# Boundary Exponential Stabilization of the KP-II Equation

F.A. Gallego<EMAIL_ADDRESS>J.R. Muñoz<EMAIL_ADDRESS>
###### Abstract

In this paper, we delve into the intricacies of boundary stabilization for the
KP-II Equation within the constraints of a bounded domain. Our primary aim is
to engineer a parameterized set of feedback laws that not only lead to the
existence of solutions originating from minuscule initial data but also
facilitate their uniform exponential decay within the energy space. As far as
we know, the present work is the first result on boundary feedback
stabilization related to the Kadomtsev-Petviashvili II equation posed in a
rectangle $\Omega=(0,L)\times(0,L)$, $L>0$.

###### keywords:

KP-II system , Boundary stabilization , Exponential decay

††journal: Systems and Control Letters

[label1]organization=Departamento de Matematica, Universidad Nacional de
Colombia (UNAL), addressline=Cra 27 No. 64-60, city=Manizales,
postcode=170003, state=Caldas, country=Colombia

[label2]organization=Departamento de Matematica, Universidade Federal de
Pernambuco, addressline=S/N Cidade Universitaria, city=Recife,
postcode=50740-545, state=PE, country=Brazil

##  1 Introduction

The KP equation, short for the Kadomtsev-Petviashvili equation, holds great
significance in the realm of partial differential equations. It emerges
prominently when investigating nonlinear waves and solitons, with applications
spanning water wave dynamics, plasma physics, and various theoretical physics
and mathematical disciplines. This equation, independently discovered by B.B.
Kadomtsev and V.I. Petviashvili in 1970 (see [19]), takes center stage in the
study of weakly nonlinear, long waves within dispersive media. Its general
form is typically expressed as follows:

$u_{t}+uu_{x}+u_{xxx}+\gamma\partial_{x}^{-1}(u_{yy})=0,\quad\gamma=\pm 1.$

When $\gamma=-1$, we denote it as KP-I, and when $\gamma=1$, it’s KP-II,
inspired by the KP equation’s convention. KP-II describes long gravity waves
with subtle surface tension, while KP-I models capillary gravity waves,
especially with strong surface tension. Notably, the KP-I equation also arises
in the transonic limit (under appropriate scaling) of the defocusing nonlinear
Schrödinger equation, as documented in [10].

In this paper, we explore boundary stabilization of the Kadomtsev-Petviashvili
(KP) equation, focusing on control strategies and dissipative boundary
conditions to ensure stability near domain boundaries. This is crucial in
applications like fluid dynamics to prevent wave or soliton instability.
Researchers use energy methods, feedback control, and simulations to achieve
global solution existence and decay. The integrable nature of the KP equation
adds to its intrigue, making it a challenging problem in partial differential
equations and mathematical physics.

###  1.1 Setting of the problem

We focus on two boundary stabilization scenarios for the KP-II equation in a
bounded domain. Specifically, we investigate the implications of applying
Neumann dissipation laws at the domain boundaries, aiming to understand their
impact on achieving exponential stabilization and the interplay between
temporal and spatial constraints. We consider parameters $L>0$ for a positive
length, $\Omega:=(0,L)\times(0,L)$ to define a rectangular region in
$\mathbb{R}^{2}$, and $T>0$ for a positive time interval. Within this context,
we explore closed-loop systems to address these boundary stabilization
challenges:

$\begin{cases}u_{t}(x,y,t)+u_{xxx}(x,y,t)+\frac{1}{2}(u^{2}(x,y,t))_{x}+\gamma\partial_{x}^{-1}u_{yy}(x,y,t)=0,\\\
u(0,y,t)=u(L,y,t)=u(x,0,t)=u(x,L,t)=0,\\\ u_{x}(L,y,t)=\alpha u_{x}(0,y,t),\\\
u(x,y,0)=u_{0}(x,y),\end{cases}$ (1)

and

$\begin{cases}u_{t}(x,y,t)+u_{xxx}(x,y,t)+\frac{1}{2}(u^{2}(x,y,t))_{x}+\gamma\partial_{x}^{-1}u_{yy}(x,y,t)=0,\\\
u(0,y,t)=u(L,y,t)=0\,\,\ u_{y}(x,L,t)=\beta u_{x}(x,L,t),\\\
u_{x}(L,y,t)=\alpha u_{x}(0,y,t),\,\,u_{y}(x,0,t)=0,\\\
u(x,y,0)=u_{0}(x,y)\end{cases}$ (2)

where $(x,y)\in\Omega$, $t\in(0,T)$, $0<|\alpha|<1$, $\beta>0$, the operator
$\partial_{x}^{-1}$ is defined by
$\partial_{x}^{-1}\varphi(x,y,t)=\psi(x,y,t)$ such that $\psi(L,y,t)=0$ and
$\psi_{x}(x,y,t)=\varphi(x,y,t)$. It can be shown that the definition of
operator $\partial_{x}^{-1}$ is equivalent to
$\partial_{x}^{-1}u(x,y,t)=-\int_{x}^{L}u(s,y,t)\,ds$.

In this context, our goal is to demonstrate that the energy associated of the
systems described by equations (1) and (2) defined by

$\begin{split}E(t)=&\frac{1}{2}\int_{\Omega}u^{2}(t)dxdy,\end{split}$ (3)

has an exponential decay. To visualize the dissipation effects resulting from
the boundary conditions, we multiply the first equation by $u$ and we
integrate over $\Omega$. Moreover, in order to control the term
$\partial_{x}^{-1}u_{yy}$, we consider $u_{y}(x,y)=f_{x}(x,y)$ with
$f(L,y)=0$. Thus, it follows that

$\frac{d}{dt}E(t)=\frac{1}{2}\int_{0}^{L}\left[u_{x}^{2}(x,y,t)-2u(x,y,t)u_{xx}(x,y,t)+\gamma\left(\partial_{x}^{-1}u_{y}(x,y,t)\right)^{2}-\frac{2}{3}u^{3}(x,y,t)\right]_{x=0}^{x=L}dy\\\
-\gamma\int_{0}^{L}\left[u(x,y,t)f(x,y,t)\right]_{y=0}^{y=L}dx$

Note that the energy of equation already has a natural dissipations mechanism
given by $\left(\partial_{x}^{-1}u_{y}(0,y,t)\right)^{2}$. Now, by using the
boundary conditions (1), we note that the energy of the full system is
dissipative,

$\displaystyle\frac{d}{dt}E(t)=-\frac{(1-\alpha^{2})}{2}\int_{0}^{L}u_{x}^{2}(0,y,t)dy-\frac{\gamma}{2}\int_{0}^{L}(\partial_{x}^{-1}u_{y})^{2}(0,y,t)\,dy\leq
0.$

Here, we add feedback at the Neumann boundary in the $x$-variable while
maintaining homogeneous Dirichlet conditions across the entire domain, denoted
as $\Omega.$ In contrast, using homogeneous Dirichlet conditions in the
$x$-variable would require two types of feedback dissipations at the Neumann
boundary throughout the domain. When employing the boundary feedback law
represented by equation (2), energy dissipation occurs, at least in the
context of the associated linear system:

$\displaystyle\frac{d}{dt}E(t)=\frac{-(1-\alpha^{2})}{2}\int_{0}^{L}u_{x}^{2}(0,y,t)\,dy-\frac{\gamma}{2}\int_{0}^{L}\left(\partial_{x}^{-1}u_{y}(0,y)\right)^{2}\,dy-\gamma\beta\int_{0}^{L}u^{2}(x,L)\,dx\leq
0.$ (4)

This observation highlights the significance of boundary conditions as
feedback damping mechanisms. Consequently, it prompts the following questions:

Does $E(t)\rightarrow 0$ as $t\rightarrow\infty$? If it is the case, can we
give the decay rate?

The main result of this paper provides a positive answer for these questions.

###  1.2 State of the Art

Significant advancements in control theory have focused on understanding how
damping mechanisms impact energy dynamics in systems governed by partial
differential equations, particularly in dispersive equations related to water
waves in bounded domains. Well-established results demonstrate the exponential
stability of equations like KdV [14], Boussinesq-KdV [15], and Kawahara [1],
among others. These analyses use the Compactness-Uniqueness framework by J.L.
Lions [13]. Noteworthy findings in [5] and [6] employ Urquiza and Backstepping
approaches. These results utilize damping mechanisms either within the
equations or at the boundaries to achieve control.

In two noteworthy studies, [8] and [9], exponential decay was observed in the
KP-II and K-KP-II equations. These investigations established equation
regularity and well-posedness while showing that introducing a suitable
damping term ($a(x,y)u(x,y,t)$) leads to exponential energy decay. Notably,
the first exploration of stabilization techniques for two-dimensional
nonlinear dispersive equations in bounded domains, following the methodology
presented here, is found in [9], which specifically examines the KP-II
equation. In [9], external damping defined by a non-negative function $a(x,y)$
results in exponential energy decay, even when $a$ vanishes on a compact set
$\Theta\subset(\delta,L-\delta)\times(\delta,L-\delta)$ for some $\delta>0$.
The proof relies on an observation inequality for the KP-II equation,
established through the unique continuation principle in [16] and energy
estimates using multiplier techniques. In contrast, [8] employs the classical
dissipation-observability method and Bourgain’s unique continuation principle
[3], utilizing an anisotropic Gagliardo-Nirenberg inequality to drop the
$y$-derivative of the norm for a key estimate. In [7], the authors investigate
the K-KP-II system with both a damping mechanism $a(x,y)u$ acting as a
feedback-damping mechanism, as seen in [8], and an anti-damping mechanism,
preventing energy decrease.

To the best of our knowledge, this current study represents the inaugural
achievement in boundary feedback stabilization pertaining to the Kadomtsev-
Petviashvili II equation within a rectangular domain
$\Omega=(0,L)\times(0,L)$, where $L>0$.

