¯
v
u
x
}
\+
δ
\|
u
\|
2
v
\+
\|
v
\|
2
v
. We define the Banach space
B
s
,
M
as the closed ball
B
s
,
R
\=
{
(
u
,
v
)
∈
X
s
,
T
∗
×
X
s
,
T
∗
:
u
X
s
,
T
∗
≤
R
,
v
X
s
,
T
∗
≤
R
}
such that
Ψ
1
:
B
s
,
R
→
B
s
,
R
and
Ψ
2
:
B
s
,
R
→
B
s
,
R
,
where R is a fixed constant,
T
∗
∈
(
0
,
T
)
and
X
s
,
T
∗
\=
C
0
,
T
∗
;
H
s
(
0
,
1
)
∩
L
2
0
,
T
∗
;
H
s
\+
1
(
0
,
1
)
.
We deduce from combining Lemmas 4–7 that
Ψ
1
X
s
,
T
∗
≤
C
(
T
∗
)
u
0
H
s
(
0
,
1
)
\+
ϑ
→
Z
s
\+
C
(
T
∗
)
u
X
s
,
T
∗
3
\+
C
(
T
∗
)
u
X
s
,
T
∗
v
X
s
,
T
∗
2
≤
C
(
T
∗
)
u
0
H
s
(
0
,
1
)
\+
ϑ
→
Z
s
\+
C
(
T
∗
)
R
3
\+
C
(
T
∗
)
R
3
≤
R
2
\+
R
8
≤
R
,
and
Ψ
2
X
s
,
T
∗
≤
C
(
T
∗
)
v
0
H
s
(
0
,
1
)
\+
ϱ
→
Z
s
\+
C
(
T
∗
)
v
X
s
,
T
∗
3
\+
C
(
T
∗
)
u
X
s
,
T
∗
2
v
X
s
,
T
∗
≤
C
(
T
∗
)
v
0
H
s
(
0
,
1
)
\+
ϱ
→
Z
s
\+
C
(
T
∗
)
R
3
\+
C
(
T
∗
)
R
3
≤
R
2
\+
R
8
≤
R
,
where
T
∗
∈
(
0
,
T
)
is chosen small enough to ensure that
C
(
T
∗
)
(
u
0
H
s
(
0
,
1
)
\+
v
0
H
s
(
0
,
1
)
\+
ϑ
→
Z
s
\+
ϱ
→
Z
s
)
≤
R
2
and
C
(
T
∗
)
R
2
≤
1
16
. For any
u
1
,
u
2
,
v
1
,
v
2
∈
B
s
,
R
, by using the definitions of
Ψ
1
,
Ψ
2
, it follows that
Ψ
1
−
Ψ
2
\=
S
u
0
,
ϑ
→
(
Q
1
(
u
1
)
)
−
S
u
0
,
ϑ
→
(
Q
1
(
u
2
)
)
\=
S
0
,
0
→
(
(
u
1
−
u
2
)
,
u
1
,
∂
x
u
1
)
\+
S
0
,
0
→
(
u
2
,
(
u
1
−
u
2
)
,
∂
x
u
1
)
\+
S
0
,
0
→
(
u
2
,
(
u
1
−
u
2
)
x
,
u
2
)
\+
S
0
,
0
→
(
(
v
1
−
v
2
)
,
v
1
,
∂
x
u
1
)
\+
S
0
,
0
→
(
v
2
,
(
v
1
−
v
2
)
,
∂
x
u
1
)
\+
S
0
,
0
→
(
v
2
,
v
2
,
(
u
1
−
u
2
)
x
)
\+
S
0
,
0
→
(
(
u
1
−
u
2
)
,
v
1
,
∂
x
v
1
)
\+
S
0
,
0
→
(
u
2
,
(
v
1
−
v
2
)
,
∂
x
v
1
)
\+
S
0
,
0
→
(
u
2
,
v
2
,
(
v
1
−
v
2
)
x
)
\+
S
0
,
0
→
(
(
u
1
−
u
2
)
,
u
1
,
u
1
)
\+
S
0
,
0
→
(
u
2
,
(
u
1
−
u
2
)
,
u
1
)
\+
S
0
,
0
→
(
u
2
,
u
2
,
(
u
1
−
u
2
)
)
\+
S
0
,
0
→
(
(
v
1
−
v
2
)
,
v
1
,
u
1
)
\+
S
0
,
0
→
(
v
2
,
(
v
1
−
v
2
)
,
u
1
)
\+
S
0
,
0
→
(
v
2
,
v
2
,
(
u
1
−
u
2
)
)
.
(40)
Then, we infer from gathering ([40](https://www.mdpi.com/2075-1680/15/3/230#FD40-axioms-15-00230)) and Lemmas 6 and 7 that
∥
Ψ
1
−
Ψ
2
∥
X
s
,
T
∗
\=
∥
S
u
0
,
ϑ
→
(
Q
1
(
u
1
,
v
1
)
)
−
S
u
0
,
ϑ
→
(
Q
1
(
u
2
,
v
2
)
)
∥
X
s
,
T
∗
\=
S
u
0
,
ϑ
→
(
u
1
,
u
1
,
u
1
)
−
S
u
0
,
ϑ
→
(
u
2
,
u
2
,
u
2
)
X
s
,
T
∗
≲
S
0
,
0
→
(
(
u
1
−
u
2
)
,
u
1
,
∂
x
u
1
)
X
s
,
T
∗
\+
S
0
,
0
→
(
u
2
,
(
u
1
−
u
2
)
,
∂
x
u
1
)
X
s
,
T
∗
\+
(
S
0
,
0
→
(
u
2
,
(
u
1
−
u
2
)
x
,
u
2
)
X
s
,
T
∗
\+
S
0
,
0
→
(
(
v
1
−
v
2
)
,
v
1
,
∂
x
u
1
)
X
s
,
T
∗
\+
S
0
,
0
→
(
v
2
,
(
v
1
−
v
2
)
,
∂
x
u
1
)
X
s
,
T
∗
\+
S
0
,
0
→
(
v
2
,
v
2
,
(
u
1
−
u
2
)
x
)
X
s
,
T
∗
\+
S
0
,
0
→
(
(
u
1
−
u
2
)
,
v
1
,
∂
x
v
1
)
X
s
,
T
∗
\+
S
0
,
0
→
(
u
2
,
(
v
1
−
v
2
)
,
∂
x
v
1
)
X
s
,
T
∗
\+
S
0
,
0
→
(
u
2
,
v
2
,
(
v
1
−
v
2
)
x
)
X
s
,
T
∗
\+
S
0
,
0
→
(
(
u
1
−
u
2
)
,
u
1
,
u
1
)
X
s
,
T
∗
\+
S
0
,
0
→
(
u
2
,
(
u
1
−
u
2
)
,
u
1
)
X
s
,
T
∗
\+
S
0
,
0
→
(
u
2
,
u
2
,
(
u
1
−
u
2
)
)
X
s
,
T
∗
\+
S
0
,
0
→
(
(
v
1
−
v
2
)
,
v
1
,
u
1
)
X
s
,
T
∗
\+
S
0
,
0
→
(
v
2
,
(
v
1
−
v
2
)
,
u
1
)
X
s
,
T
∗
\+
S
0
,
0
→
(
v
2
,
v
2
,
(
u
1
−
u
2
)
)
X
s
,
T
∗
≤
C
(
T
∗
)
(
u
1
X
s
,
T
∗
2
\+
u
2
X
s
,
T
∗
2
\+
u
1
X
s
,
T
∗
u
2
X
s
,
T
∗
\+
v
1
X
s
,
T
∗
2
\+
v
2
X
s
,
T
∗
2
)
∥
u
1
−
u
2
∥
X
s
,
T
∗
\+
(
u
1
X
s
,
T
∗
v
1
X
s
,
T
∗
\+
u
2
X
s
,
T
∗
v
2
X
s
,
T
∗
\+
u
1
X
s
,
T
∗
v
2
X
s
,
T
∗
\+
u
2
X
s
,
T
∗
v
1
X
s
,
T
∗
\+
u
1
X
s
,
T
∗
v
2
X
s
,
T
∗
\+
v
1
X
s
,
T
∗
u
1
X
s
,
T
∗
)
∥
v
1
−
v
2
∥
X
s
,
T
∗
≤
5
M
2
C
(
T
∗
)
∥
u
1
−
u
2
∥
X
s
,
T
∗
\+
6
M
2
C
(
T
∗
)
∥
v
1
−
v
2
∥
X
s
,
T
∗
≤
5
16
∥
u
1
−
u
2
∥
X
s
,
T
∗
\+
3
8
∥
v
1
−
v
2
∥
X
s
,
T
∗
,
where
C
(
T
∗
)
M
2
≤
1
16
. Let
(
u
,
v
)
be the unique solution to the IBVP ([1](https://www.mdpi.com/2075-1680/15/3/230#FD1-axioms-15-00230)) corresponding to the data
(
u
0
,
v
0
,
ϑ
→
,
ϱ
→
)
, whose existence is guaranteed by the fixed-point principle. Consider now sequences
u
0
n
→
u
0
,
v
0
n
→
v
0
in
H
s
(
0
,
1
)
and
ϑ
→
n
→
ϑ
→
,
ϱ
→
n
→
ϱ
→
in
Z
s
,
and denote by the
u
n
,
v
n
associated solutions with data
(
u
0
n
,
v
0
n
,
ϑ
→
n
,
ϱ
→
n
)
. Invoking the estimates from Lemmas 4–7, we deduce that
(
u
n
,
v
n
)
→
(
u
,
v
)
in
X
s
,
T
∗
×
X
s
,
T
∗
as
n
→
∞
.
Gathering the contraction mapping framework with the a priori estimates we rigorously established earlier naturally gives rise to the continuous dependence of the solutions. More precisely, it follows from the local well-posedness theory that there exist two pairs of solutions,
(
u
,
v
)
\=
(
Ψ
1
(
u
0
,
ϑ
→
)
,
Ψ
2
(
v
0
,
ϱ
→
)
)
∈
B
s
,
M
×
B
s
,
M
and
(
u
˜
,
v
˜
)
\=
(
Ψ
1
(
u
˜
0
,
ϑ
→
˜
)
,
Ψ
2
(
v
˜
0
,
ϱ
→
˜
)
)
∈
B
s
,
M
×
B
s
,
M
, to the IBVP ([1](https://www.mdpi.com/2075-1680/15/3/230#FD1-axioms-15-00230)) corresponding to the data
(
(
u
0
,
v
0
)
,
ϑ
→
,
ϱ
→
)
and
(
(
u
˜
0
,
v
˜
0
)
,
ϑ
→
˜
,
ϱ
→
˜
)
in
H
s
(
0
,
1
)
×
Z
s
×
Z
s
that satisfy the s\-compatibility conditions, respectively. Furthermore, the difference between the two solution pairs satisfies the corresponding initial and boundary values. Specifically, the initial and boundary values of the difference are simply the differences in the respective data. A direct computation gives that
(
Ψ
1
(
u
0
,
ϑ
→
)
,
Ψ
2
(
v
0
,
ϱ
→
)
)
−
(
Ψ
1
(
u
˜
0
,
ϑ
→
˜
)
,
Ψ
2
(
v
˜
0
,
ϱ
→
˜
)
)
X
s
,
T
∗
×
X
s
,
T
∗
\=
(
u
,
v
)
−
(
u
˜
,
v
˜
)
X
s
,
T
∗
×
X
s
,
T
∗
≲
(
S
u
0
,
ϑ
→
(
u
)
,
S
v
0
,
ϱ
→
(
v
)
)
−
(
S
u
˜
0
,
ϑ
→
˜
(
u
)
,
S
v
˜
0
,
ϑ
→
˜
(
v
)
)
X
s
,
T
∗
×
X
s
,
T
∗
≲
S
u
0
−
u
˜
0
,
ϑ
→
−
ϑ
→
˜
(
(
u
1
−
u
2
)
,
u
1
,
∂
x
u
1
)
X
s
,
T
∗
\+
S
u
0
−
u
˜
0
,
ϑ
→
−
ϑ
→
˜
(
u
2
,
(
u
1
−
u
2
)
,
∂
x
u
1
)
X
s
,
T
∗
\+
(
S
u
0
−
u
˜
0
,
ϑ
→
−
ϑ
→
˜
(
u
2
,
(
u
1
−
u
2
)
x
,
u
2
)
X
s
,
T
∗
\+
S
u
0
−
u
˜
0
,
ϑ
→
−
ϑ
→
˜
(
(
v
1
−
v
2
)
,
v
1
,
∂
x
u
1
)
X
s
,
T
∗
\+
S
u
0
−
u
˜
0
,
ϑ
→
−
ϑ
→
˜
(
v
2
,
(
v
1
−
v
2
)
,
∂
x
u
1
)
X
s
,
T
∗
\+
S
u
0
−
u
˜
0
,
ϑ
→
−
ϑ
→
˜
(
v
2
,
v
2
,
(
u
1
−
u
2
)
x
)
X
s
,
T
∗
\+
S
u
0
−
u
˜
0
,
ϑ
→
−
ϑ
→
˜
(
(
u
1
−
u
2
)
,
v
1
,
∂
x
v
1
)
X
s
,
T
∗
\+
S
u
0
−
u
˜
0
,
ϑ
→
−
ϑ
→
˜
(
u
2
,
(
v
1
−
v
2
)
,
∂
x
v
1
)
X
s
,
T
∗
\+
S
u
0
−
u
˜
0
,
ϑ
→
−
ϑ
→
˜
(
u
2
,
v
2
,
(
v
1
−
v
2
)
x
)
X
s
,
T
∗
\+
S
u
0
−
u
˜
0
,
ϑ
→
−
ϑ
→
˜
(
(
u
1
−
u
2
)
,
u
1
,
u
1
)
X
s
,
T
∗
\+
S
u
0
−
u
˜
0
,
ϑ
→
−
ϑ
→
˜
(
u
2
,
(
u
1
−
u
2
)
,
u
1
)
X
s
,
T
∗
\+
S
u
0
−
u
˜
0
,
ϑ
→
−
ϑ
→
˜
(
u
2
,
u
2
,
(
u
1
−
u
2
)
)
X
s
,
T
∗
\+
S
u
0
−
u
˜
0
,
ϑ
→
−
ϑ
→
˜
(
(
v
1
−
v
2
)
,
v
1
,
u
1
)
X
s
,
T
∗
\+
S
u
0
−
u
˜
0
,
ϑ
→
−
ϑ
→
˜
(
v
2
,
(
v
1
−
v
2
)
,
u
1
)
X
s
,
T
∗
\+
S
u
0
−
u
˜
0
,
ϑ
→
−
ϑ
→
˜
(
v
2
,
v
2
,
(
u
1
−
u
2
)
)
X
s
,
T
∗
≤
C
(
T
∗
)
u
0
−
u
˜
0
H
s
\+
v
0
−
v
˜
0
H
s
\+
ϑ
→
−
ϑ
→
˜
Z
s
\+
ϱ
→
−
ϱ
→
˜
Z
s
\+
1
2
(
u
,
v
)
−
(
u
˜
,
v
˜
)
X
s
,
T
∗
×
X
s
,
T
∗
,
where the last term depends on
C
(
T
∗
)
M
2
≤
1
16
,
C
(
T
∗
)
M
2
≤
1
16
for small
T
∗
.