###  1.3 Notations and Main Results

Before presenting answers to stabilization question, let us introduce the
functional space that will be necessary for our analysis. Given
$\Omega\subset\mathbb{R}^{2}$, let us define $X^{k}(\Omega)$ to be the Sobolev
space

$\displaystyle X^{k}(\Omega):=\left\\{\varphi\in
H^{k}(\Omega)\colon{\partial_{x}^{-1}}\varphi(x,y)=\psi(x,y)\in
H^{k}(\Omega),\psi(L,y)=0,\partial_{x}\psi(x,y)=\varphi(x,y)\right\\}$

endowed with the norm
$\left\lVert\varphi\right\rVert_{X^{k}(\Omega)}^{2}=\left\lVert\varphi\right\rVert_{H^{k}(\Omega)}^{2}+\left\lVert{\partial_{x}^{-1}}\varphi\right\rVert_{H^{k}(\Omega)}^{2}.$
Let us denote $\mathcal{H}=L^{2}(\Omega)$ and we also define the normed space
$H_{x}^{k}(\Omega)$ as

$H_{x}^{k}(\Omega):=\left\\{\varphi\colon\partial_{x}^{j}\varphi\in
L^{2}(\Omega),\ \text{for }0\leq j\leq k\right\\}$ (5)

with the norm
$\left\lVert\varphi\right\rVert_{H_{x}^{k}(\Omega)}^{2}=\sum_{j=0}^{k}\left\lVert\partial_{x}^{j}\varphi\right\rVert_{L^{2}(\Omega)}^{2}$.
Similarly, we consider the space

$\displaystyle X_{x}^{k}(\Omega):=\left\\{\varphi\in
H_{x}^{k}(\Omega)\colon{\partial_{x}^{-1}}\varphi(x,y)=\psi(x,y)\in
H_{x}^{k}(\Omega),\psi(L,y)=0,\partial_{x}\psi(x,y)=\varphi(x,y)\right\\}$

with norm
$\left\lVert\varphi\right\rVert_{X_{x}^{k}(\Omega)}^{2}=\left\lVert\varphi\right\rVert_{H_{x}^{k}(\Omega)}^{2}+\left\lVert{\partial_{x}^{-1}}\varphi\right\rVert_{H_{x}^{k}(\Omega)}^{2}.$
Finally, $H_{x0}^{k}(\Omega)$ will denote the closure of
$C_{0}^{\infty}(\Omega)$ in $H_{x}^{k}(\Omega)$. Thus, the main results of
this work provide an answer to the stabilization problem proposed at the
beginning of this paper. One of them is related to the sole feedback mechanism
in the Neumann boundary condition only in the $x$-variable:

######  Theorem 1.1 (Local Stabilization - One Dissipation).

Let $L>0$ and $r>0$. Let us suppose that

$\theta<\frac{\mu}{2(1+\mu
L)L^{2}}\left[\frac{17}{3}-(Cr)^{4/3}L^{2}\right],\quad\kappa=1+\mu L,$

where $\mu$ is a positive constants such that
$0<\mu<\frac{1-\alpha^{2}}{L\alpha^{2}}$, with $C$ is the constant given by
the anisotropic Gagliardo-Nirenberg inequality. Then, for any $u_{0}\in
L^{2}(\Omega)$ satisfying $\left\lVert{u_{0}}\right\rVert_{L^{2}(\Omega)}\leq
r$, the energy associated to closed-loop system (1) defined by (5) decays
exponentially. More precisely, there exist two positives constants $\theta$
and $\kappa$ such that

$E(t)\leq\kappa E(0)e^{-2\theta t},\quad t>0.$

With one boundary dissipation, there are restrictions on initial data and
length parameter $L>0,$ but with two Neumann boundary dissipations, there are
no such restrictions. To summarize:

######  Theorem 1.2 (Local Uniform Stabilization - Two Dissipations).

Let $L>0$. Then, there exists $r>0$ such that for every $u_{0}\in
L^{2}(\Omega)$ satisfying $\left\lVert{u_{0}}\right\rVert_{L^{2}(\Omega)}\leq
r$, the energy associated to closed-loop system (2) defined by (5) decays
exponentially. More precisely, there exists two positives constants $\theta$
and $\kappa$ such that

$E(t)\leq\kappa E(0)e^{-2\theta t},\quad t>0.$

Please note that the decay rate $\theta$ in Theorem 2.7 remains independent of
the initial value’s magnitude, $u_{0}$, within $L^{2}(\Omega)$. Consequently,
we can conclude that the system (2) exhibits uniform exponential stability. To
prove Theorem 1.1, we’ll employ the Lyapunov approach, a control theory
technique for stability analysis. It involves selecting a Lyapunov function
characterizing system energy, followed by analyzing its time derivative along
system trajectories. Additionally, the last result removes the need for small
initial data, using J.-L. Lions’ compactness-uniqueness argument [12] to
simplify the task to proving an observability inequality. The structure of
this work is as follows: In Section 2, we discuss exponential stabilization
with a single feedback mechanism, including parameter constraints. Section 3
explores the system’s asymptotic behavior with two dissipative mechanisms,
without parameter restrictions.

##  2 One Dissipation Mechanism: Exponential Stabilization with Restriction

In this section, we present the results pertaining to exponential
stabilization in the presence of a single dissipation mechanism, acting in the
Neumann Boundary Condition. To achieve this, we employ a Lyapunov method.

From now on, we will denote by $C>0$ a general constant which may vary from
line to line.

###  2.1 Preliminary estimates

Here, we establish a series of linear estimates used thereafter. To begin, we
apply the classical semigroup theory to the linearized system associated to
(1):

$\begin{cases}\begin{aligned}
u_{t}(x,y,t)+u_{xxx}(x,y,t)+\gamma\partial_{x}^{-1}u_{yy}(x,y,t)&=0,\end{aligned}&\\\
u(0,y,t)=u(L,y,t)=u(x,0,t)=u(x,L,t)=0,&\\\ u_{x}(L,y,t)=\alpha
u_{x}(0,y,t),&\\\ u(x,y,0)=u_{0}(x,y),&\end{cases}$ (6)

with $(x,y)\in\Omega$, $t\in(0,T)$. Using the notation of the operator $A$, we
rewrite the system (6) as an abstract equation $\frac{du}{dt}=Au$,
$u(0)=u_{0}$ where $A\colon D(A)\subset\mathcal{H}\to\mathcal{H}$ is defined
by

$Au=-u_{xxx}-\gamma{\partial_{x}^{-1}}u_{yy}$ (7)

on the dense domain is given by

$\displaystyle D(A):=\left\\{u\in H_{0}^{1}(\Omega)\cap X^{2}(\Omega)\colon
u_{x}(L,y)=\alpha u_{x}(0,y),\lvert\alpha\rvert<1\right\\}.$ (8)

To establish the well-posedness of the linear system, we present some useful
results.

######  Lemma 2.3.

The operator $A$ is closed and the adjoint $A^{\ast}\colon
D(A^{\ast})\subset\mathcal{H}\to\mathcal{H}$ is given by
$A^{\ast}u=v_{xxx}+\gamma\left(\partial_{x}^{-1}\right)^{\ast}v_{yy}$ with
dense domain

$\displaystyle D(A^{\ast}):=\left\\{u\in H_{0}^{1}(\Omega)\cap
X^{2}(\Omega)\colon u_{x}(0,y)=\alpha u_{x}(L,y),\
\lvert\alpha\rvert<1\right\\}.$ (9)

######  Remark 2.4.

Observe that for the adjoin problem, the operator
$\left(\partial_{x}^{-1}\right)^{\ast}$ is defined as

$\partial_{x}^{-1}\varphi(x,y)=\psi(x,y)\text{ such that }\psi(0,y)=0\text{
and }\psi_{x}(x,y)=\varphi(x,y).$

This definition is equivalent to
$\left(\partial_{x}^{-1}\right)^{\ast}u(x,y,t)=\int_{0}^{x}u(s,y,t)\,ds$.

###### Proof.

Let $u\in D(A)$ and $v\in D(A^{\ast})$. We consider two functions, $f,g$ such
that $u_{y}(x,y)=f_{x}(x,y)$ and $v_{y}(x,y)=g_{x}(x,y)$, with $f(L,y)=0$ and
$g(0,y)=0$. Firstly, we estimate the product of the $v$ by the nonlocal term
${\partial_{x}^{-1}}u_{yy}$:

$\displaystyle-\int_{\Omega}v{\partial_{x}^{-1}}u_{yy}dxdy$
$\displaystyle=\int_{\Omega}v_{y}fdxdy=-\int_{\Omega}gu_{y}dxdy$
$\displaystyle=\int_{\Omega}u{\partial_{x}^{-1}}v_{yy}dxdy.$

Consequently, we can estimate the duality product $\left\langle
Au,v\right\rangle_{\mathcal{H}}$ as follows

$\displaystyle\left\langle Au,v\right\rangle_{\mathcal{H}}$
$\displaystyle=\int_{0}^{L}[u_{x}(L,y)v_{x}(L,y)-u_{x}(0,y)v_{x}(0,y)]dy+\int_{\Omega}uv_{xxx}dxdy+\gamma\int_{\Omega}u{\partial_{x}^{-1}}v_{yy}dxdy=\left\langle
u,A^{\ast}v\right\rangle_{\mathcal{H}}.$

Finally, note that $A^{\ast\ast}=A$ then $A$ is closed. ∎

######  Proposition 2.5.

The operator $A$ is the infinitesimal generator of a $C_{0}$-semigroup in
$\mathcal{H}$.

###### Proof.

Let $u\in D(A)$ and call $f_{x}(x,y)=u_{y}(x,y)$ with $f(L,y)=0$. From (8) it
follows that

$\displaystyle\left\langle Au,u\right\rangle_{\mathcal{H}}$
$\displaystyle=\frac{1}{2}\int_{\Omega}(u_{x}^{2})_{x}dxdy-\gamma\int_{0}^{L}[f(x,y)u(x,y)]_{y=0}^{y=L}\,dx+\gamma\int_{\Omega}fu_{y}dydx=\int_{0}^{L}\left[\frac{u_{x}^{2}(x,y)}{2}\right]_{x=0}^{x=L}dy+\gamma\int_{\Omega}ff_{x}dxdy$

Finally, we have that

$\displaystyle\left\langle
Au,u\right\rangle_{\mathcal{H}}=-\frac{(1-\alpha^{2})}{2}\int_{0}^{L}u_{x}^{2}(0,y)dy-\frac{\gamma}{2}\int_{0}^{L}(\partial_{x}^{-1}u_{y})^{2}(0,y)dy\leq
0.$

Similarly, let $v\in D(A^{\ast})$ and $g_{x}(x,y)=v_{y}(x,y)$ with $g(0,y)=0$.
Then

$\left\langle
v,A^{\ast}v\right\rangle_{\mathcal{H}}=-\frac{1}{2}\int_{\Omega}(v_{x}^{2})_{x}dxdy+\gamma\int_{0}^{L}[g(x,y)v(x,y)]_{y=0}^{y=L}dx\\\
-\gamma\int_{\Omega}gv_{y}dydx=-\frac{1}{2}\int_{0}^{L}[v_{x}^{2}(x,y)]_{x=0}^{x=L}dy-\gamma\int_{\Omega}gg_{x}dxdy$

From (9), it yields that

$\displaystyle\left\langle
v,A^{\ast}v\right\rangle_{\mathcal{H}}=-\frac{(1-\alpha^{2})}{2}\int_{0}^{L}v_{x}^{2}(0,y)dy-\frac{\gamma}{2}\int_{0}^{L}(\partial_{x}^{-1}u_{y})^{2}(L,y)\,dy\leq
0.$

Thus, from [17, Corollary 4.4, page 15], the Proposition holds. ∎

Thus, we can now state the following theorem regarding the existence of
solutions for the Cauchy abstract problem:

######  Theorem 2.6.

Then, for each initial data $u_{0}\in\mathcal{H}$ there exists a unique mild
solution $u\in C\left([0,\infty),\mathcal{H}\right)$ for the system (6).
Moreover, if the initial data $u_{0}\in D(A)$ the solutions are classical such
that $u\in C\left([0,\infty),D(A)\right)\cap
C^{1}\left([0,\infty),\mathcal{H}\right).$

###### Proof.

From Proposition 2.5, it follows that $A$ generates a strongly continuous
semigroup of contractions $\\{S(t)\\}_{t\geq 0}$ in $\mathcal{H}$ (see [17,
Corollary 1.4.4]). ∎

######  Proposition 2.7.

For any mild solution of (6), the energy $E(t)$ is non-increasing and there
exists a constant $C>0$ such that

$\displaystyle\frac{d}{dt}E(t)$
$\displaystyle\leq-C\left[\int_{0}^{L}u_{x}^{2}(0,y,t)\,dy+\int_{0}^{L}({\partial_{x}^{-1}}u_{y}(0,y,t))^{2}\,dy\right]$
(10)

where $C=C(\alpha,\gamma)$ is given by
$C=\frac{1}{2}\min\left\\{1-\alpha^{2},\gamma\right\\}.$

###### Proof.

Note that

$\displaystyle\frac{d}{dt}E(t)$
$\displaystyle=\int_{\Omega}u(t)\left(-u_{xxx}(t)-\gamma{\partial_{x}^{-1}}u_{yy}(t)\right)dxdy=\left\langle
Au(t),u(t)\right\rangle_{\mathcal{H}}.$

From the proof of Proposition 2.5, we conclude that

$\displaystyle\frac{d}{dt}E(t)=-\frac{(1-\alpha^{2})}{2}\int_{0}^{L}u_{x}^{2}(0,y,t)dy-\frac{\gamma}{2}\int_{0}^{L}(\partial_{x}^{-1}u_{y})^{2}(0,y,t)\,dy.$

∎

The next result will be repeatedly referenced in this article, and it is known
as the anisotropic Gagliardo-Nirenberg inequality:

######  Theorem 2.8 ([2, Theorem 15.7]).

Let $\beta$ and $\alpha^{(j)}$, for $j=1,\dots,N$, denote $n$\- dimensional
multi-indices with non-negative-integer-valued components. Suppose that
$1<p^{(j)}<\infty$, $1<q<\infty$, $0<\mu_{j}<1$ with

$\sum_{j=1}^{N}\mu_{j}=1,\quad\frac{1}{q}\leq\sum_{j=1}^{N}\frac{\mu_{j}}{p^{(j)}},\quad\beta-\frac{1}{q}=\sum_{j=1}^{N}\mu_{j}\left(\alpha^{(j)}-\frac{1}{p^{(j)}}\right).$

Then, for $f(x)\in C_{0}^{\infty}(\mathbb{R}^{n})$, $\left\lVert
D^{\beta}f\right\rVert_{q}\leq C\prod_{j=1}^{N}\left\lVert
D^{\alpha^{(j)}}f\right\rVert_{p^{(j)}}^{\mu_{j}},$ where for non-negative
multi-index $\beta=(\beta_{1},\dots,\beta_{N})$ we denote $D^{\beta}$ by
$D^{\beta}=D_{x_{1}}^{\beta_{1}}\dots D_{x_{n}}^{\beta_{n}}$ and
$D_{x_{i}}^{\beta_{i}}=\frac{\partial^{\beta_{i}}}{\partial x_{i}^{k_{i}}}$.

The proposition provides valuable estimates for mild solutions of equation
(6), including standard energy assessments and revealing a Kato smoothing
effect.

######  Proposition 2.9.

Let $u_{0}\in\mathcal{H}$, then the mild solution $u$ of system (6) satisfies

$\|u\|_{L^{2}(0,T,H^{1}(\Omega))}\leq C\|u_{0}\|^{2}_{\mathcal{H}}$

for some positive constant $C>0$.

###### Proof.

We multiply the equation (6) by $xu$ and integrating in $\Omega\times(0,T)$,
we obtain that

$\int_{0}^{T}\int_{\Omega}xu(t)\left(u_{t}(t)+u_{xxx}(t)+\gamma{\partial_{x}^{-1}}u_{yy}(t)\right)dxdydt=0.$

Now, estimating term by term, it follows that

$\displaystyle\int_{0}^{T}\int_{\Omega}xu(t)u_{t}(t)dxdydt=\frac{1}{2}\int_{0}^{T}\int_{\Omega}x(u^{2}(t))_{t}dxdydt=\frac{1}{2}\int_{\Omega}xu^{2}(T)dxdy-\frac{1}{2}\int_{\Omega}xu_{0}^{2}dxdy.$

The next term is

$\displaystyle\int_{0}^{T}\int_{\Omega}xu(t)u_{xxx}(t)dxdydt=\frac{3}{2}\int_{0}^{T}\int_{\Omega}u_{x}^{2}(t)dxdydt-\frac{L}{2}\int_{0}^{T}\int_{0}^{L}u_{x}^{2}(L,y,t)dydt$

For the next term, as earlier, call $f_{x}(x,y)=u_{y}(x,y)$ with $f(L,y)=0$.
Thus, it follows that

$\gamma\int_{0}^{T}\int_{\Omega}xu(t){\partial_{x}^{-1}}u_{yy}(t)dxdydt=\frac{\gamma}{2}\int_{0}^{T}\int_{\Omega}({\partial_{x}^{-1}}u_{y}(t))^{2}dxdydt.$

Now, adding the above estimates, we deduce that

$\frac{3}{2}\int_{0}^{T}\int_{\Omega}u_{x}^{2}(t)dxdydt=-\frac{1}{2}\int_{\Omega}xu^{2}(T)dxdy+\frac{1}{2}\int_{\Omega}xu_{0}^{2}dxdy\\\
+\frac{L}{2}\int_{0}^{T}\int_{0}^{L}u_{x}^{2}(L,y,t)dydt-\frac{\gamma}{2}\int_{0}^{T}\int_{\Omega}({\partial_{x}^{-1}}u_{y}(t))^{2}dxdydt\\\
\leq\frac{1}{2}\int_{\Omega}xu_{0}^{2}dxdy+\frac{L}{2}\int_{0}^{T}\int_{0}^{L}u_{x}^{2}(L,y,t)dydt.$

Finally, we obtain

$\displaystyle\int_{0}^{T}\|u(t)\|_{H_{x}^{1}(\Omega)}^{2}dt$
$\displaystyle\leq\frac{L}{3}\|u_{0}\|^{2}_{\mathcal{H}}+\frac{L}{3}\int_{0}^{T}\int_{0}^{L}u_{x}^{2}(L,y,t)dydt.$
(11)

On the other hand, recall that

$\lVert u\rVert_{H^{1}(\Omega)}^{2}=\lVert u\rVert_{L^{2}(\Omega)}^{2}+\lVert
u_{x}\rVert_{L^{2}(\Omega)}^{2}+\lVert u_{y}\rVert_{L^{2}(\Omega)}^{2}$

By using Theorem 2.8 can be show that there exists $C>0$ such that

$\lVert u_{y}\rVert_{L^{2}(\Omega)}\leq C\lVert u_{x}\rVert_{L^{2}(\Omega)}.$

Thus, we deduce from (11), the following estimate

$\displaystyle\int_{0}^{T}\|u(t)\|_{H^{1}(\Omega)}^{2}dt$
$\displaystyle\leq\frac{CL}{3}\|u_{0}\|^{2}_{\mathcal{H}}+\frac{CL}{3}\int_{0}^{T}\int_{0}^{L}u_{x}^{2}(L,y,t)dydt.$

Moreover, by using the boundary conditions and (10), it follows that

$\displaystyle\int_{0}^{T}\|u(t)\|_{H^{1}(\Omega)}^{2}dt$
$\displaystyle\leq\frac{CL}{3}\|u_{0}\|^{2}_{\mathcal{H}}+\frac{CL\alpha^{2}}{3}\int_{0}^{T}\int_{0}^{L}u_{x}^{2}(0,y,t)dydt\leq\frac{CL}{3}\|u_{0}\|^{2}_{\mathcal{H}}-\frac{L\alpha^{2}}{3}E(T)+\frac{L\alpha^{2}}{3}E(0)$
$\displaystyle\leq\frac{CL}{3}\|u_{0}\|^{2}_{\mathcal{H}}+\frac{2CL\alpha^{2}}{3}\|u_{0}\|^{2}_{\mathcal{H}}.$

∎

###  2.2 Wellposedness

Now, we address the well-posedness of the system (1), allowing us to consider
the nonlinear model. Before presenting it, we introduce the following
definition:

######  Definition 2.1.

A mild solution of (1) will be a function $u\in C([0,T];L^{2}(\Omega))\cap
L^{2}(0,T;\Omega)),$ with $T>0,$ which satisfies

$\displaystyle u(t)$
$\displaystyle=S(t)u_{0}-\int_{0}^{t}S(t-s)u(s)u_{x}(s)ds,\quad t\in[0,T],$

where $\\{S(t)\\}_{t\geq 0}$ is the semigroup generated by the operator $A$
given by (7) - (8). A global mild solution of (1) will be a function $u$
defined in $[0,\infty)$ whose restriction to every bounded interval $[0,T]$ is
a mild solution of (1).