The proof of the Lipschitz continuous for ([1](https://www.mdpi.com/2075-1680/15/3/230#FD1-axioms-15-00230)) is now complete.
### 4\.2. Proof of Corollary 1
This section is concerned with the proof of Corollary 1. In the course of proving Theorem 1, the nonlinear estimates we use impose a limitation: they only work for
s
∈
\[
0
,
5
2
)
, which motivates our method for the case
s
\>
5
2
. We deduce from Lemmas 4 and 5 that
(
S
u
0
,
ϑ
→
(
0
)
,
S
v
0
,
ϱ
→
(
0
)
)
X
s
,
T
≲
u
0
H
s
\+
v
0
H
s
\+
ϑ
→
Z
s
\+
ϱ
→
Z
s
,
(41)
where
s
≥
0
and the solution corresponding to the problem ([1](https://www.mdpi.com/2075-1680/15/3/230#FD1-axioms-15-00230)) admits
(
u
(
x
,
t
)
,
v
(
x
,
t
)
)
\=
(
S
u
0
,
ϑ
→
(
0
)
,
S
v
0
,
ϱ
→
(
0
)
)
\+
(
S
0
,
0
→
(
Q
1
)
,
S
0
,
0
→
(
Q
2
)
)
.
(42)
where
Q
1
(
u
)
\=
i
χ
{
2
\|
u
\|
2
\+
\|
v
\|
2
u
x
\+
u
v
¯
v
x
}
\+
δ
\|
u
\|
2
\+
\|
v
\|
2
u
and
Q
2
(
u
,
v
)
\=
i
χ
{
(
\|
u
\|
2
\+
2
\|
v
\|
2
)
v
x
\+
u
¯
v
u
x
}
\+
δ
\|
u
\|
2
\+
\|
v
\|
2
v
. In the case of homogeneous initial-boundary data, we define
ω
1
\=
∂
t
S
0
,
0
→
(
Q
1
)
,
ω
2
\=
∂
t
S
0
,
0
→
(
Q
2
)
, which in turn yields that
i
∂
t
ω
1
\+
α
1
∂
x
2
ω
1
\+
i
ς
∂
x
3
ω
1
\+
L
u
,
v
(
ω
1
,
ω
2
)
\=
0
,
i
∂
t
ω
2
\+
α
2
∂
x
2
ω
2
\+
i
ς
∂
x
3
ω
2
\+
L
˜
u
,
v
(
ω
1
,
ω
2
)
\=
0
,
ω
1
(
x
,
0
)
\=
0
,
ω
2
(
x
,
0
)
\=
0
,
ω
1
(
0
,
t
)
\=
0
,
ω
1
(
1
,
t
)
\=
0
,
ψ
1
x
(
1
,
t
)
\=
0
,
ω
2
(
0
,
t
)
\=
0
,
ω
2
(
1
,
t
)
\=
0
,
ψ
2
x
(
1
,
t
)
\=
0
,
(43)
where the operators
L
u
,
v
(
ω
1
,
ω
2
)
and
L
˜
u
,
v
(
ω
1
,
ω
2
)
are nonlinear terms related to
ω
1
,
ω
2
.
Since the linearization operators
L
u
,
v
(
ω
1
,
ω
2
)
and
L
˜
u
,
v
(
ω
1
,
ω
2
)
are linear with respect to
ω
1
,
ω
2
, the standard linear energy estimates can be directly applied. It follows from Lemma 7 that
ω
1
X
s
,
T
≤
(
u
X
s
,
T
2
\+
v
X
s
,
T
2
)
∂
t
u
X
s
,
T
,
(44)
and
ω
2
X
s
,
T
≤
(
v
X
s
,
T
2
\+
u
X
s
,
T
2
)
∂
t
v
X
s
,
T
,
(45)
for
s
∈
\[
0
,
1
)
. Furthermore, it relies on the key observation that
∂
t
u
X
s
,
T
\=
∂
t
S
u
0
,
ϑ
→
(
0
)
\+
∂
t
S
0
,
0
→
(
Q
1
)
X
s
,
T
≲
∂
t
S
u
0
,
ϑ
→
(
0
)
X
s
,
T
\+
ω
1
X
s
,
T
,
(46)
∂
t
v
X
s
,
T
\=
∂
t
S
v
0
,
ϱ
→
(
0
)
\+
∂
t
S
0
,
0
→
(
Q
2
)
X
s
,
T
≲
∂
t
S
v
0
,
ϱ
→
(
0
)
X
s
,
T
\+
ω
2
X
s
,
T
.
(47)
After setting
s
\=
0
in ([44](https://www.mdpi.com/2075-1680/15/3/230#FD44-axioms-15-00230)) and ([45](https://www.mdpi.com/2075-1680/15/3/230#FD45-axioms-15-00230)), we substitute this result into ([46](https://www.mdpi.com/2075-1680/15/3/230#FD46-axioms-15-00230)) and ([47](https://www.mdpi.com/2075-1680/15/3/230#FD47-axioms-15-00230)), respectively, thereby obtaining that
ω
1
X
0
,
T
≲
(
u
X
0
,
T
2
\+
v
X
0
,
T
2
)
(
∂
t
S
u
0
,
ϑ
→
(
0
)
X
0
,
T
\+
ω
1
X
0
,
T
)
,
ω
2
X
0
,
T
≲
(
v
X
0
,
T
2
\+
u
X
0
,
T
2
)
(
∂
t
S
u
0
,
ϱ
→
(
0
)
X
0
,
T
\+
ω
2
X
0
,
T
)
.
Theorem 1 yields the key estimate
u
X
0
,
T
\+
v
X
0
,
T
≲
u
0
L
2
\+
v
0
L
2
\+
ϑ
→
Z
0
\+
ϱ
→
Z
0
.
For
T
∗
∈
(
0
,
T
)
to be chosen, it must be sufficiently small to ensure that
C
(
T
∗
)
(
u
X
0
,
T
2
\+
v
X
0
,
T
2
)
≤
1
8
. Under this condition, we deduce
ω
1
X
0
,
T
∗
≲
∂
t
S
u
0
,
ϑ
→
(
0
)
X
0
,
T
∗
,
(48)
ω
2
X
0
,
T
∗
≲
∂
t
S
u
0
,
ϑ
→
(
0
)
X
0
,
T
∗
.
(49)
Take the
X
0
,
T
∗
norm on both sides of ([43](https://www.mdpi.com/2075-1680/15/3/230#FD43-axioms-15-00230)), and apply the triangle inequality for the following norms:
∥
∂
t
u
∥
X
0
,
T
∗
≲
α
1
∥
∂
x
2
u
∥
X
0
,
T
∗
\+
ς
∥
∂
x
3
u
∥
X
0
,
T
∗
\+
∥
Q
1
(
u
,
v
)
∥
X
0
,
T
∗
,
(50)
∥
∂
t
v
∥
X
0
,
T
∗
≲
α
2
∥
∂
x
2
v
∥
X
0
,
T
∗
\+
ς
∥
∂
x
3
v
∥
X
0
,
T
∗
\+
∥
Q
2
(
u
,
v
)
∥
X
0
,
T
∗
.
(51)
It follows from gathering ([41](https://www.mdpi.com/2075-1680/15/3/230#FD41-axioms-15-00230)), ([42](https://www.mdpi.com/2075-1680/15/3/230#FD42-axioms-15-00230)), and ([48](https://www.mdpi.com/2075-1680/15/3/230#FD48-axioms-15-00230))–(51) that
∂
t
u
X
0
,
T
∗
≲
∂
t
S
u
0
,
ϑ
→
(
0
)
X
0
,
T
∗
\+
ω
1
X
0
,
T
∗
≲
∂
t
S
u
0
,
ϑ
→
(
0
)
X
0
,
T
∗
≲
∂
t
u
L
2
(
0
)
\+
∂
t
ϑ
→
Z
0
≲
u
0
H
3
\+
ϑ
→
Z
3
,
(52)
and
∂
t
v
X
0
,
T
∗
≲
v
0
H
3
\+
ϱ
→
Z
3
.
(53)
Moreover, by using ([1](https://www.mdpi.com/2075-1680/15/3/230#FD1-axioms-15-00230)), we have
ς
∂
x
3
u
\=
−
u
t
\+
i
α
1
∂
x
2
u
\+
i
Q
1
(
u
,
v
)
,
ς
∂
x
3
v
\=
−
v
t
\+
i
α
2
∂
x
2
v
\+
i
Q
2
(
u
,
v
)
,
which, when combined with Young’s inequality and the Gagliardo–Nirenberg inequality, can be used to derive that
u
X
3
,
T
∗
≲
∂
t
u
X
0
,
T
∗
\+
∂
x
3
u
X
0
,
T
∗
\+
∂
x
2
u
X
0
,
T
∗
≲
u
0
H
3
\+
ϑ
→
Z
3
,
and
v
X
3
,
T
∗
≲
v
0
H
3
\+
ϱ
→
Z
3
.
Then, the solution mappings
Ψ
1
,
Ψ
2
for (1) satisfy
Ψ
1
,
Ψ
2
:
H
3
×
Z
3
→
X
3
,
T
∗
,
Ψ
1
X
3
,
T
∗
≲
u
0
H
3
\+
ϑ
→
Z
3
,
Ψ
2
X
3
,
T
∗
≲
v
0
H
3
\+
ϱ
→
Z
3
,
and for the case
s
\=
0
, we further have that
Ψ
1
,
Ψ
2
:
L
2
×
Z
0
→
X
0
,
T
∗
,
Ψ
1
(
u
0
,
ϑ
→
)
−
Ψ
1
(
u
0
,
ϑ
→
˜
)
X
0
,
T
∗
≲
u
0
−
u
˜
0
L
2
\+
ϑ
→
−
ϑ
→
˜
Z
0
,
Ψ
2
(
v
0
,
ϱ
→
)
−
Ψ
2
(
v
0
,
ϱ
→
˜
)
X
0
,
T
∗
≲
v
0
−
v
˜
0
L
2
\+
ϱ
→
−
ϱ
→
˜
Z
0
.
Moreover, we deduce from Lemma 2 that
Ψ
1
,
Ψ
2
:
H
3
(
1
−
θ
)
×
Z
3
(
1
−
θ
)
→
X
3
(
1
−
θ
)
,
T
∗
,
Ψ
1
(
u
0
,
ϑ
→
)
X
3
(
1
−
θ
)
,
T
∗
≲
u
0
H
3
(
1
−
θ
)
\+
ϑ
→
Z
3
(
1
−
θ
)
,
Ψ
2
(
v
0
,
ϱ
→
)
X
3
(
1
−
θ
)
,
T
∗
≲
v
0
H
3
(
1
−
θ
)
\+
ϱ
→
Z
3
(
1
−
θ
)
,
(54)
where
3
(
1
−
θ
)
\>
5
2
,
θ
∈
(
0
,
1
6
)
. We deduce from Lemma 2 and setting
s
\=
3
k
(
1
−
θ
)
,
k
\=
2
,
3
,
…
,
for
θ
∈
(
0
,
1
)
that
Ψ
1
,
Ψ
2
:
H
s
×
Z
s
→
X
s
,
T
∗
,
Ψ
1
(
u
0
,
ϑ
→
)
X
s
,
T
∗
≲
u
0
H
s
\+
ϑ
→
Z
s
,
Ψ
2
(
v
0
,
ϱ
→
)
X
s
,
T
∗
≲
v
0
H
s
\+
ϱ
→
Z
s
.
(55)
This finishes the proof of Corollary 1, gathering ([54](https://www.mdpi.com/2075-1680/15/3/230#FD54-axioms-15-00230)) and (55).
## 5\. Conclusions
We consider the well-posedness of ([1](https://www.mdpi.com/2075-1680/15/3/230#FD1-axioms-15-00230)) on a bounded interval. By utilizing the Laplace transform, which is well suited to the mixed dispersive structure of the equation, we derive a solution operator formula for the linear nonhomogeneous coupled system. Combining the fixed-point principle with energy estimates, we establish the local well-posedness of the problem in
H
s
(
0
,
1
)
when
s
∈
\[
0
,
3
/
2
)
is rigorously proven. Furthermore, a corollary extending the local result to the case
s
≥
3
/
2
is proved. It is clarified that when the initial values
(
u
0
,
v
0
)
∈
H
s
(
0
,
1
)
×
H
s
(
0
,
1
)
and the boundary values belong to space
Z
s
and satisfy compatibility conditions, ([1](https://www.mdpi.com/2075-1680/15/3/230#FD1-axioms-15-00230)) admits a unique solution and the associated solution map is Lipschitz continuous in
X
s
,
T
×
X
s
,
T
. During the research process, by decomposing the solution operator into an initial value operator and boundary operator, linear and nonlinear estimates were established, respectively. Tools such as Cauchy–Schwarz inequality, Gagliardo–Nirenberg inequality, and interpolation theorem are utilized to complete a priori estimates. This approach addresses the balance between the smoothing effect of dispersion brought by third-order derivatives and the loss of spatial derivatives in nonlinear terms, and also compensates for the limitations of the Duhamel forcing operator method in dealing with the mixed dispersion problem of this equation. The research results provide a reference framework for the study of dispersion equations involving third-order derivatives. Within this context, modern mesh-free methods demonstrate significant potential for solving high-order boundary value problems. For instance, Karageorghis, Noorizadegan, and Chen have recently employed Radial Basis Function (RBF) methods, such as the fictitious center RBF method, which have been successfully applied to high-order boundary value problems, as demonstrated in \[34\]. Future research could explore the application and adaptation of such mesh-free techniques to numerically solve the IBVP of (cH), for which our analytical results on solution stability and regularity provide a crucial theoretical underpinning.
## Author Contributions
S.W. and H.W. contributed to the draft of the manuscript. All authors have read and agreed to the published version of the manuscript.
## Funding
This research received no external funding.
## Data Availability Statement
The data presented in this study are available on request from the corresponding author.
## Conflicts of Interest
The authors declare no conflicts of interest.
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| Readable Markdown | - Article
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20 March 2026
and
School of Mathematics, Statistics and Mechanics, Beijing University of Technology, Beijing 100021, China
\*
Author to whom correspondence should be addressed.
## Abstract
In this work, we address the nonhomogeneous initial-boundary value problem for the coupled Hirota equation posed on the finite interval \[ 0 , L \]. To investigate the well-posedness of this problem, we first adopt an appropriate transformation, namely the Laplace transform, which is tailored to the specific characteristics of the problem, and further obtain an explicit solution formula for the linear inhomogeneous coupled system. Subsequently, the local well-posedness of the original nonhomogeneous initial-boundary value problem in X s , T × X s , T X s , T \= C ( 0 , T ; H s ( 0 , 1 ) ) ∩ L 2 ( 0 , T ; H s \+ 1 ( 0 , 1 ) ) is rigorously established through the combination of this explicit formula, the contraction mapping principle and energy estimates.
MSC:
35Q55; 35Q53; 35G61
## 1\. Introduction
We study the coupled Hirota (cH) equation
i u t \+ α 1 ∂ x 2 u \+ i ς ∂ x 3 u \+ i χ 2 \| u \| 2 u x \+ \| v \| 2 u x \+ u v ¯ v x \+ δ \| u \| 2 u \+ \| v \| 2 u \= 0 , i v t \+ α 2 ∂ x 2 v \+ i ς ∂ x 3 v \+ i χ \| u \| 2 v x \+ 2 \| v \| 2 v x \+ u ¯ v u x \+ δ \| u \| 2 v \+ \| v \| 2 v \= 0 ,
(cH)
where α i ( i \= 1 , 2 ) , ς ∈ R \+ , χ , δ ∈ R are prescribed real constants, and u , v denote complex-valued functions. Tasgal and Potasek \[1\] first derived the coupled Hirota equation from the single Hirota equation, showing that third-order dispersion and self-steepening can support the simultaneous propagation of two ultrashort pulses while suppressing inelastic Raman scattering effect.
The Hirota equation
i u t \+ α ∂ x 2 u \+ i ς ∂ x 3 u \+ i χ ∂ x ( \| u \| 2 u ) \+ δ \| u \| 2 u \= 0 , α ς χ δ ≠ 0 ,
proposed by R. Hirota in 1973 \[2\], stands as a fundamental generalization of the integrable nonlinear Schrödinger (NLS) equation. It provides an accurate description of ultrashort optical pulse propagation in single-mode fibers. To be precise, the NLS equation only accounts for second-order dispersion and the Kerr effect (for picosecond pulses). Extra effects have to be incorporated when the pulse length is on the order of the wavelength or the pulse width reaches the femtosecond regime. In such cases, the NLS equation is generalized to the Hirota equation. The Hirota equation, also known as the Airy–Schrödinger equation, includes the classic NLS equation for α \= 1 , ς \= χ \= 0, and the derivative NLS equation for α \= 1 , ς \= δ \= 0.
The Hirota equation is completely integrable for some special parameters \[2\],
i u t \+ ∂ x 2 u \+ i ς ∂ x 3 u \+ 6 i ς \| u \| 2 u x \+ 2 \| u \| 2 u \= 0 ,
which possesses infinite conservation laws with respect to the Cauchy problem. Employing the conservation laws, one can readily derive the global well-posedness of the Hirota equation in H s for s ∈ N. Evidently, the fact that the Hirota equation is completely integrable does not affect its well-posedness, allowing us to investigate the Hirota equations with more general parameters. Previously, numerous studies have focused on the Hirota equation and the coupled Hirota equation \[3,4,5,6,7,8\]. The earlier results on higher regularity for the Hirota equation were established by Guo and Tan \[4\] in H s for s ≥ 3. Following this, Ref. \[3\] generalized the earlier local result to H s for s \> 3 4 and the global result to H s for s \= 1 and s ≥ 2 . In \[5\], Staffilani also proved the low regularity in H s for s \> 1 4 . By using more meticulous methods, the results of well-posedness enter the discussion of low-regularity cases (see \[6,7,8\]). The optimal regularity result for the Hirota equation was achieved in \[7\], which established the local result in H s for s \> − 1 4 and global result in H s for s \> 0 . In addition, the only existing result concerning the Cauchy problem of (cH) in H s × H s for s ≥ 1 4 was rigorously demonstrated in \[9\].
This manuscript is devoted to investigating the IBVP for (cH) defined on a finite interval:
i u t \+ α 1 ∂ x 2 u \+ i ς ∂ x 3 u \+ i χ 2 \| u \| 2 u x \+ \| v \| 2 u x \+ u v ¯ v x \+ δ \| u \| 2 u \+ \| v \| 2 u \= 0 , i v t \+ α 2 ∂ x 2 v \+ i ς ∂ x 3 v \+ i χ ( \| u \| 2 v x \+ 2 \| v \| 2 v x ) \+ u ¯ v u x \+ δ \| u \| 2 v \+ \| v \| 2 v \= 0 , u ( x , 0 ) \= u 0 ( x ) , v ( x , 0 ) \= v 0 ( x ) , x ∈ R \+ , u ( 0 , t ) \= ϑ 1 ( t ) , u ( L , t ) \= ϑ 2 ( t ) , u x ( L , t ) \= ϑ 3 ( t ) , t ∈ R \+ , v ( 0 , t ) \= ϱ 1 ( t ) , v ( L , t ) \= ϱ 2 ( t ) , v x ( L , t ) \= ϱ 3 ( t ) , t ∈ R \+ ,
(1)
where L is positive nonzero constants, and without loss of generality, let L \= 1.
Recall the result of the IBVP concerning R \+ or a interval \[ 0 , L \] . Early research on IBVP primarily focused on high regularity due to the energy method. For the IBVP posed on R \+, Colliander, Kenig \[10\], and Holmer \[11,12\] extended the well-posedness to the low-regularity case. Their proof relied on the Duhamel boundary operator and Bourgain space X s , b, which was proposed by Bourgain in \[13,14\]. For the IBVP posed on a finite interval, due to the Duhamel boundary operator being unsuitable for equations exhibiting mixed dispersion effects, Bona et al. \[15,16,17\] proved the low-regularity existence for the KdV and NLS equation via the Laplace transform. Reference is made to \[16,18,19,20,21,22,23,24,25,26,27,28,29,30,31\] for results involving the IBVP for other dispersive equations.
Our primary goal herein is to analyze the IBVP of the cH equation within a bounded interval framework. In doing so, we resolve the core issues of existence and uniqueness, and further demonstrate the continuous dependence on the prescribed data under suitable regularity constraints. The Laplace transform is specially selected in this work to address the core challenges brought by the mixed dispersion inherent in the coupled Hirota equation. The third-order spatial derivative term complicates the standard Bourgain space methods for addressing the associated IBVP, making them far more intricate than the approaches applied to the second-order NLS equation. It is evident from the proof that existence can be obtained on the initial data ( u 0 ( x ) , v 0 ( x ) ) ∈ H s ( 0 , 1 ) : \= H s ( 0 , 1 ) × H s ( 0 , 1 ) and boundary data
ϑ 1 ( t ) , ϑ 2 ( t ) , ϑ 3 ( t ) , ϱ 1 ( t ) , ϱ 2 ( t ) , ϱ 3 ( t ) ∈ Z s : \= H s \+ 1 3 ( R \+ ) × H s \+ 1 3 ( R \+ ) × H s 3 ( R \+ ) .
In order to address the well-posedness of IBVP, the given data must satisfy the following compatibility condition and definition:
u 0 , ι ( 0 ) \= ϑ 1 ( ι ) ( 0 ) , u 0 , ι ( 1 ) \= ϑ 2 ( ι ) ( 0 ) , u 0 , ι ′ ( 1 ) \= ϑ 3 ( ι ) ( 0 ) , v 0 , ι ( 0 ) \= ϱ 1 ( ι ) ( 0 ) , v 0 , ι ( 1 ) \= ϱ 2 ( ι ) ( 0 ) , v 0 , ι ′ ( 1 ) \= ϱ 3 ( ι ) ( 0 ) ,
where u 0 , ι ( 0 ) \= u 0 ( x ) , v 0 , ι ( 0 ) \= v 0 ( x ) if ι \= 0 and for ι ≥ 1 ,
u 0 , ι ( x ) \= i α 1 u 0 , ι − 1 ″ − ς u 0 , ι − 1 ‴ \+ i δ ∑ j \= 0 ι − 1 ( \| u 0 , j \| 2 \+ \| v 0 , j \| 2 ) u 0 , ι − 1 − j − χ ∑ j \= 0 ι − 1 { ( 2 \| u 0 , j \| 2 \+ \| v 0 , j \| 2 ) u 0 , ι − 1 − j ′ \+ u 0 , j v ¯ 0 , ι − 1 − j v 0 , ι − 1 − j ′ } , v 0 , ι ( x ) \= i α 2 v 0 , ι − 1 ″ − ς v 0 , ι − 1 ‴ \+ i δ ∑ j \= 0 ι − 1 ( \| u 0 , j \| 2 \+ \| v 0 , j \| 2 ) v 0 , ι − 1 − j − χ ∑ j \= 0 ι − 1 { ( \| u 0 , j \| 2 \+ 2 \| v 0 , j \| 2 ) v 0 , ι − 1 − j ′ \+ u ¯ 0 , j u 0 , j ′ v 0 , ι − 1 − j } .
**Definition** **1\.**
For arbitrary s ∈ \[ 0 , \+ ∞ ) and T ∈ ( 0 , \+ ∞ ) , the compatibility condition for u 0 ( x ) , v 0 ( x ) , ϑ → ( t ) , ϱ → ( t ) \= u 0 ( x ) , v 0 ( x ) , ϑ 1 ( t ) , ϑ 2 ( t ) , ϑ 3 ( t ) , ϱ 1 ( t ) , ϱ 2 ( t ) , ϱ 3 ( t ) ∈ H s ( 0 , 1 ) × Z s × Z s is as follows:
1\.
If s − 3 s 3 ≤ 1 2 ,
u 0 , ι ( 0 ) \= ϑ 1 ( ι ) ( 0 ) , u 0 , ι ( 1 ) \= ϑ 2 ( ι ) ( 0 ) , v 0 , ι ( 0 ) \= ϱ 1 ( ι ) ( 0 ) , v 0 , ι ( 1 ) \= ϱ 2 ( ι ) ( 0 ) ,
(2)
is valid for ι \= 0 , 1 , ⋯ , s 3 − 1 .
2\.
If 1 2 \< s − 3 s 3 \< 3 2 , ([2](https://www.mdpi.com/2075-1680/15/3/230#FD2-axioms-15-00230)) is valid for ι \= 0 , 1 , ⋯ , s 3 .
3\.
If s − 3 s 3 \> 3 2 ,
u 0 , ι ( 0 ) \= ϑ 1 ( ι ) ( 0 ) , u 0 , ι ( 1 ) \= ϑ 2 ( ι ) ( 0 ) , u 0 , ι ′ ( 1 ) \= ϑ 3 ( ι ) ( 0 ) v 0 , ι ( 0 ) \= ϱ 1 ( ι ) ( 0 ) , v 0 , ι ( 1 ) \= ϱ 2 ( ι ) ( 0 ) , v 0 , ι ′ ( 1 ) \= ϱ 3 ( ι ) ( 0 ) .
is valid for ι \= 0 , 1 , ⋯ , s 3 − 1 .
4\.
We agree that ([2](https://www.mdpi.com/2075-1680/15/3/230#FD2-axioms-15-00230)) is vacuous if s 3 − 1 \< 0 .
It is obvious that higher-order time derivatives cannot be directly observed from boundary value, and they must be computed recursively through the equation itself along with its initial conditions. In light of the previously introduced compatibility conditions, we now formulate a theorem, which constitutes the central focus of our investigation.
**Theorem** **1\.**
For s ∈ \[ 0 , 3 2 ) and T \> 0 , we assume that
( U 0 , ϱ → ) : \= u 0 ( x ) , v 0 ( x ) , ϑ → ( t ) , ϱ → ( t ) \= u 0 ( x ) , v 0 ( x ) , ϑ 1 ( t ) , ϑ 2 ( t ) , ϑ 3 ( t ) , ϱ 1 ( t ) , ϱ 2 ( t ) , ϱ 3 ( t ) ∈ H s ( 0 , 1 ) × Z s × Z s ,
which satisfies Definition 1. Then, IBVP ([1](https://www.mdpi.com/2075-1680/15/3/230#FD1-axioms-15-00230)) admits a unique solution in
X s , T × X s , T ,
where X s , T \= C 0 , T ; H s ( 0 , 1 ) ∩ L 2 0 , T ; H s \+ 1 ( 0 , 1 ) and
T : \= T u 0 H s , v 0 H s , ϑ → Z s , ϱ → Z s .
In addition, Lipschitz continuity also holds for the given initial and boundary values.