First we consider the system with a source term $f(x,y,t)\in
L^{1}(0,T;L^{2}(\Omega))$,

$\begin{cases}\begin{aligned}
u_{t}(x,y,t)+u_{xxx}(x,y,t)+\gamma\partial_{x}^{-1}u_{yy}(x,y,t)&=f(x,y,t),\end{aligned}\\\
u(0,y,t)=u(L,y,t)=u(x,0,t)=u(x,L,t)=0,\\\ u_{x}(L,y,t)=\alpha u_{x}(0,y,t),\\\
u(x,y,0)=u_{0}(x,y).\end{cases}$ (12)

Using the semigroup properties given by our system, we obtain the following
useful estimates:

######  Proposition 2.10.

For any $u_{0}\in L^{2}(\Omega)$ and $f\in L^{1}(0,T;L^{2}(\Omega))$, there
exists a unique mild solution $u$ belongs to $C([0,T];L^{2}(\Omega))\cap
L^{2}(0,T;\Omega))$ for (12). Moreover,

$\left\lVert u\right\rVert_{C([0,T];L^{2}(\Omega))}\leq C\left(\left\lVert
u_{0}\right\rVert_{L^{2}(\Omega)}+\left\lVert
f\right\rVert_{L^{1}(0,T;L^{2}(\Omega)}\right)$ (13)

and

$\left\lVert u\right\rVert_{L^{2}(0,T;H^{1}(\Omega))}^{2}\leq
M\left(\left\lVert u_{0}\right\rVert_{L^{2}(\Omega)}^{2}+\left\lVert
f\right\rVert_{L^{1}(0,T;L^{2}(\Omega)}^{2}\right).$ (14)

where
$M=M(\alpha,L,T)=\left(\frac{C(2\alpha^{2}+1)}{3}+\frac{L}{2}T^{2}+\frac{L}{2}\right)$.

###### Proof.

Note that the operator $A$ (defined in equations (7)-(8)) is the infinitesimal
generator of the $C_{0}$-semigroup $\\{S(t)\\}_{t\geq 0}$. As per [17], the
mild solution of the system is: $u(t)=S(t)u_{0}-\int_{0}^{t}S(t-s)f(s)ds$, for
$t\in[0,T]$. As a result know, we can derive equation (13). On the other hand,
following the same steps that in the proof of Proposition 2.9, we get (14)
taking into account that

$\displaystyle\left\lvert\int_{0}^{T}\int_{\Omega}xf(t)u(t)dxdydt\right\rvert\leq\frac{L}{2}\left\lVert{u}\right\rVert_{C([0,T];L^{2}(0,L))}^{2}+\frac{L}{2}\left\lVert{f}\right\rVert_{L^{1}(0,T,L^{2}(\Omega))}^{2}.$

∎

Now we associate the nonlinear term $uu_{x}$ as the source term. Let us call
the next functional spaces

$\begin{split}\mathcal{B}_{X}=&C\left([0,T],L^{2}(\Omega)\right)\cap
L^{2}\left(0,T,X_{x0}^{1}(\Omega)\right),\\\
\mathcal{B}_{H}=&C\left([0,T],L^{2}(\Omega)\right)\cap
L^{2}\left(0,T,H_{x0}^{1}(\Omega)\right)\end{split}$

endowed with its natural norms

$\displaystyle\left\lVert{y}\right\rVert_{\mathcal{B}_{X}}$
$\displaystyle=\max_{t\in[0,T]}\left\lVert{y(t)}\right\rVert_{L^{2}(\Omega)}+\left(\int_{0}^{T}\left\lVert{y(t)}\right\rVert_{X_{x0}^{2}(\Omega)}^{2}\,dt\right)^{\frac{1}{2}},$
$\displaystyle\left\lVert{y}\right\rVert_{\mathcal{B}_{H}}$
$\displaystyle=\max_{t\in[0,T]}\left\lVert{y(t)}\right\rVert_{L^{2}(\Omega)}+\left(\int_{0}^{T}\left\lVert{y(t)}\right\rVert_{H_{x0}^{2}(\Omega)}^{2}\,dt\right)^{\frac{1}{2}}.$

Here, $X_{x0}^{k}(\Omega)$ denotes the space

$\displaystyle X_{x0}^{k}(\Omega):=\left\\{\varphi\in
H_{x0}^{k}(\Omega)\colon{\partial_{x}^{-1}}\varphi=\psi\in
H_{x0}^{k}(\Omega),\psi(L,y)=0,\partial_{x}\psi=\varphi\right\\}.$

######  Lemma 2.11.

If $u\in\mathcal{B}_{X}$ then $uu_{x}\in L^{1}(0,T;L^{2}(\Omega))$ and the map
$u\in\mathcal{B}_{X}\mapsto uu_{x}\in L^{1}(0,T;L^{2}(\Omega))$ is continuous.
In particular, there exists $C>0$, such that, for all $u,v\in\mathcal{B}_{X}$
we have

$\left\lVert{u\partial_{x}u-v\partial_{x}v}\right\rVert_{L^{1}(0,T,L^{2}(\Omega))}\leq
C\left(\left\lVert{u}\right\rVert_{\mathcal{B}_{X}}+\left\lVert{v}\right\rVert_{\mathcal{B}_{X}}\right)\left\lVert{u-v}\right\rVert_{\mathcal{B}_{X}}.$

###### Proof.

A sketch of the proof is given for the sake of completeness (see [18] for
details). The Hölder inequality and the Sobolev embedding
$H_{x0}^{1}(\Omega)\hookrightarrow L^{\infty}(\Omega)$ (for details, see [4])
gives us

$\displaystyle\left\lVert
u\partial_{x}u-v\partial_{x}v\right\rVert_{L^{1}(0,T,L^{2}(\Omega))}\leq
C_{1}\cdot C\cdot
T^{\frac{1}{4}}\left(\left\lVert{u}\right\rVert_{\mathcal{B}_{H}}+\left\lVert{v}\right\rVert_{\mathcal{B}_{H}}\right)\left\lVert{u-v}\right\rVert_{\mathcal{B}_{H}},$
(15)

for $u,v\in\mathcal{B}_{X}$. Note that, $u\in\mathcal{B}_{X}$ implies that
$u(\cdot,\cdot,t)\in H_{x0}^{1}(\Omega)$. Here, using the definition of the
operator ${\partial_{x}^{-1}}$ and the Poincaré’s inequality (see e.g. [4]) we
obtain,

$\left\lVert{\partial_{x}^{-1}}(u\partial_{x}u)\right\rVert_{L^{1}(0,T,L^{2}(\Omega))}\leq
L^{2}\left\lVert u\partial_{x}u\right\rVert_{L^{1}(0,T,L^{2}(\Omega))}$ (16)

So, from (15), with $v=0$, and (16) we get $u\partial_{x}u\in
L^{1}(0,T,L^{2}(\Omega))$ and the proof is complete. ∎

Now, we are ready to establish the well-posedness for the system (1), subject
to certain restrictions on the initial data.

######  Theorem 2.12.

Consider $L,T>0$. Then, there exists $r>0$ such that for all initial data
$u_{0}\in L^{2}(\Omega)$ satisfying $\|u_{0}\|_{L^{2}(\Omega)}\leq r,$ there
exists a unique $u\in\mathcal{B}_{X}$ mild solution of (1) such that

$\lVert u\rVert_{\mathcal{B}_{X}}\leq C\lVert u_{0}\rVert_{L^{2}(\Omega)}.$

###### Proof.

Let $u_{0}\in L^{2}(\Omega)$ such that $\lVert u_{0}\rVert_{L^{2}(\Omega)}<r$
where $r>0$ is chosen small enough later. Consider $u\in\mathcal{B}_{X}$ and
define the application $\Phi\colon\mathcal{B}_{X}\to\mathcal{B}_{X}$ given by
$\Phi(u)=\tilde{u}$ where $\tilde{u}$ is solution of

$\begin{cases}\begin{aligned}
\tilde{u}_{t}(x,y,t)+\tilde{u}_{xxx}(x,y,t)+\gamma\partial_{x}^{-1}\tilde{u}_{yy}(x,y,t),\end{aligned},\\\
\tilde{u}(0,y,t)=\tilde{u}(L,y,t)=\tilde{u}(x,0,t)=\tilde{u}(x,L,t)=0,\\\
\tilde{u}_{x}(L,y,t)=\alpha\tilde{u}_{x}(0,y,t),\\\
\tilde{u}(x,y,0)=\tilde{u}_{0}(x,y).\end{cases}$

It means that $\Phi(u)(t)=S(t)\tilde{u}_{0}-\int_{0}^{t}S(t-s)u(t)u_{x}(s)ds,$
$t\in[0,T]$. Clearly, $u\in\mathcal{B}_{X}$ is solution to (1) if and only if
$u$ is a fixed point of $\Phi$. Observe that, Proposition 2.10 and Lemma 2.11
imply that

$\displaystyle\left\lVert\Phi u\right\rVert_{\mathcal{B}_{X}}$
$\displaystyle=\left\lVert\tilde{u}\right\rVert_{\mathcal{B}_{X}}\leq
M\left(\left\lVert u_{0}\right\rVert_{L^{2}(\Omega)}+\left\lVert
uu_{x}\right\rVert_{L^{1}(0,T;L^{2}(\Omega))}\right)\leq M\left\lVert
u_{0}\right\rVert_{L^{2}(\Omega)}+MCT^{1/4}\lVert
u\rVert_{\mathcal{B}_{X}}^{2}.$

$M=\left(\frac{C(2\alpha^{2}+1)}{3}+\frac{L}{2}T^{2}+\frac{L}{2}\right)$. On
the other hand, note that $\Phi u-\Phi v$ solves the system with initial data
zero, so,

$\left\lVert\Phi u-\Phi v\right\rVert_{\mathcal{B}_{X}}\leq
CT^{1/4}M\left(\left\lVert{u}\right\rVert_{\mathcal{B}_{X}}+\left\lVert{v}\right\rVert_{\mathcal{B}_{X}}\right)\left\lVert{u-v}\right\rVert_{\mathcal{B}_{X}}.$

Limit the application of $\Phi$ to the closed ball defined as follows:
$B_{r}(0)=\left\\{u\in\mathcal{B}_{X}:\lVert u\rVert_{\mathcal{B}_{X}}\leq
r\right\\}$, where the value of $r>0$ is chosen as
$r=\frac{1}{2}\left(\frac{C(2\alpha^{2}+1)}{3}+L\right)\lVert
u_{0}\rVert_{L^{2}(\Omega)}$ with

$T<1\quad\text{and}\quad
r<\frac{3}{2CT^{1/4}\left({C(2\alpha^{2}+1)}+3L\right)},$

thus, we have that

$\displaystyle\left\lVert\Phi u\right\rVert_{\mathcal{B}_{X}}$
$\displaystyle\leq M\left(\left\lVert
u_{0}\right\rVert_{L^{2}(\Omega)}+CT^{1/4}r^{2}\right)<\frac{r}{2}+\frac{r}{2}.$