**Corollary** **1\.**
For s ≥ 3 2 and T \> 0 , assume that u 0 ( x ) , v 0 ( x ) , ϑ → ( t ) , ϱ → ( t ) ∈ H s ( 0 , 1 ) × Z s × Z s , which satisfies Definition 1. Then there exists a unique solution of ([1](https://www.mdpi.com/2075-1680/15/3/230#FD1-axioms-15-00230)) for sufficiently small T.
Compared with the scalar Hirota equation, the coupled Hirota (cH) system introduces interdependent nonlinear cross-coupling terms (i.e., u 2 v x , u v ¯ v x , v 2 u x , u ¯ v u x) and asymmetric linear dispersion terms with distinct coefficients α 1 , α 2 for the two components u and v . These coupling characteristics give rise to fundamental mathematical challenges, such as cross-dependent nonlinearities in energy estimates and compatibility conditions for the coupled system, which do not arise in the scalar case.
Theorem 1 and Corollary 1 establish the local low regularity in the energy space H s ( 0 , 1 ) × H s ( 0 , 1 ). To clarify this definition of regularity, it is necessary to specify the precise meaning of ( u , v ) as a solution to ([1](https://www.mdpi.com/2075-1680/15/3/230#FD1-axioms-15-00230)). This is particularly crucial for low regularity cases, where the solution must satisfy the equation in a suitable sense. For IBVP with a bounded interval, the concept of a mild solution is typically defined to ensure that the solution satisfies the nonlinear term along with the given data.
In earlier research of IBVP, the standard approach to handling nonhomogeneous boundary data involved transforming the dependent variable to achieve homogeneous boundary conditions. The specific approach involved solving a combination of the boundary value and the boundary conditions, thereby making the boundary conditions equal to 0, but at this point, the new variable satisfied a new form of equation. While this approach permits converting the IBVP into an initial value problem for investigation, it requires the boundary data to possess higher regularity compared to more recent methods. To achieve low-regularization of the well-posedness of IBVP, the Duhamel boundary forcing operator serves as a powerful tool. However, due to its specific construction form, it is not applicable to equations exhibiting mixed dispersion effects (e.g., Hirota equation). Proof of the main results adapt the strategy proposed by Bona et al. \[16\], who developed a strategy involving the Laplace transform to construct a solution formula that incorporates both the linear equation and the boundary operator. Specifically, the prior estimates encompass both linear and nonlinear estimates. The linear estimate is derived based on the energy method, while the nonlinear estimate is obtained by Lemma 4 and the interpolation theorem.
Below is the structure of the rest of this work. [Section 2](https://www.mdpi.com/2075-1680/15/3/230#sec2-axioms-15-00230) establishes the foundational framework: it standardizes notation, formalizes function space definitions, and deriving a solution representation that decomposes the solution into linear and nonlinear evolution. The analytical backbone of this work is presented in [Section 3](https://www.mdpi.com/2075-1680/15/3/230#sec3-axioms-15-00230), where we develop the critical estimates required to implement a contraction mapping argument. Finally, [Section 4](https://www.mdpi.com/2075-1680/15/3/230#sec4-axioms-15-00230) is dedicated to rigorously proving the main result (Theorem 1) and its associated corollary (Corollary 1).
## 2\. Preliminaries
### 2\.1. Notations
We use the following notation for function spaces:
C t 0 H x s \= C 0 , T ; H s ( 0 , 1 ) L t p H x s \= L p 0 , T ; H s ( 0 , 1 ) , p ∈ \[ 1 , ∞ ) X s , T \= C 0 , T ; H s ( 0 , 1 ) ∩ L 2 0 , T ; H s \+ 1 ( 0 , 1 ) ,
and
Z s \= Z s : \= H s \+ 1 3 ( R \+ ) × H s \+ 1 3 ( R \+ ) × H s 3 ( R \+ ) ,
where we write the inhomogeneous L 2\-based space H s \= H s ( R ) as
∥ g ∥ H s ( R ) \= ⟨ ξ ⟩ s g ^ ( ξ ) L ξ 2 .
We denote the space H 0 s as
H 0 s ( φ ) \= the closure of D ( φ ) in H s ( φ ) ,
Let φ be an open set in R 1, and let both m and n be continuous functions. Define the space D ( φ ) as the set of all functions ϕ, such that ϕ is infinitely differentiable on φ and has compact support contained in φ.
Finally, throughout this paper, unless otherwise stated, we abbreviate ∥ · ∥ H s as ∥ · ∥ H x s ( 0 , 1 ) for simplicity. In addition, the notation a ≲ ( ≳ ) b is used to denote that a ≤ ( ≥ ) C b, where C is a positive constant.
### 2\.2. Solution Formula
With the aim of constructing a solution formula for Equation ([1](https://www.mdpi.com/2075-1680/15/3/230#FD1-axioms-15-00230)) using the Laplace transform. In order to establish the solution to problem ([1](https://www.mdpi.com/2075-1680/15/3/230#FD1-axioms-15-00230)), we examine the following nonhomogeneous initial-boundary value problem:
i u t \+ α 1 ∂ x 2 u \+ i ς ∂ x 3 u \+ Q 1 ( u , v ) \= 0 , i v t \+ α 2 ∂ x 2 v \+ i ς ∂ x 3 v \+ Q 2 ( u , v ) \= 0 , u ( x , 0 ) \= u 0 ( x ) , v ( x , 0 ) \= v 0 ( x ) , u ( 0 , t ) \= ϑ 1 ( t ) , u ( 1 , t ) \= ϑ 2 ( t ) , u x ( 1 , t ) \= ϑ 3 ( t ) , t ∈ R \+ , v ( 0 , t ) \= ϱ 1 ( t ) , v ( 1 , t ) \= ϱ 2 ( t ) , v x ( 1 , t ) \= ϱ 3 ( t ) , t ∈ R \+ ,
(3)
where
Q 1 ( u ) \= i χ 2 \| u \| 2 u x \+ \| v \| 2 u x \+ u v ¯ v x \+ δ \| u \| 2 u \+ \| v \| 2 u , Q 2 ( u , v ) \= i χ ( \| u \| 2 v x \+ 2 \| v \| 2 v x ) \+ u ¯ v u x \+ δ \| u \| 2 v \+ \| v \| 2 v
and ( u 0 , v 0 , ϑ → , ϱ → ) \= ( u 0 , v 0 , ϑ 1 , ϑ 2 , ϑ 3 , ϱ 1 , ϱ 2 , ϱ 3 ) ∈ H s × Z s × Z s that satisfies Definition 1. We first decompose the IBVP ([3](https://www.mdpi.com/2075-1680/15/3/230#FD3-axioms-15-00230)) into two distinct problems: a nonlinear system which satisfies initial data
i u t \+ α 1 ∂ x 2 u \+ i ς ∂ x 3 u \+ Q 1 ( u , v ) \= 0 , i v t \+ α 2 ∂ x 2 v \+ i ς ∂ x 3 v \+ Q 2 ( u , v ) \= 0 , u ( x , 0 ) \= u 0 ( x ) , v ( x , 0 ) \= v 0 ( x ) , u ( 0 , t ) \= 0 , u ( 1 , t ) \= 0 , u x ( 1 , t ) \= 0 , t ∈ R \+ , v ( 0 , t ) \= 0 , v ( 1 , t ) \= 0 , v x ( 1 , t ) \= 0 , t ∈ R \+ ,
(4)
and a linear system which satisfies nonhomogeneous boundary conditions
i u t \+ α 1 ∂ x 2 u \+ i ς ∂ x 3 u \= 0 , i v t \+ α 2 ∂ x 2 v \+ i ς ∂ x 3 v \= 0 , u ( x , 0 ) \= 0 , v ( x , 0 ) \= 0 , u ( 0 , t ) \= ϑ 1 ( t ) , u ( 1 , t ) \= ϑ 2 ( t ) , u x ( 1 , t ) \= ϑ 3 ( t ) , v ( 0 , t ) \= ϱ 1 ( t ) , v ( 1 , t ) \= ϱ 2 ( t ) , v x ( 1 , t ) \= ϱ 3 ( t ) .
(5)
Guided by the solution construction strategy from \[16\], we thus express the solution as
( Ψ 1 ( u , v ) , Ψ 2 ( u , v ) ) \= ( u ( x , t ) , v ( x , t ) ) \= ( ϕ 1 , ϕ 2 ) \+ ( ψ 1 , ψ 2 ) ,
(6)
where Ψ 1 ( u , v ) and Ψ 2 ( u , v ) are the integral operators, the initial solution operators ϕ 1 , ϕ 2 are the solutions of ([4](https://www.mdpi.com/2075-1680/15/3/230#FD4-axioms-15-00230)) with initial value, and the boundary solution operators ψ 1 , ψ 2 are the solutions of the linear problem ([5](https://www.mdpi.com/2075-1680/15/3/230#FD5-axioms-15-00230)) with nonhomogeneous boundary conditions. In fact, to derive these two boundary operators for ([5](https://www.mdpi.com/2075-1680/15/3/230#FD5-axioms-15-00230)), it suffices to consider only one equation in the system, since the dispersion effects are identical for both equations and, consequently, the boundary operators take the same form. Therefore, we focus on the following equation:
i Υ t \+ α Υ x x \+ i ς Υ x x x \= 0 , Υ ( x , 0 ) \= 0 , Υ ( 0 , t ) \= h 1 ( t ) , Υ ( 1 , t ) \= h 2 ( t ) , Υ x ( 1 , t ) \= h 3 ( t ) ,
(7)
where α \= α 1 or α 2 , Υ \= u or v , h j ( t ) \= ϑ j ( t ) or ϱ j ( t ) for j \= 1 , 2 , 3 . Equation ([7](https://www.mdpi.com/2075-1680/15/3/230#FD7-axioms-15-00230)) can be reformulated as a family of BVP by Laplace transform in t:
i δ Υ ˜ ( x , δ ) \+ α Υ ˜ x x \+ i ς Υ ˜ x x x \= 0 Υ ˜ ( 0 , δ ) \= h ˜ 1 ( δ ) , Υ ˜ ( 1 , δ ) \= h ˜ 2 ( δ ) , Υ ˜ ( 1 , δ ) \= h ˜ 3 ( δ ) ,
(8)
where Υ ˜ ( x , δ ) \= ∫ 0 \+ ∞ e − δ t Υ ( x , t ) d t. The characteristic equation of (8) is
δ − i α φ 2 \+ ς φ 3 \= 0 .
It is clear that x 3 \= i has three roots:
cos 1 6 π \+ i sin 1 6 π , cos 5 6 π \+ i sin 5 6 π , cos 9 6 π \+ i sin 9 6 π ,
Letting δ \= ( i ξ ) 3 \= − i ξ 3 ( ξ ≥ 0 ), by means of the perturbation analysis, the roots of the characteristic equation for ξ → \+ ∞ can be derived as
φ 1 ( ξ ) \= − i ξ \+ ∘ ( 1 ξ ) , φ 2 ( ξ ) \= ξ ( cos 1 6 π \+ i sin 1 6 π ) \+ ∘ ( 1 ξ ) , φ 3 ( ξ ) \= ξ ( cos 5 6 π \+ i sin 5 6 π ) \+ ∘ ( 1 ξ ) .
The solution Υ ˜ ( x , δ ) of (8) takes the form
u ˜ ( x , δ ) \= ∑ j \= 1 3 c j ( δ ) e φ j ( δ ) x ,
where c j satisfies the following linear equations:
c 1 \+ c 2 \+ c 3 \= h ˜ 1 ( δ ) , c 1 e h 1 ( δ ) \+ c 2 e φ 2 ( δ ) \+ c 3 e φ 3 ( δ ) \= h ˜ 2 ( δ ) , c 1 φ 1 ( δ ) e φ 1 ( δ ) \+ c 2 φ 2 ( δ ) e φ 2 ( δ ) \+ c 3 φ 3 ( δ ) e φ 3 ( δ ) \= h ˜ 3 ( δ ) .
(9)
Applying Cramer’s rule, we obtain
c j \= D j ( δ ) D ( δ ) ,
where D ( s ) represents the determinant of the coefficient matrix, with D j ( s ) given by substituting the j\-th column with ( h ˜ 1 ( s ) , h ˜ 2 ( s ) , h ˜ 3 ( s ) ). Υ admits the following representation after taking the Mellin transform with a given χ \> 0:
Υ ( x , t ) \= ∑ j \= 1 3 1 2 π i ∫ χ − i ∞ χ \+ i ∞ e δ t Υ ˜ ( x , δ ) d δ \= ∑ j \= 1 3 1 2 π i ∫ χ − i ∞ χ \+ i ∞ e δ t D j ( δ ) D ( δ ) e φ j x d δ .
(10)
The solution Υ of (7) can be expressed a sum:
Υ ( x , t ) \= S 0 , h → ( 0 ) \= Υ 1 ( x , t ) \+ Υ 2 ( x , t ) \+ Υ 3 ( x , t ) ,
where Υ m is a solution to (7) with φ j \= 0 for j ≠ m. Invoking (10), we have
Υ m ( x , t ) \= ∑ j \= 1 3 1 2 π i ∫ χ − i ∞ χ \+ i ∞ e δ t D j m ( δ ) D ( δ ) e φ j ( δ ) x h ˜ m ( δ ) d δ : \= S m ( t ) φ m ,
where D j m ( s ) is derived from D j ( s ) by setting h j ˜ ( δ ) \= 1 and h k ˜ ( δ ) \= 0 for k ≠ m ( k , m \= 1 , 2 , 3 ). We fix χ \= 0, in which case the expression of Υ m can be denoted as
Υ m ( x , t ) \= ∑ j \= 1 3 1 2 π i ∫ 0 \+ i ∞ e δ t D j m \+ ( δ ) D \+ ( δ ) e φ j \+ ( δ ) x h ˜ m \+ ( δ ) d δ \+ ∑ j \= 1 3 1 2 π i ∫ − i ∞ 0 e − δ t D j m − ( δ ) D − ( δ ) e φ j − ( δ ) x h ˜ m − ( δ ) d δ : \= I m \+ I I m .
(11)
Therefore, I m and I I m can be written as
I m \= ∑ j \= 1 3 1 2 π ∫ 0 \+ ∞ e i ξ 3 t D j m \+ ( ξ ) D \+ ( ξ ) e φ j \+ ( ξ ) x h ˜ m \+ ( ξ ) ( 3 ξ 2 ) d ξ , I I m \= ∑ j \= 1 3 1 2 π ∫ 0 ∞ e − i ξ 3 t D j m − ( ξ ) D − ( ξ ) e φ j − ( ξ ) x h ˜ m − ( ξ ) ( 3 ξ 2 ) d ξ ,
where h ˜ ± ( ξ ) \= g ˜ ( ± i ξ 3 ). The quantities carrying the superscript \+, such as h ˜ m \+ ( ξ ), D j m \+ ( ξ ), D \+ ( ξ ), and φ j \+ ( ξ ), are complex conjugates of their counterparts with the superscript −. For notational simplicity, we henceforth suppress the superscript + and write h ˜ m \+ ( ξ ), D \+ ( ξ ), D j m \+ ( ξ ), and φ j \+ ( ξ ) in place of h ˜ m ( ξ ), D ( ξ ), D j m ( ξ ), and φ j ( ξ ), respectively.
**Remark** **1\.**
Indeed, since the nonlinear terms involve both u and v, the solution operators corresponding to u and v should depend on both initial data. That said, when introducing the notation, u is associated only with u 0 and v only with v 0 . This is because the final nonlinear estimates are incorporated as a whole within the linear estimates, and the detailed calculation process can be found in Lemma 4.
The solution of ([1](https://www.mdpi.com/2075-1680/15/3/230#FD1-axioms-15-00230)) is established in the space X s , T × X s , T by using the classical fixed-point principle and a priori estimates. Furthermore, this smooth solution is shown to satisfy Equation ([1](https://www.mdpi.com/2075-1680/15/3/230#FD1-axioms-15-00230)) in the classical sense. To establish the contraction mapping property, we decompose the boundary operator S 0 , h → ( 0 ) into three distinct components and derive corresponding estimates for each component. We now apply the preceding lemmas to furnish the boundary operator Υ 1. The estimates of Υ 2 and Υ 3 are demonstrated in a manner analogous to that of Υ 1.
**Lemma** **1**
(\[32\])**.** We assume that φ is a bounded open set in R n with a n − 1 dsmooth boundary. Let θ ∈ 0 , 1 , s 1 , s 2 ≥ 0 and s 1 , s 2 \= i n t e g e r \+ 1 2 .
1\.
If s 1 ≥ s 2 ≥ 0 and
( 1 − θ ) s 1 \+ θ s 2 ≠ i n t e g e r \+ 1 2 ,
\[ H 0 s 1 ( φ ) , H 0 s 2 ( φ ) \] θ \= H 0 ( 1 − θ ) s 1 \+ θ s 2 ( φ )
(12)
2\.
If s 2 ≥ s 1 ≥ 0 and
( 1 − θ ) s 1 \+ θ s 2 ≠ i n t e g e r \+ 1 2 ,
we have
\[ H 0 − s 1 ( φ ) , H 0 − s 2 ( φ ) \] θ \= H 0 − ( 1 − θ ) s 1 − θ s 2 ( φ )
(13)
where H − s ( φ ) \= ( H s ( φ ) ) ′ .
3\.
If
( 1 − θ ) s 1 \+ θ s 2 ≠ N \+ 1 2 , i n t e g e r N ≥ 0
we have
\[ H 0 s 1 ( φ ) , H 0 − s 2 ( φ ) \] θ \= H 0 ( 1 − θ ) s 1 − θ s 2 ( φ ) if ( 1 − θ ) s 1 − θ s 2 ≥ 0 H ( 1 − θ ) s 1 − θ s 2 ( φ ) if ( 1 − θ ) s 1 − θ s 2 ≤ 0
(14)
**Lemma** **2**
(\[33\], Theorem 2)**.** We assume that V 0 ⊂ V 1 and W 0 ⊂ W 1 are Banach spaces. The mapping L : V i → W i ( i \= 0 , 1 ) satisfies
∥ L u − L v ∥ W 1 ≤ m ∥ u ∥ V 1 , ∥ v ∥ V 1 ∥ u − v ∥ V 1 c , ∀ u , v ∈ V 1 , ∥ T u ∥ W 0 ≤ n ∥ u ∥ V 1 ∥ u ∥ V 0 d , ∀ u ∈ V 0 .
For any 0 \< θ \< 1 and 1 ≤ p ≤ \+ ∞ , we obtain
L : V 0 , V 1 θ , p → W 0 , W 1 κ , q .
Moreover,
∥ L u ∥ W 0 , W 1 κ , q ≤ C f ∥ u ∥ V 1 ∥ u ∥ V 0 , V 1 θ , p ( 1 − κ ) d \+ κ c ,
where
f ( t ) \= n ( 2 t ) 1 − κ m ( t , 2 t ) κ , 1 − κ κ \= 1 − θ θ · c d , q \= max 1 , p ( 1 − κ ) d \+ κ c \= max 1 , 1 − θ d \+ θ c p ,
m ( · ) and n ( · ) are continuous functions.
Throughout this paper, in Lemma 2, the parameters are specified as c \= d \= 1 , θ \= κ , and p \= q \= 2. In the process of estimating the boundary operator, we still need to introduce a technical lemma from \[16\].
**Lemma** **3**
(\[16\])**.** Let g ∈ L 2 ( 0 , ∞ ) , and define the operator L f by
L g ( x ) \= ∫ 0 ∞ e χ ( μ ) x g ( μ ) d μ ,
where χ ( μ ) : ( 0 , ∞ ) → C is a continuous function satisfying the following assumptions:
1\.
For some constant σ \> 0 and sufficient small ε \> 0 , it holds that
sup 0 \< μ \< σ \| R e χ ( μ ) \| μ ≥ ε .
2\.
For a complex number α \+ i ς , it holds that
lim μ → ∞ χ ( μ ) μ \= α \+ i ς .
Under these conditions, for all g ∈ L 2 ( 0 , \+ ∞ ) , we have
∥ L g ∥ L 2 ( 0 , 1 ) ≤ C e R e χ ( · ) g ( · ) L 2 R \+ \+ ∥ g ( · ) ∥ L 2 R \+ .
## 3\. A Prior Estimates
In this section, the prior bounds for the solution operator will be derived. The solution operator is constructed from the solution components S u 0 , 0 → ( Q 1 ), S v 0 , 0 → ( Q 2 ), and S 0 , h → ( 0 ), given in [Section 2](https://www.mdpi.com/2075-1680/15/3/230#sec2-axioms-15-00230). To initiate our analysis, we utilize energy techniques to obtain bounds for the solution operator related to the initial data, a step that in turn enables the derivation of the required nonlinear bounds.
### 3\.1. Linear Estimates
We begin by establishing the linear estimates associated with the nonlinear problem, subject to nonhomogeneous initial and homogeneous boundary conditions. Since proofs of many analogous properties have been presented in other literature (see \[31\]), we provide a concise proof of the relevant lemma here to enrich our demonstration.
**Lemma** **4\.**
For any s ≥ 0 , u 0 , v 0 ∈ H s ( 0 , 1 ) . One can find a positive constant C ( T ) \> 0 depending solely on T, such that
ϕ 1 X s , T ≤ C ( T ) u 0 H s \+ ∥ Q 1 ∥ L t 2 H s − 2 , if 0 ≤ s ≤ 2 , C ( T ) u 0 H s \+ ∥ Q 1 ∥ L t 2 H 0 s − 2 , if s \> 2 , s ≠ integer \+ 5 2 ,
(15)
and
ϕ 2 X s , T ≤ C ( T ) v 0 H s \+ ∥ Q 2 ∥ L t 2 H s − 2 , if 0 ≤ s ≤ 2 , C ( T ) v 0 H s \+ ∥ Q 2 ∥ L t 2 H 0 s − 2 , if s \> 2 , s ≠ integer \+ 5 2 .
(16)
**Proof.**
For the case s \= 0, since the dispersive effects of the two equations in the system differ only by a constant, their estimation procedures are analogous. Therefore, it suffices to consider a single equation for our analysis. Suppose that Υ ( x , t ) solves the homogeneous boundary value problem
i ∂ t Υ \+ α ∂ x x Υ \+ i ς ∂ x x x Υ \= Q ( x , t ) Υ ( x , 0 ) \= Υ 0 ( x ) Υ ( 0 , t ) \= 0 , Υ ( 1 , t ) \= 0 , Υ x ( 1 , t ) \= 0 .
(17)
where Q \= Q 1 or Q 2. We multiply (17) by ( 1 \+ x ) Υ ¯, and integrate over the space–time domain \[ 0 , 1 \] × ( 0 , t ) to arrive at
1 2 ∫ 0 1 ( 1 \+ x ) Υ 2 d x \+ ς 2 ∫ 0 t Υ x 2 \| x \= 0 d s \+ 3 ς 2 ∫ 0 t ∫ 0 1 Υ x 2 d x d s \= 1 2 ∫ 0 1 ( 1 \+ x ) Υ 0 2 d x \+ ℑ ∫ 0 t ∫ 0 1 Q ( Υ ) ( 1 \+ x ) Υ ¯ d x d s ≤ Υ 0 L 2 2 \+ ℑ ∫ 0 t ∫ 0 1 Q ( 1 \+ x ) Υ ¯ d x d s \+ ℑ α ∫ 0 t ∫ 0 1 Υ x Υ ¯ d x d s ,
(18)
where the last term of ([18](https://www.mdpi.com/2075-1680/15/3/230#FD18-axioms-15-00230)) can be controlled by integration by parts together with the Cauchy–Schwarz inequality. Meanwhile, the second term on the last line of ([18](https://www.mdpi.com/2075-1680/15/3/230#FD18-axioms-15-00230)) provides the essential bridge between the linear and nonlinear estimates. Its bound is based on the observation that ∂ x − 1 u ( x ) \= ∫ 0 x u ( y ) d y. We therefore get
∫ 0 t ∫ 0 1 Q 1 ( Υ ) ( 1 \+ x ) Υ ¯ d x d s \= ∫ 0 t ∫ 0 1 ∂ x − 2 ∂ x 2 Q 1 ( Υ ) ( 1 \+ x ) Υ ¯ d x d s , ≤ ϵ ∫ 0 t ∫ 0 1 ( ∂ x Υ ) 2 d x d s \+ 1 ϵ Q 1 L 2 ( 0 , T ; H − 2 ( 0 , 1 ) ) 2 ,
where we clearly point out that the H − 2 space is specifically chosen in this work to balance the spatial derivative loss induced by the nonlinear terms in the coupled Hirota equation and the dispersive smoothing effect of the third-order spatial derivative operator. By means of the aforementioned calculations, we have
∥ Υ ∥ X 0 , T ≲ ∥ Υ 0 ∥ L 2 ( 0 , 1 ) ) \+ ∥ Q 1 ∥ L 2 ( 0 , T ; H − 2 ( 0 , 1 ) ) ,
(19)
where X 0 , T \= C ( 0 , T ; L 2 ( 0 , 1 ) ) ∩ L 2 ( 0 , T ; H 1 ( 0 , 1 ) ). In the following, we allow s ∈ ( 0 , \+ ∞ ). For s \= 3, we derive from (17) that
∂ x 3 Υ ( x , t ) ∣ x \= 0 , 1 \= ∂ x 4 Υ ( x , t ) ∣ x \= 1 \= 0 .
By applying ∂ x 3 to both sides of (17), we obtain that
Υ X 3 , T ≲ C ( T ) Υ 0 H x 3 \+ Q 1 L 2 ( 0 , T ; H 0 1 ) .
(20)
By Lemma 1, we get that
∥ Υ ∥ X 3 ( 1 − θ ) , T ≲ Υ 0 H 3 ( 1 − θ ) \+ ∥ Q 1 ∥ L 2 0 , T ; H 0 1 − 3 θ
where θ ∈ 0 , 1 3 , θ ≠ 1 6, and
∥ Υ ∥ X 3 ( 1 − θ ) , T ≲ Υ 0 H 3 ( 1 − θ ) \+ ∥ Q 1 ∥ L 2 0 , T ; H 1 − 3 θ
for θ ∈ 1 3 , 1. It follows that
S Υ 0 , 0 → ( Q 1 ) X s , T ≲ Υ 0 H s \+ ∥ Q 1 ∥ L t 2 H s − 2 , if 0 ≤ s ≤ 2 , Υ 0 H s \+ ∥ Q 1 ∥ L t 2 H 0 s − 2 , if 2 \< s ≤ 3 , s ≠ 5 2 .
(21)
We next consider the case s ≥ 3 and k ∈ Z \+, from which we obtain
Υ \[ X 3 ( k \+ 1 ) , T , X 3 k , T \] θ ≲ Υ 0 \[ L x 2 H x 3 ( k \+ 1 ) , L x 2 H x 3 k \] θ \+ Q 1 \[ L x 2 H 0 3 ( k \+ 1 ) − 2 , L x 2 H 0 3 k − 2 \] θ .
By using Lemma 2, for θ ∈ ( 0 , 1 ), we obtain
Υ X 3 ( k \+ 1 ) − 3 θ , T ≲ Υ 0 H x 3 ( k \+ 1 ) − 3 θ \+ Q 1 L 2 ( 0 , T ; H 0 3 ( k \+ 1 ) − 3 θ − 2 ) ,
where θ ∈ ( 0 , 1 ), and θ does not belong to the set { k \+ 1 6 − integer 3 } , for some k ∈ Z \+ . We deduce from setting s \= 3 ( k \+ 1 ) − 3 θ that
Υ X s , T ≲ Υ 0 H x s \+ Q 1 L 2 ( 0 , T ; H 0 s − 2 ) ,
(22)
where s ≠ integer \+ 5 2. Consequently, by combining relations (21) and (22), we obtain ([15](https://www.mdpi.com/2075-1680/15/3/230#FD15-axioms-15-00230)). Following a similar line of reasoning, we may also derive ([16](https://www.mdpi.com/2075-1680/15/3/230#FD16-axioms-15-00230)). □
The estimates for the boundary operators below are identical to those established in \[31\]; therefore, their proofs are omitted here for brevity. The boundary operator estimates for the coupled Hirota system (cH system) depend entirely on the linear terms of the equations (i.e., i u t \+ α 1 ∂ x 2 u \+ i ς ∂ x 3 u \= 0 and i v t \+ α 2 ∂ x 2 v \+ i ς ∂ x 3 u \= 0) rather than the nonlinear coupling terms. For the two components u and v, their linear parts share the fundamental third-order dispersion term i ς ∂ x 3, differing only in the second-order dispersion coefficients α 1 and α 2, which do not affect the boundary operator. We outline the key steps: (1) decompose the boundary operator ψ 1 , ψ 2 into three components by analogy with the scalar case; (2) derive the integral representations for each component using Laplace transforms and inverse Laplace transforms; (3) prove the X s , T norm estimates for each component using the Lemma 3; (4) derive a unified estimate for the coupled system by summing the bounds of the three components, where the constant coefficients α 1 , α 2 introduce only finite constant bounds in the estimate.