The above estimate allows us to conclude that

$\Phi\colon B_{r}(0)\subset\mathcal{B}_{X}\to B_{r}(0).$

Moreover, note that

$\displaystyle\left\lVert\Phi u-\Phi v\right\rVert_{\mathcal{B}_{X}}$
$\displaystyle\leq
2rCT^{1/4}M\left\lVert{u-v}\right\rVert_{\mathcal{B}_{X}}\leq
K\left\lVert{u-v}\right\rVert_{\mathcal{B}_{X}}.$

with $K<1$. Hence, $\Phi\colon B_{r}(0)\to B_{r}(0)$ is a contraction and, by
Banach fixed-point theorem, we obtain a unique $u\in B_{r}(0)$, such that
$\Phi u=u\in\mathcal{B}_{X}$. The proof is complete. ∎

###  2.3 Exponential Stability - Lyapunov’s approach

In this section, we establish exponential stabilization using Lyapunov’s
approach, where we show that the Lyapunov function’s rate of change is bounded
by a negative constant times the Lyapunov function itself, ensuring
exponential energy decrease, system stability, and convergence, with certain
parameter constraints. Here’s the proof of one central result:

######  Proof of the Theorem 1.1.

Firstly, let us consider the following Lyapunov’s functional $V(t)=E(t)+\mu
V_{1}(t)$, where $\mu$ is a constant to be fixed later, $E(t)$ is given by (3)
and

$V_{1}(t)=\frac{1}{2}\int_{\Omega}xu^{2}(x,y,t)\,dx\,dy.$

Note that $E(t)$ and $V(t)$ are equivalent in the following sense $E(t)\leq
V(t)\leq(1+\mu L)E(t)$ and observe that

$\displaystyle\frac{d}{dt}V_{1}(t)$
$\displaystyle=\int_{\Omega}xu(t)u_{t}(t)dxdy=-\frac{3}{2}\int_{\Omega}u_{x}^{2}(t)dxdy+\frac{L\alpha^{2}}{2}\int_{0}^{L}u_{x}^{2}(0,y,t)\,dy$
$\displaystyle\quad-\frac{\gamma}{2}\int_{\Omega}\left(\partial_{x}^{-1}u_{y}(t)\right)^{2}dxdy+\frac{1}{3}\int_{\Omega}u^{3}(t)dxdy$

and remember that
$\frac{d}{dt}E(t)=-\frac{(1-\alpha^{2})}{2}\int_{0}^{L}u_{x}^{2}(0,y,t)\,dy-\frac{\gamma}{2}\int_{0}^{L}\left(\partial_{x}^{-1}u_{y}(0,y,t)\right)^{2}\,dx\,dy.$
Therefore,

$\frac{d}{dt}V(t)=\frac{1}{2}\left[L\alpha^{2}\mu+\alpha^{2}-1\right]\int_{0}^{L}u_{x}^{2}(0,y,t)\,dy\\\
-\frac{\gamma}{2}\int_{0}^{L}(\partial_{x}^{-1}u_{y}(0,y,t))^{2}\,dy-\frac{\gamma}{2}\mu\int_{\Omega}(\partial_{x}^{-1}u_{y}(t))^{2}dxdy\\\
-\frac{3}{2}\mu\int_{\Omega}u_{x}^{2}(t)dxdy+\frac{1}{3}\mu\int_{\Omega}u^{3}(t)dxdy.$

Consequently, straightforward calculations, it follows that

$\frac{d}{dt}V(t)+2\theta
V(t)\leq\frac{1}{2}\left[L\alpha^{2}\mu+\alpha^{2}-1\right]\int_{0}^{L}u_{x}^{2}(0,y,t)\,dy\\\
-\frac{3}{2}\mu\int_{\Omega}u_{x}^{2}(t)dxdy+\theta(1+\theta\mu
L)\int_{\Omega}u^{2}(t)dxdy+\frac{1}{3}\mu\int_{\Omega}u^{3}(t)dxdy,$ (17)

and applying the Poincaré inequality, it yields that

$\frac{d}{dt}V(t)+2\theta
V(t)\leq\frac{1}{2}\left[L\alpha^{2}\mu+\alpha^{2}-1\right]\int_{0}^{L}u_{x}^{2}(0,y,t)\,dy+\left[\theta(1+\mu
L)L^{2}-\frac{3}{2}\mu\right]\int_{\Omega}u_{x}^{2}(t)dxdy+\frac{1}{3}\mu\int_{\Omega}u^{3}(t)dxdy$

From Theorem 2.8, Young’s inequality, Proposition 2.7 and considering initial
data $\lVert u_{0}\rVert_{L^{2}(\Omega)}\leq r$, we deduce that

$\displaystyle\left\lVert u\right\rVert_{L^{3}(\Omega)}^{3}$
$\displaystyle\leq C\left\lVert
u\right\rVert_{L^{2}(\Omega)}^{\frac{5}{2}}\left\lVert
u_{x}\right\rVert_{L^{2}(\Omega)}^{\frac{1}{2}}\leq\frac{1}{4}\left\lVert
u_{x}\right\rVert_{L^{2}(\Omega)}^{2}+\frac{3C^{4/3}}{4}\left\lVert
u\right\rVert_{L^{2}(\Omega)}^{\frac{10}{3}}$
$\displaystyle\leq\frac{1}{4}\left\lVert
u_{x}\right\rVert_{L^{2}(\Omega)}^{2}+\frac{3C^{4/3}}{4}r^{\frac{4}{3}}\left\lVert
u\right\rVert_{L^{2}(\Omega)}^{2}.$

By employing the inequality mentioned above in equation (17), and applying the
Poincaré inequality once more, we obtain:

$\frac{d}{dt}V(t)+2\theta
V(t)\leq\frac{1}{2}\left[L\alpha^{2}\mu+\alpha^{2}-1\right]\int_{0}^{L}u_{x}^{2}(0,y,t)\,dy+\left[\theta(1+\mu
L)L^{2}-\frac{3}{2}\mu+\frac{1}{12}\mu+\frac{1}{4}(Cr)^{\frac{4}{3}}L^{2}\mu\right]\int_{\Omega}u_{x}^{2}(t)dxdy.$

∎

##  3 Two Dissipation Mechanisms: Exponential Stabilization without
Restrictions

In this section, we delve into the findings regarding exponential
stabilization when confronted with a dual dissipation mechanism. This
mechanism operates within the Neumann Boundary Condition and the homogeneous
Dirichlet boundary condition specifically along the $x$-variable. To
accomplish this, we utilize an observability inequality.

###  3.1 Preliminar estimates

In this context, we initiate a sequence of linear estimates that will be
subsequently applied. Initially, we utilize classical semigroup theory in
connection with the linearized system linked to (2),

$\begin{cases}\begin{aligned}
u_{t}(x,y,t)+u_{xxx}(x,y,t)+\gamma\partial_{x}^{-1}u_{yy}(x,y,t)&=0,\end{aligned},\\\
u(0,y,t)=u(L,y,t)=0\,\,\ u_{y}(x,L,t)=\beta u_{x}(x,L,t),\\\
u_{x}(L,y,t)=\alpha u_{x}(0,y,t),\,\,u_{y}(x,0,t)=0,\\\
u(x,y,0)=u_{0}(x,y),\end{cases}$ (18)

Using the notation of the operator $\tilde{A}$ we rewrite the system (18) as
an abstract evolution equation $\frac{du}{dt}=\tilde{A}u$, $u(0)=u_{0}$ where
$\tilde{A}\colon D(\tilde{A})\subset\mathcal{H}\to\mathcal{H}$ is defined by

$\tilde{A}u=-u_{xxx}-\gamma{\partial_{x}^{-1}}u_{yy}$ (19)

on the dense domain given by

$\displaystyle D(\tilde{A}):=\left\\{u\in H_{x}^{3}(\Omega)\cap
X^{2}(\Omega)\colon\,u(0,y)=u(L,y)=0,u_{y}(x,L)=\beta
u_{x}(x,L),\,u_{x}(L,y)=\alpha u_{x}(0,y)\,u_{y}(x,0)=0\right\\},$

with $|\alpha|<1,\beta>0$. In order to ascertain the soundness of the linear
system, we introduce several valuable findings.

######  Lemma 3.13.

The operator $\tilde{A}$ is closed and the adjoint $\tilde{A}^{\ast}\colon
D(\tilde{A}^{\ast})\subset\mathcal{H}\to\mathcal{H}$ is given by
$\tilde{A}^{\ast}u=v_{xxx}+\gamma\left(\partial_{x}^{-1}\right)^{\ast}v_{yy}$
with dense domain

$\displaystyle D(\tilde{A}^{\ast}):=\left\\{v\in H_{x}^{3}(\Omega)\cap
X^{2}(\Omega)\colon v(0,y)=v(L,y)=0,v_{y}(x,L)=-\beta
v_{x}(x,L),v_{x}(0,y)=\alpha v_{x}(L,y),\,\,v_{y}(x,0)=0\right\\}$

with $|\alpha|<1,\beta>0$ and the operator
$\left(\partial_{x}^{-1}\right)^{\ast}$ is defined in Remark 2.4.