**Lemma** **5\.**
For any s ≥ 0 . Then, for any ϑ → \= ( ϑ 1 , ϑ 2 , ϑ 3 ) , ϱ → \= ( ϱ 1 , ϱ 2 , ϱ 3 ) satisfy ϑ → , ϱ → ∈ Z s , we have
ψ 1 X s , T ≤ C ( T ) ϑ → Z s and ψ 2 X s , T ≤ C ( T ) ϱ → Z s .
(23)
### 3\.2. Nonlinear Estimates
We now turn to the nonlinear estimates for ([1](https://www.mdpi.com/2075-1680/15/3/230#FD1-axioms-15-00230)) involving the nonlinear term Q 1 ( u ) \= i χ 2 \| u \| 2 \+ \| v \| 2 u x \+ u v ¯ v x \+ δ \| u \| 2 \+ \| v \| 2 u, Q 2 ( u , v ) \= i χ ( \| u \| 2 \+ 2 \| v \| 2 ) v x \+ u ¯ v u x \+ δ \| u \| 2 \+ \| v \| 2 v and the parameters χ , δ. Note that L t 2 H 0 s − 2 \= L t 2 H s − 2 for \[ 2 , 5 2 ). It suffices to estimate the nonlinear terms in L t 2 H x s − 2 for \[ 2 , 5 2 ), and this also remains the case for the terms established in this section.
**Lemma** **6\.**
For arbitrary u , v ∈ X s , T for s ∈ \[ 2 , 5 2 ) and T ∈ ( 0 , \+ ∞ ) , we obtain
Q 1 ( u , v ) L t 2 H s − 2 ≲ C ( T ) u X s , T 3 \+ u X s , T v X s , T 2 ,
(24)
and
Q 2 ( u , v ) L t 2 H s − 2 ≲ C ( T ) v X s , T 3 \+ u X s , T 2 v X s , T ,
(25)
where C ( T ) → 0 as T → 0 .
**Proof.**
We first focus on the case s \= 2. The Sobolev embedding theorem H k \+ 3 2 \+ ϵ ( 0 , 1 ) ↪ C k ( 0 , k ) , k ∈ Z \+ , ϵ \> 0 is crucial for our analysis; it transforms the H s Sobolev norm of solutions into a unified L ∞ norm. This move aims to constrain the nonlinear product terms arising from the dot product between the solution and its derivatives, such as \| u \| 2 u x and \| v \| 2 u x. By the definition of X 2 , T \= C 0 , T ; H 2 ( 0 , 1 ) ∩ L 2 0 , T ; H 3 ( 0 , 1 ), all dot-product nonlinear terms are thereby explicitly defined and possess boundedness. Using the Sobolev embedding theorem and the definition of X 2 , T, we have
Q 1 ( u , v ) L t 2 L x 2 2 \= u 2 u x L t 2 L x 2 2 \+ u x v 2 L t 2 L x 2 2 \+ u v ¯ v x L t 2 L x 2 2 \+ u 2 u L t 2 L x 2 2 \+ v 2 u L t 2 L x 2 2 \= ∫ 0 T u 2 u x L 2 2 d t \+ ∫ 0 T u x v 2 L 2 2 d t \+ ∫ 0 T u v ¯ v x L 2 2 d t \+ ∫ 0 T u 2 u L 2 2 d t \+ ∫ 0 T v 2 u L 2 2 d t ≲ ∫ 0 T sup x ∈ \[ 0 , 1 \] u 4 u x L 2 2 d t \+ ∫ 0 T sup x ∈ \[ 0 , 1 \] v 4 u x L 2 2 d t \+ ∫ 0 T sup x ∈ \[ 0 , 1 \] u 2 v ¯ v x L 2 2 d t \+ ∫ 0 T sup x ∈ \[ 0 , 1 \] u 4 u L 2 2 d t \+ ∫ 0 T sup x ∈ \[ 0 , 1 \] v 4 u L 2 2 d t ≲ ∫ 0 T sup x ∈ \[ 0 , 1 \] u 4 ( u x L 2 2 \+ u L 2 2 ) \+ sup x ∈ \[ 0 , 1 \] v 4 ( u x L 2 2 \+ u L 2 2 ) d t \+ ∫ 0 T sup x ∈ \[ 0 , 1 \] u 2 ( v L 2 4 \+ v x L 2 4 ) d t ≲ u X 0 , T 6 \+ u X 0 , T 2 v X 0 , T 4 .
(26)
In the case s \= 3, one can find
u 2 u x L t 2 H x 1 \+ u x v 2 L t 2 H x 1 \+ u v ¯ v x L t 2 H x 1 \+ u 2 u L t 2 H x 1 \+ v 2 u L t 2 H x 1 \= ∂ x ( u 2 ) u x L t 2 L x 2 2 \+ u 2 u x x L t 2 L x 2 2 \+ u x x v 2 L t 2 L x 2 2 \+ u x ∂ x ( v 2 ) L t 2 L x 2 2 \+ u x v ¯ v x \+ u v ¯ x v x \+ u v ¯ v x x L t 2 L x 2 2 \+ ∂ x ( u 2 ) u L t 2 L x 2 2 \+ u 2 u x L t 2 L x 2 2 \+ v 2 u x L t 2 L x 2 2 \+ ∂ x ( v 2 ) u L t 2 L x 2 2 ≲ u x 2 u x L t 2 L x 2 2 \+ u x 2 v ¯ L t 2 L x 2 2 \+ u 2 u x x L t 2 L x 2 2 \+ u x x v 2 L t 2 L x 2 2 \+ u x v ¯ v x L t 2 L x 2 2 \+ u x v ¯ x v L t 2 L x 2 2 \+ u v x 2 L t 2 L x 2 2 \+ u v ¯ v x x L t 2 L x 2 2 \+ u 2 u x L t 2 L x 2 2 \+ u 2 u ¯ x L t 2 L x 2 2 \+ u x v 2 L t 2 L x 2 2 \+ u v x v ¯ L t 2 L x 2 2 \+ u v v ¯ x L t 2 L x 2 2 ≲ ∫ 0 T u H 1 6 d t \+ ∫ 0 T u H 2 2 u H 1 4 d t \+ ∫ 0 T u H 2 2 v H 1 4 d t \+ ∫ 0 T u H 1 2 v H 1 4 d t \+ ∫ 0 T u H 1 2 v H 2 4 d t ≲ ∫ 0 T u H 1 6 d t \+ u X 2 , T 2 u X 1 , T 4 \+ u X 1 , T 2 v X 1 , T 4 \+ u X 1 , T 2 v X 2 , T 4 d t .
(27)
Employing the Gagliardo–Nirenberg inequality, we can deduce that
u H 1 ≤ u L 2 \+ u L 2 1 2 u x x L 2 1 2 ,
(28)
This inequality interpolates between the low-order L 2 norm and the high-order H 2 norm, serving as a crucial step in defining the nonlinear term within the L t 2 H x s − 2 space. It enables the H 1 norm to be controlled by the solution space X s , T. Equation ([28](https://www.mdpi.com/2075-1680/15/3/230#FD28-axioms-15-00230)) gives that
∫ 0 T u H 1 6 d t ≲ ∫ 0 T ( u L 2 \+ u L 2 1 2 u x x L 2 1 2 ) 6 d t ≲ ∫ 0 T u L 2 6 d t \+ ∫ 0 T u L 2 3 u x x L 2 3 d t ≲ C ( T ) ( u X 0 , T 6 \+ u X 0 , T 3 u X 1 , T 3 ) .
(29)
We deduce from (26) and (29) that
u 2 u x L t 2 H x 1 \+ u x v 2 L t 2 H x 1 \+ u v ¯ v x L t 2 H x 1 \+ u 2 u L t 2 H x 1 \+ v 2 u L t 2 H x 1 ≲ ( u X 0 , T 6 \+ u X 0 , T 3 u X 1 , T 3 ) \+ u X 2 , T 2 u X 1 , T 2 \+ u X 2 , T 2 v X 1 , T 4 \+ u X 1 , T 2 v X 1 , T 4 \+ u X 1 , T 2 v X 2 , T 4 ≲ ( u X 3 , T 6 \+ u X 3 , T 2 u X 3 , T 4 ) .
(30)
where C ( T ) → 0 as T → 0 . This shows that, in the case s \= 3, relation ([24](https://www.mdpi.com/2075-1680/15/3/230#FD24-axioms-15-00230)) holds true. By Tatar’s interpolation theorem, we obtain that ([24](https://www.mdpi.com/2075-1680/15/3/230#FD24-axioms-15-00230)) holds for s ∈ \[ 2 , 5 2 ). Indeed, the estimation procedures for Q 1 and Q 2 are similar, so the estimate for Q 2 follows naturally by the same argument. □
We now prove the nonlinear estimates result for s \< 2 .
**Lemma** **7\.**
For arbitrary u , v ∈ X s , T with 0 ≤ s \< 2 and T ∈ ( 0 , \+ ∞ ) , we obtain
S 0 , 0 → ( Q 1 ) X s , T ≲ u X s , T 3 \+ u X s , T v X s , T 2 ,
(31)
and
S 0 , 0 → ( Q 2 ) X s , T ≤ v X s , T 3 \+ u X s , T 2 v X s , T .
(32)
**Proof.**
We begin with the case s \= 0 . Let Λ \= S 0 , 0 → ( Q 1 ). Then, Λ satisfies
i Λ t \+ α 1 ∂ x 2 Λ \+ i ς ∂ x 3 Λ \= Q 1 ( u , v ) , Λ ( x , 0 ) \= 0 , Λ ( 0 , t ) \= 0 , Λ ( L , t ) \= 0 , Λ x ( L , t ) \= 0 , t ∈ R \+ .
(33)
An application of integration by parts, after multiplying ([33](https://www.mdpi.com/2075-1680/15/3/230#FD33-axioms-15-00230)) by ( 1 \+ x ) Λ ¯, and taking the imaginary part, leads to
1 2 d d t ∫ 0 1 Λ 2 ( 1 \+ x ) d x \+ ς 2 Λ x 2 \| x \= 0 \+ 3 2 ς ∫ 0 1 Λ x 2 d x \= ℑ ∫ 0 1 ( 1 \+ x ) Λ ¯ Q 1 d x \+ ℑ α ∫ 0 1 Λ Λ ¯ d x
where
∫ 0 1 ( 1 \+ x ) Λ ¯ Q 1 d x ≲ Λ L ∞ u L 2 3 \+ Λ L 2 u L ∞ 2 u H 1 \+ Λ L 2 v L ∞ 2 u H 1 \+ v L ∞ ∫ 0 1 u Λ v x d x \+ Λ L 2 v L ∞ 2 u L 2 .
An argument analogous to that in Lemma 4, combined with integration in t, yields
Λ L 2 2 \+ Λ x L t 2 L x 2 2 ≲ ∫ 0 T Λ L ∞ u L 2 3 \+ Λ L 2 u L ∞ 2 u H 1 \+ Λ L 2 v L ∞ u H 1 \+ v L ∞ u L ∞ Λ L 2 v x L 2 \+ v L ∞ 2 u L 2 Λ L 2 d t : \= Q 1 \+ Q 2 \+ Q 3 \+ Q 4 \+ Q 5 .
(34)
We deduce from Lemma 2 and the Young’s inequality that
Q 1 ≤ ∫ 0 1 Λ L ∞ 3 d t 1 3 ∫ 0 1 u L 2 9 2 d t 2 3 ≤ T 2 3 Λ L t 2 L x ∞ u L t ∞ L x 2 3 ≤ T 2 3 Λ X 0 , T u X 0 , T 3 ,
(35)
and
Q 2 ≤ ∫ 0 T Λ L 2 6 d t 1 6 ∫ 0 T u L ∞ 6 d t 1 3 ∫ 0 T u H 1 2 d t 1 2 ≤ T 1 6 Λ X 0 , T u X 0 , T 3 .
(36)
Sobolev embedding theorem and Young’s inequality are employed here to separate the norms of Λ and u, which is crucial for the fixed-point argument. We need to constrain the norm of the nonlinear operator using the norm of the solution itself. The estimates for the remaining three terms follow arguments similar to those used for Q 1 and Q 2, from which we directly obtain
Q 3 ≤ T 1 6 Λ X 0 , T u X 0 , T v X 0 , T 2 ,
(37)
Q 4 ≤ T 1 2 Λ X 0 , T v X 0 , T 2 u X 0 , T ,
(38)
Q 5 ≤ T 1 6 v X 0 , T 2 Λ X 0 , T u X 0 , T .
(39)
It follows from ([34](https://www.mdpi.com/2075-1680/15/3/230#FD34-axioms-15-00230))–(39) that
Λ L 2 2 \+ Λ x L t 2 L x 2 2 ≲ Λ X 0 , T ( u X 0 , T 3 \+ u X 0 , T v X 0 , T 2 ) ,
where 0 ≤ t ≤ T. We deduce that
Λ X 0 , T 2 ≲ Λ X 0 , T ( u X 0 , T 3 \+ u X 0 , T v X 0 , T 2 ) ,
which leads to
Λ X 0 , T ≲ u X 0 , T 3 \+ u X 0 , T v X 0 , T 2 .
Furthermore, in analogy with Lemma 6, for s \= 2, we obtain
Λ X 2 , T ≲ u X 2 , T 3 \+ u X 2 , T v X 2 , T 2 .
We deduce from Lemmas 2 and 3 that
Λ X 2 θ , T ≲ u X 2 θ , T 3 \+ u X 2 θ , T v X 2 θ , T 2 , θ ∈ ( 0 , 1 ) ,
which completes the proof of ([31](https://www.mdpi.com/2075-1680/15/3/230#FD31-axioms-15-00230)).
Indeed, the estimation procedures for S 0 , 0 → ( Q 1 ) and S 0 , 0 → ( Q 2 ) are similar, both relying on the same energy estimates. Thus, the estimate for S 0 , 0 → ( Q 2 ) follows naturally by the same argument, and we therefore omit its proof here. □
## 4\. Proof of Main Results
### 4\.1. Proof of Theorem 1
In the current section, we carry out the proof of local well-posedness associated with system ([6](https://www.mdpi.com/2075-1680/15/3/230#FD6-axioms-15-00230)), with the aid of the estimates from [Section 3](https://www.mdpi.com/2075-1680/15/3/230#sec3-axioms-15-00230). The preceding estimates reveal a fundamental link between the regularity of the solution operator and that of the initial and boundary conditions. The solution operator for the IBVP ([1](https://www.mdpi.com/2075-1680/15/3/230#FD1-axioms-15-00230)) admits the representation
( u ( x , t ) , v ( x , t ) ) \= ( Ψ 1 ( u , v ) , Ψ 2 ( u , v ) ) \= ( ϕ 1 , ϕ 2 ) \+ ( ψ 1 , ψ 2 ) ,
where Q 1 ( u ) = i χ { 2 \| u \| 2 u x \+ \| v \| 2 u x \+ u v ¯ v x } \+ δ \| u \| 2 u \+ \| v \| 2 u and Q 2 ( u , v ) = i χ { ( \| u \| 2 v x \+ 2 \| v \| 2 v x ) \+ u ¯ v u x } \+ δ \| u \| 2 v \+ \| v \| 2 v. We define the Banach space B s , M as the closed ball
B s , R \= { ( u , v ) ∈ X s , T ∗ × X s , T ∗ : u X s , T ∗ ≤ R , v X s , T ∗ ≤ R }
such that
Ψ 1 : B s , R → B s , R and Ψ 2 : B s , R → B s , R ,
where R is a fixed constant, T ∗ ∈ ( 0 , T ) and X s , T ∗ \= C 0 , T ∗ ; H s ( 0 , 1 ) ∩ L 2 0 , T ∗ ; H s \+ 1 ( 0 , 1 ) . We deduce from combining Lemmas 4–7 that
Ψ 1 X s , T ∗ ≤ C ( T ∗ ) u 0 H s ( 0 , 1 ) \+ ϑ → Z s \+ C ( T ∗ ) u X s , T ∗ 3 \+ C ( T ∗ ) u X s , T ∗ v X s , T ∗ 2 ≤ C ( T ∗ ) u 0 H s ( 0 , 1 ) \+ ϑ → Z s \+ C ( T ∗ ) R 3 \+ C ( T ∗ ) R 3 ≤ R 2 \+ R 8 ≤ R ,
and
Ψ 2 X s , T ∗ ≤ C ( T ∗ ) v 0 H s ( 0 , 1 ) \+ ϱ → Z s \+ C ( T ∗ ) v X s , T ∗ 3 \+ C ( T ∗ ) u X s , T ∗ 2 v X s , T ∗ ≤ C ( T ∗ ) v 0 H s ( 0 , 1 ) \+ ϱ → Z s \+ C ( T ∗ ) R 3 \+ C ( T ∗ ) R 3 ≤ R 2 \+ R 8 ≤ R ,
where T ∗ ∈ ( 0 , T ) is chosen small enough to ensure that C ( T ∗ ) ( u 0 H s ( 0 , 1 ) \+ v 0 H s ( 0 , 1 ) \+ ϑ → Z s \+ ϱ → Z s ) ≤ R 2 and C ( T ∗ ) R 2 ≤ 1 16. For any u 1 , u 2 , v 1 , v 2 ∈ B s , R, by using the definitions of Ψ 1 , Ψ 2, it follows that
Ψ 1 − Ψ 2 \= S u 0 , ϑ → ( Q 1 ( u 1 ) ) − S u 0 , ϑ → ( Q 1 ( u 2 ) ) \= S 0 , 0 → ( ( u 1 − u 2 ) , u 1 , ∂ x u 1 ) \+ S 0 , 0 → ( u 2 , ( u 1 − u 2 ) , ∂ x u 1 ) \+ S 0 , 0 → ( u 2 , ( u 1 − u 2 ) x , u 2 ) \+ S 0 , 0 → ( ( v 1 − v 2 ) , v 1 , ∂ x u 1 ) \+ S 0 , 0 → ( v 2 , ( v 1 − v 2 ) , ∂ x u 1 ) \+ S 0 , 0 → ( v 2 , v 2 , ( u 1 − u 2 ) x ) \+ S 0 , 0 → ( ( u 1 − u 2 ) , v 1 , ∂ x v 1 ) \+ S 0 , 0 → ( u 2 , ( v 1 − v 2 ) , ∂ x v 1 ) \+ S 0 , 0 → ( u 2 , v 2 , ( v 1 − v 2 ) x ) \+ S 0 , 0 → ( ( u 1 − u 2 ) , u 1 , u 1 ) \+ S 0 , 0 → ( u 2 , ( u 1 − u 2 ) , u 1 ) \+ S 0 , 0 → ( u 2 , u 2 , ( u 1 − u 2 ) ) \+ S 0 , 0 → ( ( v 1 − v 2 ) , v 1 , u 1 ) \+ S 0 , 0 → ( v 2 , ( v 1 − v 2 ) , u 1 ) \+ S 0 , 0 → ( v 2 , v 2 , ( u 1 − u 2 ) ) .
(40)
Then, we infer from gathering ([40](https://www.mdpi.com/2075-1680/15/3/230#FD40-axioms-15-00230)) and Lemmas 6 and 7 that
∥ Ψ 1 − Ψ 2 ∥ X s , T ∗ \= ∥ S u 0 , ϑ → ( Q 1 ( u 1 , v 1 ) ) − S u 0 , ϑ → ( Q 1 ( u 2 , v 2 ) ) ∥ X s , T ∗ \= S u 0 , ϑ → ( u 1 , u 1 , u 1 ) − S u 0 , ϑ → ( u 2 , u 2 , u 2 ) X s , T ∗ ≲ S 0 , 0 → ( ( u 1 − u 2 ) , u 1 , ∂ x u 1 ) X s , T ∗ \+ S 0 , 0 → ( u 2 , ( u 1 − u 2 ) , ∂ x u 1 ) X s , T ∗ \+ ( S 0 , 0 → ( u 2 , ( u 1 − u 2 ) x , u 2 ) X s , T ∗ \+ S 0 , 0 → ( ( v 1 − v 2 ) , v 1 , ∂ x u 1 ) X s , T ∗ \+ S 0 , 0 → ( v 2 , ( v 1 − v 2 ) , ∂ x u 1 ) X s , T ∗ \+ S 0 , 0 → ( v 2 , v 2 , ( u 1 − u 2 ) x ) X s , T ∗ \+ S 0 , 0 → ( ( u 1 − u 2 ) , v 1 , ∂ x v 1 ) X s , T ∗ \+ S 0 , 0 → ( u 2 , ( v 1 − v 2 ) , ∂ x v 1 ) X s , T ∗ \+ S 0 , 0 → ( u 2 , v 2 , ( v 1 − v 2 ) x ) X s , T ∗ \+ S 0 , 0 → ( ( u 1 − u 2 ) , u 1 , u 1 ) X s , T ∗ \+ S 0 , 0 → ( u 2 , ( u 1 − u 2 ) , u 1 ) X s , T ∗ \+ S 0 , 0 → ( u 2 , u 2 , ( u 1 − u 2 ) ) X s , T ∗ \+ S 0 , 0 → ( ( v 1 − v 2 ) , v 1 , u 1 ) X s , T ∗ \+ S 0 , 0 → ( v 2 , ( v 1 − v 2 ) , u 1 ) X s , T ∗ \+ S 0 , 0 → ( v 2 , v 2 , ( u 1 − u 2 ) ) X s , T ∗ ≤ C ( T ∗ ) ( u 1 X s , T ∗ 2 \+ u 2 X s , T ∗ 2 \+ u 1 X s , T ∗ u 2 X s , T ∗ \+ v 1 X s , T ∗ 2 \+ v 2 X s , T ∗ 2 ) ∥ u 1 − u 2 ∥ X s , T ∗ \+ ( u 1 X s , T ∗ v 1 X s , T ∗ \+ u 2 X s , T ∗ v 2 X s , T ∗ \+ u 1 X s , T ∗ v 2 X s , T ∗ \+ u 2 X s , T ∗ v 1 X s , T ∗ \+ u 1 X s , T ∗ v 2 X s , T ∗ \+ v 1 X s , T ∗ u 1 X s , T ∗ ) ∥ v 1 − v 2 ∥ X s , T ∗ ≤ 5 M 2 C ( T ∗ ) ∥ u 1 − u 2 ∥ X s , T ∗ \+ 6 M 2 C ( T ∗ ) ∥ v 1 − v 2 ∥ X s , T ∗ ≤ 5 16 ∥ u 1 − u 2 ∥ X s , T ∗ \+ 3 8 ∥ v 1 − v 2 ∥ X s , T ∗ ,
where C ( T ∗ ) M 2 ≤ 1 16. Let ( u , v ) be the unique solution to the IBVP ([1](https://www.mdpi.com/2075-1680/15/3/230#FD1-axioms-15-00230)) corresponding to the data ( u 0 , v 0 , ϑ → , ϱ → ), whose existence is guaranteed by the fixed-point principle. Consider now sequences u 0 n → u 0 , v 0 n → v 0 in H s ( 0 , 1 ) and ϑ → n → ϑ → , ϱ → n → ϱ → in Z s , and denote by the u n , v n associated solutions with data ( u 0 n , v 0 n , ϑ → n , ϱ → n ). Invoking the estimates from Lemmas 4–7, we deduce that ( u n , v n ) → ( u , v ) in X s , T ∗ × X s , T ∗ as n → ∞.
Gathering the contraction mapping framework with the a priori estimates we rigorously established earlier naturally gives rise to the continuous dependence of the solutions. More precisely, it follows from the local well-posedness theory that there exist two pairs of solutions, ( u , v ) \= ( Ψ 1 ( u 0 , ϑ → ) , Ψ 2 ( v 0 , ϱ → ) ) ∈ B s , M × B s , M and ( u ˜ , v ˜ ) \= ( Ψ 1 ( u ˜ 0 , ϑ → ˜ ) , Ψ 2 ( v ˜ 0 , ϱ → ˜ ) ) ∈ B s , M × B s , M, to the IBVP ([1](https://www.mdpi.com/2075-1680/15/3/230#FD1-axioms-15-00230)) corresponding to the data ( ( u 0 , v 0 ) , ϑ → , ϱ → ) and ( ( u ˜ 0 , v ˜ 0 ) , ϑ → ˜ , ϱ → ˜ ) in H s ( 0 , 1 ) × Z s × Z s that satisfy the s\-compatibility conditions, respectively. Furthermore, the difference between the two solution pairs satisfies the corresponding initial and boundary values. Specifically, the initial and boundary values of the difference are simply the differences in the respective data. A direct computation gives that
( Ψ 1 ( u 0 , ϑ → ) , Ψ 2 ( v 0 , ϱ → ) ) − ( Ψ 1 ( u ˜ 0 , ϑ → ˜ ) , Ψ 2 ( v ˜ 0 , ϱ → ˜ ) ) X s , T ∗ × X s , T ∗ \= ( u , v ) − ( u ˜ , v ˜ ) X s , T ∗ × X s , T ∗ ≲ ( S u 0 , ϑ → ( u ) , S v 0 , ϱ → ( v ) ) − ( S u ˜ 0 , ϑ → ˜ ( u ) , S v ˜ 0 , ϑ → ˜ ( v ) ) X s , T ∗ × X s , T ∗ ≲ S u 0 − u ˜ 0 , ϑ → − ϑ → ˜ ( ( u 1 − u 2 ) , u 1 , ∂ x u 1 ) X s , T ∗ \+ S u 0 − u ˜ 0 , ϑ → − ϑ → ˜ ( u 2 , ( u 1 − u 2 ) , ∂ x u 1 ) X s , T ∗ \+ ( S u 0 − u ˜ 0 , ϑ → − ϑ → ˜ ( u 2 , ( u 1 − u 2 ) x , u 2 ) X s , T ∗ \+ S u 0 − u ˜ 0 , ϑ → − ϑ → ˜ ( ( v 1 − v 2 ) , v 1 , ∂ x u 1 ) X s , T ∗ \+ S u 0 − u ˜ 0 , ϑ → − ϑ → ˜ ( v 2 , ( v 1 − v 2 ) , ∂ x u 1 ) X s , T ∗ \+ S u 0 − u ˜ 0 , ϑ → − ϑ → ˜ ( v 2 , v 2 , ( u 1 − u 2 ) x ) X s , T ∗ \+ S u 0 − u ˜ 0 , ϑ → − ϑ → ˜ ( ( u 1 − u 2 ) , v 1 , ∂ x v 1 ) X s , T ∗ \+ S u 0 − u ˜ 0 , ϑ → − ϑ → ˜ ( u 2 , ( v 1 − v 2 ) , ∂ x v 1 ) X s , T ∗ \+ S u 0 − u ˜ 0 , ϑ → − ϑ → ˜ ( u 2 , v 2 , ( v 1 − v 2 ) x ) X s , T ∗ \+ S u 0 − u ˜ 0 , ϑ → − ϑ → ˜ ( ( u 1 − u 2 ) , u 1 , u 1 ) X s , T ∗ \+ S u 0 − u ˜ 0 , ϑ → − ϑ → ˜ ( u 2 , ( u 1 − u 2 ) , u 1 ) X s , T ∗ \+ S u 0 − u ˜ 0 , ϑ → − ϑ → ˜ ( u 2 , u 2 , ( u 1 − u 2 ) ) X s , T ∗ \+ S u 0 − u ˜ 0 , ϑ → − ϑ → ˜ ( ( v 1 − v 2 ) , v 1 , u 1 ) X s , T ∗ \+ S u 0 − u ˜ 0 , ϑ → − ϑ → ˜ ( v 2 , ( v 1 − v 2 ) , u 1 ) X s , T ∗ \+ S u 0 − u ˜ 0 , ϑ → − ϑ → ˜ ( v 2 , v 2 , ( u 1 − u 2 ) ) X s , T ∗ ≤ C ( T ∗ ) u 0 − u ˜ 0 H s \+ v 0 − v ˜ 0 H s \+ ϑ → − ϑ → ˜ Z s \+ ϱ → − ϱ → ˜ Z s \+ 1 2 ( u , v ) − ( u ˜ , v ˜ ) X s , T ∗ × X s , T ∗ ,
where the last term depends on C ( T ∗ ) M 2 ≤ 1 16, C ( T ∗ ) M 2 ≤ 1 16 for small T ∗ . The proof of the Lipschitz continuous for ([1](https://www.mdpi.com/2075-1680/15/3/230#FD1-axioms-15-00230)) is now complete.