###### Proof.

Similarly as the proof of the Lemma 2.3, we consider $u\in D(\tilde{A})$,
$v\in D(\tilde{A}^{\ast})$ and two funcitons $f,g$ such that
$u_{y}(x,y)=f_{x}(x,y)$ and $v_{y}(x,y)=g_{x}(x,y)$ with $f(L,y)=g(0,y)=0$, it
follows that

$\displaystyle-\int_{\Omega}v{\partial_{x}^{-1}}u_{yy}dxdy=-\int_{0}^{L}\left[v(x,y)f(x,y)\right]_{y=0}^{y=L}dx-\int_{0}^{L}\left[g(x,y)u(x,y)\right]_{y=0}^{y=L}dx+\int_{\Omega}g_{y}udxdy=I+\int_{\Omega}u{\partial_{x}^{-1}}v_{yy}dxdy,$

where $I$ is given by the first two terms of the preceding equation. By using
the definition of the domains, it follows that

$\begin{cases}f(x,0)=-\int_{x}^{L}u_{y}(s,0)ds=0\\\
f(x,L)=-\int_{x}^{L}u_{y}(s,L)ds=-\beta\int_{x}^{L}u_{x}(s,L)ds=\beta
u(x,L)\\\ \end{cases}$ (20)

and

$\begin{cases}g(x,0)=\int_{0}^{x}v_{y}(s,0)ds=0\\\
g(x,L)=\int_{0}^{x}v_{y}(s,L)ds=-\beta\int_{0}^{x}v_{x}(s,L)ds=-\beta
v(x,L)\\\ \end{cases}$ (21)

Then, estimating $I$, we have that

$\displaystyle I$ $\displaystyle=-\int_{0}^{L}v(x,L)(\beta
u(x,L))\,dx-\int_{0}^{L}(-\beta v(x,L))u(x,L)\,dx=0$

and

$\displaystyle-\int_{\Omega}$ $\displaystyle
v(x,y){\partial_{x}^{-1}}u_{yy}(x,y)\,dx\,dy=\int_{\Omega}u(x,y){\partial_{x}^{-1}}v_{yy}(x,y)\,dx\,dy.$

Consequently, we can estimate the duality product
$\left\langle\tilde{A}u,v\right\rangle_{\mathcal{H}}$ to obtain

$\displaystyle\left\langle\tilde{A}u,v\right\rangle_{\mathcal{H}}$
$\displaystyle=\left\langle u,\tilde{A}^{\ast}v\right\rangle_{\mathcal{H}}$

Finally, note that $\tilde{A}^{\ast\ast}=\tilde{A}$ then $\tilde{A}$ is
closed. ∎

######  Proposition 3.14.

The operator $\tilde{A}$ is the infinitesimal generator of a $C_{0}$-semigroup
in $\mathcal{H}$.

###### Proof.

Similarly as the proof of the Lemma 2.5, we consider $u\in D(\tilde{A})$ and
call $f_{x}(x,y)=u_{y}(x,y)$ with $f(L,y)=0$, it follows that

$\displaystyle\left\langle\tilde{A}u,u\right\rangle_{\mathcal{H}}$
$\displaystyle=\frac{1}{2}\int_{0}^{L}[u_{x}^{2}(x,y)]_{x=0}^{x=L}dy-\gamma\int_{0}^{L}[f(x,y)u(x,y)]_{y=0}^{y=L}dx+\gamma\int_{\Omega}ff_{x}dxdy.$

From (20), we conclude

$\displaystyle\left\langle\tilde{A}u,u\right\rangle_{\mathcal{H}}=-\frac{(1-\alpha^{2})}{2}\int_{0}^{L}u_{x}^{2}(0,y)dy-\beta\gamma\int_{0}^{L}u^{2}(x,L)dx-\frac{\gamma}{2}\int_{0}^{L}(\partial_{x}^{-1}u_{y})^{2}(0,y)dy.$

Finally, we have that $\left\langle\tilde{A}u,u\right\rangle_{\mathcal{H}}\leq
0.$ Similarly, let $v\in D(\tilde{A}^{\ast})$ and $g_{x}(x,y)=v_{y}(x,y)$ with
$g(0,y)=0$. Then,

$\displaystyle\left\langle
v,\tilde{A}^{\ast}v\right\rangle_{\mathcal{H}}=-\frac{1}{2}\int_{0}^{L}[v_{x}^{2}(x,y)]_{x=0}^{x=L}dy+\gamma\int_{0}^{L}[g(x,y)v(x,y)]_{y=0}^{y=L}dx-\gamma\int_{\Omega}gg_{x}dxdy$

From (21), it yields that $\left\langle
v,\tilde{A}^{\ast}v\right\rangle_{\mathcal{H}}\leq 0$. Thus, from [17,
Corollary 4.4, page 15], the Proposition holds. ∎

Therefore, we are in a position to present the subsequent theorem concerning
the existence of solutions for the Cauchy abstract problem:

######  Theorem 3.15.

For each initial data $u_{0}\in\mathcal{H}$, there exists a unique mild
solution $u\in C\left([0,\infty),\mathcal{H}\right)$ for the system (18).
Moreover, if the initial data $u_{0}\in D(\tilde{A})$ the solutions are
classical such that

$u\in C\left([0,\infty),D(\tilde{A})\right)\cap
C^{1}\left([0,\infty),\mathcal{H}\right).$

###### Proof.

From Proposition 3.14, it follows that $\tilde{A}$ generates a strongly
continuous semigroup of contractions $\\{\tilde{S}(t)\\}_{t\geq 0}$ in
$\mathcal{H}$ (see Corollary 1.4.4 in [17]). ∎

######  Proposition 3.16.

For any mild solution of (18) the energy $E(t)$, defined by (3), is non-
increasing and there exists a constant $C>0$ such that

$\displaystyle\frac{d}{dt}E(t)\leq-C\left[\int_{0}^{L}u_{x}^{2}(0,y,t)\,dy+\int_{0}^{L}u^{2}(x,L)\,dx+\int_{0}^{L}({\partial_{x}^{-1}}u_{y}(0,y,t))^{2}\,dy\right]$
(22)

where $C=C(\alpha,\gamma)$ is given by
$C=\min\left\\{\frac{(1-\alpha^{2})}{2},\gamma,\beta\gamma\right\\}.$

###### Proof.

From proof of Proposition 3.14, (22) holds. ∎

The following proposition provides useful estimates for the mild solutions of
os (18). The first ones are standard energy estimates, while the last one
reveals a Kato smoothing effect.

######  Proposition 3.17.

Let $u_{0}\in\mathcal{H}$, then the mild solution $u$ of system (18) satisfies

$\|u\|_{L^{2}(0,T,H^{1}(\Omega))}\leq C\|u_{0}\|^{2}_{\mathcal{H}}$ (23)

for some positive constant $C>0$.

###### Proof.

Similar that the proof of Proposition 2.9, we multiply the equation (18) by
$xu$ and integrating in $\Omega\times(0,T)$, we obtain that
$\int_{0}^{T}\int_{\Omega}xu(t)\left(u_{t}(t)+u_{xxx}(t)+\gamma{\partial_{x}^{-1}}u_{yy}(t)\right)dxdydt=0.$
Now, estimating term by term, it follows that the first two terms are given by

$\begin{cases}\int_{0}^{T}\int_{\Omega}xu(t)u_{t}(t)dxdydt=\frac{1}{2}\int_{\Omega}xu(T)^{2}dxdy-\frac{1}{2}\int_{\Omega}xu_{0}^{2}dxdy,\\\
\\\
\int_{0}^{T}\int_{\Omega}xu(t)u_{xxx}(t)dxdydt=\frac{3}{2}\int_{0}^{T}\int_{\Omega}u_{x}(t)^{2}dxdydt-\frac{L}{2}\int_{0}^{T}\int_{0}^{L}u_{x}(L,y,t)^{2}dydt\end{cases}$

Hovewer, for the last term, we call $f_{x}(x,y)=u_{y}(x,y)$ with $f(L,y)=0$.
Thus, it follows that

$\gamma\int_{0}^{T}\int_{\Omega}xu(t){\partial_{x}^{-1}}u_{yy}(t)dxdydt=\gamma\int_{0}^{T}\int_{\Omega}xu(t)f_{y}(t)dydxdt\\\
=\gamma\int_{0}^{T}\int_{0}^{L}[xu(t)f(t)]_{y=0}^{y=L}\,dx\,dt+\frac{\gamma}{2}\int_{0}^{T}\int_{\Omega}f^{2}(t)dxdydt.$

From (20), it follows that

$\displaystyle\gamma\int_{0}^{T}\int_{\Omega}xu(t){\partial_{x}^{-1}}u_{yy}(t)dxdydt=\gamma\beta\int_{0}^{T}\int_{0}^{L}xu^{2}(x,L,t)\,dx\,dt+\frac{\gamma}{2}\int_{0}^{T}\int_{\Omega}({\partial_{x}^{-1}}u_{y}(t))^{2}dxdydt.$

Now, adding the above estimates, we deduce that

$\displaystyle\frac{3}{2}\int_{0}^{T}\int_{\Omega}u_{x}^{2}(t)dxdydt\leq\frac{1}{2}\int_{\Omega}xu_{0}^{2}dxdy+\frac{L}{2}\int_{0}^{T}\int_{0}^{L}u_{x}^{2}(L,y,t)dydt.$

In conclusion, following the same methodology employed in the proof of
Proposition 2.9, we can establish the completeness of this proof. ∎

The forthcoming proposition furnishes valuable estimates for the mild
solutions of (18), exemplifying a standard energy estimation estimate.

######  Proposition 3.18.

Let $u_{0}\in\mathcal{H}$, then the mild solution $u$ of system (18) satisfies

$T\|u_{0}\|_{\mathcal{H}}^{2}=\int_{0}^{T}\|u(t)\|_{\mathcal{H}}^{2}dt+2\gamma\beta\int_{0}^{T}\int_{0}^{L}(T-t)u^{2}(x,L,t)dxdt\\\
+(1-\alpha^{2})\int_{0}^{T}\int_{0}^{L}(T-t)(u_{x}(0,y,t))^{2}dydt\\\
+\gamma\int_{0}^{T}\int_{0}^{L}(T-t)({\partial_{x}^{-1}}u_{y}(0,y,t))^{2}dydt$
(24)

###### Proof.

Multiplying (18) by $(T-t)u$ and integrating in $\Omega\times(0,T)$, we obtain
that

$\displaystyle\int_{0}^{T}\int_{\Omega}(T-t)u(t)\left(u_{t}(t)+u_{xxx}(t)+\gamma{\partial_{x}^{-1}}u_{yy}(t)\right)dxdydt=0.$

Now, estimating term by term, it follows that

$\displaystyle\int_{0}^{T}\int_{\Omega}(T-t)u(t)u_{t}(t)dxdydt=-\frac{T}{2}\int_{\Omega}u_{0}^{2}dxdy+\frac{1}{2}\int_{0}^{T}\int_{\Omega}u^{2}(t)dxdydt$

and the another terms are given by

$\displaystyle\int_{0}^{T}\int_{\Omega}(T-t)u(t)u_{xxx}(t)dxdydt=\frac{(1-\alpha^{2})}{2}\int_{0}^{T}\int_{0}^{L}(T-t)(u_{x}(0,y,t))^{2}dydt$

On the other hand, note that calling $f_{x}(x,y)=u_{y}(x,y)$ with $f(L,y)=0$
and from (20), it follows that

$\displaystyle\gamma\int_{0}^{T}\int_{\Omega}(T-t)u(t){\partial_{x}^{-1}}u_{yy}(t)dxdydt=\gamma\beta\int_{0}^{T}\int_{0}^{L}(T-t)u^{2}(x,L,t)dxdt+\frac{\gamma}{2}\int_{0}^{T}\int_{0}^{L}(T-t)({\partial_{x}^{-1}}u_{y}(0,y,t))^{2}dydt.$

Now, adding the above estimates, we deduce (24). ∎

###  3.2 Exponential Stability - Linear System

We are now prepared to establish the exponential stability of the linearized
system associated with (2). Consequently, the central result in this context
is as follows:

######  Theorem 3.19.

There exist two constant $C_{0},\mu_{0}>0$ such that for any
$u_{0}\in\mathcal{H}$, the solution of $\eqref{eq:KP_1}$ satisfies

$\|u(t)\|_{\mathcal{H}}\leq
C_{0}e^{-\mu_{0}t}\|u_{0}\|_{\mathcal{H}},\quad\forall t\geq 0.$

###### Proof.

By leveraging energy dissipation, specifically the estimate (22), in
conjunction with a classical argument, our task simplifies to proving the
following _observability inequality_ :

$\displaystyle\|u_{0}\|_{\mathcal{H}}\leq
C\left[\int_{0}^{T}\int_{0}^{L}u_{x}^{2}(0,y,t)dydt+\int_{0}^{T}\int_{0}^{L}u^{2}(x,L,t)dxdt+\int_{0}^{T}\int_{0}^{L}({\partial_{x}^{-1}}u_{y}(0,y,t))^{2}dydt\right]$
(25)

where $u$ denotes the solution of (18).