### 4\.2. Proof of Corollary 1
This section is concerned with the proof of Corollary 1. In the course of proving Theorem 1, the nonlinear estimates we use impose a limitation: they only work for s ∈ \[ 0 , 5 2 ), which motivates our method for the case s \> 5 2. We deduce from Lemmas 4 and 5 that
( S u 0 , ϑ → ( 0 ) , S v 0 , ϱ → ( 0 ) ) X s , T ≲ u 0 H s \+ v 0 H s \+ ϑ → Z s \+ ϱ → Z s ,
(41)
where s ≥ 0 and the solution corresponding to the problem ([1](https://www.mdpi.com/2075-1680/15/3/230#FD1-axioms-15-00230)) admits
( u ( x , t ) , v ( x , t ) ) \= ( S u 0 , ϑ → ( 0 ) , S v 0 , ϱ → ( 0 ) ) \+ ( S 0 , 0 → ( Q 1 ) , S 0 , 0 → ( Q 2 ) ) .
(42)
where Q 1 ( u ) \= i χ { 2 \| u \| 2 \+ \| v \| 2 u x \+ u v ¯ v x } \+ δ \| u \| 2 \+ \| v \| 2 u and Q 2 ( u , v ) \= i χ { ( \| u \| 2 \+ 2 \| v \| 2 ) v x \+ u ¯ v u x } \+ δ \| u \| 2 \+ \| v \| 2 v. In the case of homogeneous initial-boundary data, we define ω 1 \= ∂ t S 0 , 0 → ( Q 1 ), ω 2 \= ∂ t S 0 , 0 → ( Q 2 ), which in turn yields that
i ∂ t ω 1 \+ α 1 ∂ x 2 ω 1 \+ i ς ∂ x 3 ω 1 \+ L u , v ( ω 1 , ω 2 ) \= 0 , i ∂ t ω 2 \+ α 2 ∂ x 2 ω 2 \+ i ς ∂ x 3 ω 2 \+ L ˜ u , v ( ω 1 , ω 2 ) \= 0 , ω 1 ( x , 0 ) \= 0 , ω 2 ( x , 0 ) \= 0 , ω 1 ( 0 , t ) \= 0 , ω 1 ( 1 , t ) \= 0 , ψ 1 x ( 1 , t ) \= 0 , ω 2 ( 0 , t ) \= 0 , ω 2 ( 1 , t ) \= 0 , ψ 2 x ( 1 , t ) \= 0 ,
(43)
where the operators L u , v ( ω 1 , ω 2 ) and L ˜ u , v ( ω 1 , ω 2 ) are nonlinear terms related to ω 1 , ω 2 . Since the linearization operators L u , v ( ω 1 , ω 2 ) and L ˜ u , v ( ω 1 , ω 2 ) are linear with respect to ω 1 , ω 2, the standard linear energy estimates can be directly applied. It follows from Lemma 7 that
ω 1 X s , T ≤ ( u X s , T 2 \+ v X s , T 2 ) ∂ t u X s , T ,
(44)
and
ω 2 X s , T ≤ ( v X s , T 2 \+ u X s , T 2 ) ∂ t v X s , T ,
(45)
for s ∈ \[ 0 , 1 ). Furthermore, it relies on the key observation that
∂ t u X s , T \= ∂ t S u 0 , ϑ → ( 0 ) \+ ∂ t S 0 , 0 → ( Q 1 ) X s , T ≲ ∂ t S u 0 , ϑ → ( 0 ) X s , T \+ ω 1 X s , T ,
(46)
∂ t v X s , T \= ∂ t S v 0 , ϱ → ( 0 ) \+ ∂ t S 0 , 0 → ( Q 2 ) X s , T ≲ ∂ t S v 0 , ϱ → ( 0 ) X s , T \+ ω 2 X s , T .
(47)
After setting s \= 0 in ([44](https://www.mdpi.com/2075-1680/15/3/230#FD44-axioms-15-00230)) and ([45](https://www.mdpi.com/2075-1680/15/3/230#FD45-axioms-15-00230)), we substitute this result into ([46](https://www.mdpi.com/2075-1680/15/3/230#FD46-axioms-15-00230)) and ([47](https://www.mdpi.com/2075-1680/15/3/230#FD47-axioms-15-00230)), respectively, thereby obtaining that
ω 1 X 0 , T ≲ ( u X 0 , T 2 \+ v X 0 , T 2 ) ( ∂ t S u 0 , ϑ → ( 0 ) X 0 , T \+ ω 1 X 0 , T ) , ω 2 X 0 , T ≲ ( v X 0 , T 2 \+ u X 0 , T 2 ) ( ∂ t S u 0 , ϱ → ( 0 ) X 0 , T \+ ω 2 X 0 , T ) .
Theorem 1 yields the key estimate
u X 0 , T \+ v X 0 , T ≲ u 0 L 2 \+ v 0 L 2 \+ ϑ → Z 0 \+ ϱ → Z 0 .
For T ∗ ∈ ( 0 , T ) to be chosen, it must be sufficiently small to ensure that C ( T ∗ ) ( u X 0 , T 2 \+ v X 0 , T 2 ) ≤ 1 8. Under this condition, we deduce
ω 1 X 0 , T ∗ ≲ ∂ t S u 0 , ϑ → ( 0 ) X 0 , T ∗ ,
(48)
ω 2 X 0 , T ∗ ≲ ∂ t S u 0 , ϑ → ( 0 ) X 0 , T ∗ .
(49)
Take the X 0 , T ∗ norm on both sides of ([43](https://www.mdpi.com/2075-1680/15/3/230#FD43-axioms-15-00230)), and apply the triangle inequality for the following norms:
∥ ∂ t u ∥ X 0 , T ∗ ≲ α 1 ∥ ∂ x 2 u ∥ X 0 , T ∗ \+ ς ∥ ∂ x 3 u ∥ X 0 , T ∗ \+ ∥ Q 1 ( u , v ) ∥ X 0 , T ∗ ,
(50)
∥ ∂ t v ∥ X 0 , T ∗ ≲ α 2 ∥ ∂ x 2 v ∥ X 0 , T ∗ \+ ς ∥ ∂ x 3 v ∥ X 0 , T ∗ \+ ∥ Q 2 ( u , v ) ∥ X 0 , T ∗ .
(51)
It follows from gathering ([41](https://www.mdpi.com/2075-1680/15/3/230#FD41-axioms-15-00230)), ([42](https://www.mdpi.com/2075-1680/15/3/230#FD42-axioms-15-00230)), and ([48](https://www.mdpi.com/2075-1680/15/3/230#FD48-axioms-15-00230))–(51) that
∂ t u X 0 , T ∗ ≲ ∂ t S u 0 , ϑ → ( 0 ) X 0 , T ∗ \+ ω 1 X 0 , T ∗ ≲ ∂ t S u 0 , ϑ → ( 0 ) X 0 , T ∗ ≲ ∂ t u L 2 ( 0 ) \+ ∂ t ϑ → Z 0 ≲ u 0 H 3 \+ ϑ → Z 3 ,
(52)
and
∂ t v X 0 , T ∗ ≲ v 0 H 3 \+ ϱ → Z 3 .
(53)
Moreover, by using ([1](https://www.mdpi.com/2075-1680/15/3/230#FD1-axioms-15-00230)), we have
ς ∂ x 3 u \= − u t \+ i α 1 ∂ x 2 u \+ i Q 1 ( u , v ) , ς ∂ x 3 v \= − v t \+ i α 2 ∂ x 2 v \+ i Q 2 ( u , v ) ,
which, when combined with Young’s inequality and the Gagliardo–Nirenberg inequality, can be used to derive that
u X 3 , T ∗ ≲ ∂ t u X 0 , T ∗ \+ ∂ x 3 u X 0 , T ∗ \+ ∂ x 2 u X 0 , T ∗ ≲ u 0 H 3 \+ ϑ → Z 3 ,
and
v X 3 , T ∗ ≲ v 0 H 3 \+ ϱ → Z 3 .
Then, the solution mappings Ψ 1 , Ψ 2 for (1) satisfy
Ψ 1 , Ψ 2 : H 3 × Z 3 → X 3 , T ∗ , Ψ 1 X 3 , T ∗ ≲ u 0 H 3 \+ ϑ → Z 3 , Ψ 2 X 3 , T ∗ ≲ v 0 H 3 \+ ϱ → Z 3 ,
and for the case s \= 0, we further have that
Ψ 1 , Ψ 2 : L 2 × Z 0 → X 0 , T ∗ , Ψ 1 ( u 0 , ϑ → ) − Ψ 1 ( u 0 , ϑ → ˜ ) X 0 , T ∗ ≲ u 0 − u ˜ 0 L 2 \+ ϑ → − ϑ → ˜ Z 0 , Ψ 2 ( v 0 , ϱ → ) − Ψ 2 ( v 0 , ϱ → ˜ ) X 0 , T ∗ ≲ v 0 − v ˜ 0 L 2 \+ ϱ → − ϱ → ˜ Z 0 .
Moreover, we deduce from Lemma 2 that
Ψ 1 , Ψ 2 : H 3 ( 1 − θ ) × Z 3 ( 1 − θ ) → X 3 ( 1 − θ ) , T ∗ , Ψ 1 ( u 0 , ϑ → ) X 3 ( 1 − θ ) , T ∗ ≲ u 0 H 3 ( 1 − θ ) \+ ϑ → Z 3 ( 1 − θ ) , Ψ 2 ( v 0 , ϱ → ) X 3 ( 1 − θ ) , T ∗ ≲ v 0 H 3 ( 1 − θ ) \+ ϱ → Z 3 ( 1 − θ ) ,
(54)
where 3 ( 1 − θ ) \> 5 2 , θ ∈ ( 0 , 1 6 ). We deduce from Lemma 2 and setting s \= 3 k ( 1 − θ ) , k \= 2 , 3 , … , for θ ∈ ( 0 , 1 ) that
Ψ 1 , Ψ 2 : H s × Z s → X s , T ∗ , Ψ 1 ( u 0 , ϑ → ) X s , T ∗ ≲ u 0 H s \+ ϑ → Z s , Ψ 2 ( v 0 , ϱ → ) X s , T ∗ ≲ v 0 H s \+ ϱ → Z s .
(55)
This finishes the proof of Corollary 1, gathering ([54](https://www.mdpi.com/2075-1680/15/3/230#FD54-axioms-15-00230)) and (55).
## 5\. Conclusions
We consider the well-posedness of ([1](https://www.mdpi.com/2075-1680/15/3/230#FD1-axioms-15-00230)) on a bounded interval. By utilizing the Laplace transform, which is well suited to the mixed dispersive structure of the equation, we derive a solution operator formula for the linear nonhomogeneous coupled system. Combining the fixed-point principle with energy estimates, we establish the local well-posedness of the problem in H s ( 0 , 1 ) when s ∈ \[ 0 , 3 / 2 ) is rigorously proven. Furthermore, a corollary extending the local result to the case s ≥ 3 / 2 is proved. It is clarified that when the initial values ( u 0 , v 0 ) ∈ H s ( 0 , 1 ) × H s ( 0 , 1 ) and the boundary values belong to space Z s and satisfy compatibility conditions, ([1](https://www.mdpi.com/2075-1680/15/3/230#FD1-axioms-15-00230)) admits a unique solution and the associated solution map is Lipschitz continuous in X s , T × X s , T. During the research process, by decomposing the solution operator into an initial value operator and boundary operator, linear and nonlinear estimates were established, respectively. Tools such as Cauchy–Schwarz inequality, Gagliardo–Nirenberg inequality, and interpolation theorem are utilized to complete a priori estimates. This approach addresses the balance between the smoothing effect of dispersion brought by third-order derivatives and the loss of spatial derivatives in nonlinear terms, and also compensates for the limitations of the Duhamel forcing operator method in dealing with the mixed dispersion problem of this equation. The research results provide a reference framework for the study of dispersion equations involving third-order derivatives. Within this context, modern mesh-free methods demonstrate significant potential for solving high-order boundary value problems. For instance, Karageorghis, Noorizadegan, and Chen have recently employed Radial Basis Function (RBF) methods, such as the fictitious center RBF method, which have been successfully applied to high-order boundary value problems, as demonstrated in \[34\]. Future research could explore the application and adaptation of such mesh-free techniques to numerically solve the IBVP of (cH), for which our analytical results on solution stability and regularity provide a crucial theoretical underpinning.
## Author Contributions
S.W. and H.W. contributed to the draft of the manuscript. All authors have read and agreed to the published version of the manuscript.
## Funding
This research received no external funding.
## Data Availability Statement
The data presented in this study are available on request from the corresponding author.
## Conflicts of Interest
The authors declare no conflicts of interest.
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