To establish the observability inequality (25), we approach it in three
distinct steps:

### STEP 1. (Compactness-Uniqueness Argument)

We argue by contradiction, applying the compactness uniqueness argument due to
E. Zuazua (see [12]). If (25) is false, then we may find a sequence
$\\{u^{n}_{0}\\}_{n\in\mathbb{N}}$ in $\mathcal{H}$ such that
$\|u_{0}^{n}\|_{\mathcal{H}}=1$ and

$\displaystyle\|u_{0}^{n}\|_{\mathcal{H}}\geq
n\left[\int_{0}^{T}\int_{0}^{L}(u^{n}_{x})^{2}(0,y,t)dydt+\int_{0}^{T}\int_{0}^{L}(u^{n})^{2}(x,L,t)dxdt+\int_{0}^{T}\int_{0}^{L}({\partial_{x}^{-1}}u^{n}_{y}(0,y,t))^{2}dydt\right].$
(26)

On the other hand, it follows from (23) that
$\\{u^{n}\\}_{n\in\mathbb{N}}=\\{\tilde{S}(\cdot)u_{0}^{n}\\}_{n\in\mathbb{N}}$
is bounded in $L^{2}(0,T;\mathcal{H})$. By (18),
$\\{u^{n}_{t}\\}_{n\in\mathbb{N}}$ is bounded in $L^{2}(0,T;H^{-2}(\Omega))$,
Indeed, by (18)m $u_{t}^{n}=-u_{xxx}^{n}-\gamma\partial_{x}^{-1}u_{yy}^{n}$.
As $\left\\{u^{n}\right\\}_{n}$ is bounded in $L^{2}(0,T;H^{1}(\Omega))$
implies that $\left\\{u_{xxx}^{n}\right\\}_{n}$ is bounded in
$L^{2}(0,T;H^{-2}(\Omega))$. Moreover, we have that
$\left\\{u_{x}^{n}\right\\}_{n}$ and $\left\\{u_{y}^{n}\right\\}_{n}$ are
bounded in $L^{2}(0,T;L^{2}(\Omega))$. Define $f$ such that
$u_{y}^{n}=f_{x}^{n}$ wich implies $\partial_{x}^{-1}u_{yy}^{n}=f_{y}^{n}.$
Therefore, using Gagliardo-Niremberg anisotropic, Theorem 2.8, it yields that
$\lVert\partial_{x}^{-1}u_{yy}^{n}\rVert_{L^{2}}=\lVert
f_{y}^{n}\rVert_{L^{2}}\leq C\lVert f_{x}^{n}\rVert_{L^{2}}=C\lVert
u_{y}^{n}\rVert_{L^{2}}.$ Recognizing that $H_{0}^{1}\subset L^{2}\subset
H^{-1}$, we select $\xi$ with sufficient regularity to ensure that.

$\displaystyle\left\lvert\left\langle\partial_{x}^{-1}u_{yy}^{n},\xi\right\rangle_{H^{-1},H_{0}^{1}}\right\rvert=\left\lvert\left\langle
f_{y}^{n},\xi\right\rangle_{H^{-1},H_{0}^{1}}\right\rvert=\left\lvert\left\langle\xi,f_{y}^{n}\right\rangle_{L^{2},L^{2}}\right\rvert\leq\lVert\xi\rVert_{L^{2}}\cdot\lVert
f_{y}^{n}\rVert_{L^{2}}\leq C\lVert\xi\rVert_{L^{2}}\cdot\lVert
f_{x}^{n}\rVert_{L^{2}}=C\lVert\xi\rVert_{L^{2}}\cdot\lVert
u_{y}^{n}\rVert_{L^{2}}.$

Then,

$\displaystyle\lVert\partial_{x}^{-1}u_{yy}^{n}\rVert_{L^{2}(0,T;H^{-1})}^{2}\leq\int_{0}^{T}\lVert\partial_{x}^{-1}u_{yy}^{n}(\cdot,\cdot,t)\rVert_{H^{-1}(\Omega)}^{2}\,dt\leq
C^{2}\int_{0}^{T}\lVert u_{y}^{n}(\cdot,\cdot,t)\rVert_{L^{2}(\Omega)}^{2}\leq
C^{2}\lVert u^{n}\rVert_{L^{2}(0,T;H^{1}(\Omega))}.$

Hence, applying Aubin’s lemma, we see that a subsequence of
$\\{u^{n}\\}_{n\in\mathbb{N}}$, again denoted by
$\\{u^{n}\\}_{n\in\mathbb{N}}$, converges strongly in $L^{2}(0,T;\mathcal{H})$
towards some $u$. Moreover, using (24) and (26), we see that
$\\{u_{0}^{n}\\}_{n\in\mathbb{N}}$ is a Cauchy sequence in $\mathcal{H}$,
hence for some $u_{0}\in\mathcal{H}$. It means that $u_{0}^{n}\rightarrow
u_{0}$ in $\mathcal{H}$. Clearly, $u=\tilde{S}(\cdot)u_{0}$, and we infer from
(26) that

$u_{x}(0,y,t)=u(x,L,t)=\partial_{x}^{-1}u_{y}(0,y,t)=0\quad\text{for all
$x,y\in[0,L]$}$ (27)

and that $\|u_{0}\|_{\mathcal{H}}=1$.

### STEP 2. (Reduction to a Spectral Problem)

We eliminate the time in following the procedure described in [18] for the
Korteweg–de Vries equation

######  Lemma 3.20.

For any $T>0$, let $N_{T}$ denote the space of all the initial states,
$u_{0}\in\mathcal{H}$ for which the solution $u(t)=S(t)u_{0}$ of (18)
satisfies (27). Then $N_{T}=\\{0\\}$.

###### Proof.

The proof uses the same arguments as those given in [18, Lemma 3.4],
Therefore, if $N_{T}\neq\emptyset$, the map $u_{0}\in
N_{T}\mapsto\tilde{A}(N_{T})\subset CN_{T}$ (where $CN_{T}$ denote the
complexification of $N_{T}$) has (at least) one eigenvalue; hence, there exist
$\lambda\in\mathcal{C}$ and $u_{0}\in H^{3}_{x}(\Omega)\cap H^{2}_{y}(\Omega)$
such that

$\begin{cases}\lambda
u_{0}(x,y)+u_{xxx,0}(x,y)+\gamma\partial_{x}^{-1}u_{yy,0}(x,y)=0,\\\
u_{0}(0,y)=u_{0}(L,y)=0,\quad u_{y,0}(x,L)=\beta u_{x,0}(x,L),\\\
u_{x,0}(L,y)=\alpha u_{x,0}(0,y)=0,\quad u_{y,0}(x,0)=0\\\
u_{x,0}(0,y)=u_{0}(x,L)=\partial_{x}^{-1}u_{y,0}(0,y)=0\end{cases}$ (28)

with $(x,y)\in\Omega$. To conclude the proof of the Lemma, we prove that this
does not hold. ∎

In order to obtain the contradiction, it remains to prove that a duple
$(\lambda,u_{0})$ as above does not exist.

### Step 3. (No Nontrivial Solution for the Spectral Problem)

######  Lemma 3.21.

Let $\lambda\in\mathbb{C}$ and $u_{0}\in H^{3}_{x}(\Omega)\cap
H^{2}_{y}(\Omega)$ fulfilling (28). Then $u_{0}=0$.

###### Proof.

Let us introduce the notation $u(x,y)=u_{0}(x,y)$ and define
$v_{x}(x,y)=u_{y,0}(x,y)$ such that $v(L,y)=0$. Thus, we can rewrite our
system (28) as:

$\begin{cases}-u_{y}(x,y)+v_{x}(x,y)=0\\\ \gamma
v_{y}(x,y)+u_{xxx}(x,y)=-\lambda u(x,y)\end{cases}$ (29)

with boundary conditions,

$\begin{cases}u(0,y)=u(L,y)=0,\\\ u(x,L)=0,\,u_{y}(x,L)=\beta u_{x}(x,L)=0,\\\
u_{x}(L,y)=\alpha u_{x}(0,y)=0,\,u_{y}(x,0)=0,\\\ v(0,y)=0.\end{cases}$

The existence of a solution is guaranteed by Lemma 3.20. Now, we focus on
establishing uniqueness for system (29). We assume that both $(u_{1},v_{1})$
and $(u_{2},v_{2})$ are solutions and define $u:=u_{1}-u_{2}$ and
$v:=v_{1}-v_{2}$. We analyze two cases: one where $\lambda=0$ and another
where $\lambda\neq 0$. When $\lambda=0$, we have that
$-u_{y}(x,y)+v_{x}(x,y)=0$ and $\gamma v_{y}(x,y)+u_{xxx}(x,y)=0.$ We use an
approach _via_ energy functional. Let us define

$\tilde{E}(u,v)(y):=\frac{1}{2}\int_{0}^{L}\left(\gamma
v^{2}(x,y)+u_{x}^{2}(x,y)\right)dx,\quad y\in(0,L),$

the energy associated to the system (29). Note that, by multiplying the first
and second equations of (29) by $u_{xx}(x,y)$ and by $v(x,y)$, respectively,
integrating over $x$ in the interval $(0,L)$, and summing the results, we can
conclude that:

$\displaystyle\int_{0}^{L}u_{xx}(x,y)\left(-u_{y}(x,y)+v_{x}(x,y)\right)+v(x,y)\left(\gamma
v_{y}(x,y)+u_{xxx}(x,y)\right)\,dx=0.$

Straighforward calculations and the boundary conditions imply that

$\frac{\partial}{\partial
y}\tilde{E}(u,v)=\left[u_{x}(x,y)u_{y}(x,y)-u_{xx}(x,y)v(x,y)\right]_{x=0}^{x=L}=0.$

Then,

$\tilde{E}(u,v)(y)=\frac{1}{2}\int_{0}^{L}v^{2}(x,y)+u_{x}^{2}(x,y)\,dx=C,\quad
y\in(0,L).$

Note that, by definition
$v(x,y)=\partial_{x}^{-1}u_{y}(x,y)=-\int_{x}^{L}u_{y}(s,y)\,ds.$ Thus, by the
boundary condition ($u_{y}(x,L)=0$)

$v(x,L)=-\int_{x}^{L}u_{y}(s,L)\,ds=0.$

Since $v(x,L)=0$ and $u_{x}(x,L)=0$ it follows that $C=0$. Therefore,

$v(x,y)=0\quad\text{and}\quad u_{x}(x,y)=0,\quad\forall x,y\in(0,L).$

In particular, it follows that $v_{1}=v_{2}$ and notice that when
$u_{x}(x,y)=0,$ it implies that $u(x,y)=g(y)$. If we differentiate with
respect to $y,$ we obtain $u_{y}(x,y)=g_{y}(y)$ for all $x,y\in(0,L).$ Then,
we have that

$0=\partial_{x}^{-1}u_{y}(x,y)=\int_{x}^{L}u_{y}(s,y)\,ds=\int_{x}^{L}g_{y}(y)\,ds=(L-x)g_{y}(y).$

Consolidating all information, we deduce that $g_{y}(y)=0$, implying $u(x,y)$
is constant. This follows directly from the boundary conditions $u(x,y)=0$,
which implies $u_{1}=u_{2}$.

In the next step, let us consider the case where $\lambda\neq 0.$ Observe that
by multiplying the second equation in (29) by $u(x,y,t),$ employing the
variable change defined in (29), and integrating by parts over the domain
$\Omega,$ we derive the following result:

$\displaystyle-\int_{\Omega}\lambda u^{2}(t)dxdy$
$\displaystyle=\gamma\int_{\Omega}u(t)v_{y}(t)dxdy+\int_{\Omega}u(t)u_{xxx}(t)dxdy=-\frac{\gamma}{2}\int_{\Omega}\left(v^{2}(t)\right)_{x}dxdy-\frac{1}{2}\int_{0}^{L}\left[u_{x}^{2}(x,y,t)\right]_{x=0}^{x=L}dy$
$\displaystyle=-\frac{\gamma}{2}\int_{0}^{L}\left[v^{2}(x,y,t)\right]_{x=0}^{x=L}dy=0.$

This implies $u=0$, consequently, $u_{1}=u_{2}$. Similarly, from the equation,
we conclude $v=0$, meaning $v_{1}=v_{2}$. Utilizing system uniqueness and
recognizing zero as a valid solution, we deduce $u_{0}=0$. ∎

With these Lemmas, we find $u_{0}=0,$ leading to a contradiction as
$\|u_{0}\|_{L^{2}(\Omega)}=1$. Thus, we have successfully proven exponential
stabilization. ∎

###  3.3 The Nonlinear System

Now, we consider the well-posedness of the system (2), allowing us to explore
the nonlinear model. Similar to our treatment of system (6), we define $u$ as
a mild solution of (2) if $u$ belongs to $C([0,T];L^{2}(\Omega))\cap
L^{2}(0,T;\Omega)$ for $T>0$ and satisfies the Duhamel formula.

$\displaystyle u(t)$
$\displaystyle=\tilde{S}(t)u_{0}-\int_{0}^{t}\tilde{S}(t-s)u(s)u_{x}(s)ds,\quad
t\in[0,T],$

where $\\{\tilde{S}(t)\\}_{t\geq 0}$ is the semigroup generated by the
operator $\tilde{A}$ given by (19). A global mild solution of (2) will be a
function $u$ defined in $[0,\infty)$ whose restriction to every bounded
interval $[0,T]$ is a mild solution of (2). The proof of the following results
follows a similar approach to that of Theorem 2.12, and therefore, we will
omit it.

######  Theorem 3.22.

Consider $L,T>0$. Then, there exists $r>0$ such that for all initial data
$u_{0}\in L^{2}(\Omega)$ satisfying $\|u_{0}\|_{L^{2}(\Omega)}\leq r,$ there
exists a unique $u\in\mathcal{B}_{X}$ mild solution of (18) such that $\lVert
u\rVert_{\mathcal{B}_{X}}\leq C\lVert u_{0}\rVert_{L^{2}(\Omega)}.$
Furthermore, it is feasible to derive the estimate

$\left\lVert u\right\rVert_{C([0,T];L^{2}(\Omega))}\leq C\left(\left\lVert
u_{0}\right\rVert_{L^{2}(\Omega)}+\left\lVert
uu_{x}\right\rVert_{L^{1}(0,T;L^{2}(\Omega)}\right).$

In the following, we provide an overview of the proof for one of the key
results presented in this paper:

###### Proof of Theorem 1.2.

Let $u_{0}\in L^{2}(\Omega)$ such that $\lVert u_{0}\rVert_{L^{2}(\Omega)}\leq
r$ with $r$ to be chosen later. Observe that the solution of (2) can be
decomposed as $u=u^{1}+u^{2}$, where $u^{1}$ is solution of

$\begin{cases}{u}^{1}_{t}(x,y,t)+{u}^{1}_{xxx}(x,y,t)+\gamma\partial_{x}^{-1}{u}^{1}_{yy}(x,y,t)=0,\\\
{u}^{1}(0,y,t)={u}^{1}(L,y,t)=0\,\,\
{u}^{1}_{y}(x,L,t)=\beta{u}^{1}_{x}(x,L,t),\\\
{u}^{1}_{x}(L,y,t)=\alpha{u}^{1}_{x}(0,y,t),\,\,{u}^{1}_{y}(x,0,t)=0,\\\
{u}^{1}(x,y,0)={u}_{0}(x,y),\end{cases}$

and $u^{2}$ is solution of

$\begin{cases}{u}^{2}_{t}(x,y,t)+{u}^{2}_{xxx}(x,y,t)+\gamma\partial_{x}^{-1}{u}^{2}_{yy}(x,y,t)=-u(x,y,t)u_{x}(x,y,t),\\\
{u}^{2}(0,y,t)={u}^{2}(L,y,t)=0\,\,\
{u}^{2}_{y}(x,L,t)=\beta{u}^{2}_{x}(x,L,t),\\\
{u}^{2}_{x}(L,y,t)=\alpha{u}^{2}_{x}(0,y,t),\,\,{u}^{2}_{y}(x,0,t)=0,\\\
{u}^{1}(x,y,0)=0.\end{cases}$

Therefore, from Lemma 2.11 and Theorems 3.19, 3.22, we get, for $0<\delta<1$,
the following

$\displaystyle\lVert u(T)\rVert_{L^{2}(\Omega)}$
$\displaystyle\leq\delta\lVert u_{0}\rVert_{L^{2}(\Omega)}+C\lVert
uu_{x}\rVert_{L^{1}(0,T;L^{2}(\Omega))}$ (30) $\displaystyle\leq\delta\lVert
u_{0}\rVert_{L^{2}(\Omega)}+CT^{\frac{1}{4}}\lVert
u\rVert_{L^{2}(0,T;H^{1}(\Omega))}^{2}.$

Now, we establish an apriori estimate for the last term. Observe that
following the same steps of the Kato smoothing effect (23) and taking into
account the nonlinearity, we have

$\int_{0}^{T}\|u(t)\|_{H^{1}(\Omega)}^{2}dt\leq\frac{C(2\alpha^{2}+1)L}{3}\|u_{0}\|^{2}_{\mathcal{H}}+\frac{2}{9}\int_{0}^{T}\int_{\Omega}u^{3}(t)dxdydt.$
(31)

Note that the last term can be controlled by

$\displaystyle\int_{0}^{T}\int_{\Omega}u^{3}dxdydt$
$\displaystyle\leq\int_{0}^{T}\|u\|_{H^{1}(\Omega)}\|u\|_{L^{2}(\Omega)}^{2}dt\leq
C\left\|u_{0}\right\|_{L^{2}(\Omega)}^{2}\sqrt{T}\|u\|_{L^{2}\left(0,T;H^{1}(\Omega)\right)}.$
(32)

Substituting (32) in (31) and by using the Young’s inequality, it yields that

$\displaystyle\|u\|_{L^{2}\left(0,T;H^{1}(\Omega)\right)}^{2}\leq\frac{C(2\alpha^{2}+1)L}{3}\left\|u_{0}\right\|_{L^{2}(\Omega)}^{2}+\frac{2CT}{81}\left\|u_{0}\right\|_{L^{2}(\Omega)}^{4}+\frac{1}{2}\|u\|_{L^{2}\left(0,T;H^{1}(\Omega)\right)}^{2}.$
(33)

Gathering (30) and (33), we obtain that $\lVert
u(T)\rVert_{L^{2}(\Omega)}\leq\lVert
u_{0}\rVert_{L^{2}(\Omega)}\left(\delta+r+\mathcal{C}r^{3}\right).$ for some
positive constant $\mathcal{C}$. Given $\varepsilon>0$ such that
$\delta+\varepsilon<1$, we can take $r>0$ satisfying
$r+r^{3}<\frac{\varepsilon}{\mathcal{C}}$ to obtain $\lVert
u(T)\rVert_{L^{2}(\Omega)}\leq\gamma\lVert u_{0}\rVert_{L^{2}(\Omega)},$
$0<\gamma<1.$ Finally, using a boot-strap argument similar as the linear case,
the result is acquired. ∎

##  4 Acknowledge

Muñoz was supported by FACEPE Ph.D. scholarship number IBPG-0909-1.01/20 and
conducted this research as part of their Ph.D. thesis at the Department of
Mathematics, UFPE. The second author conducted this research during their
visit to Universidad Nacional de Colombia Sede Manizales and expresses
gratitude for their warm hospitality.

## References

  * [1] F.D. Araruna, R.A. Capistrano–Filho, and G.G. Doronin, Energy decay for the modified Kawahara equation posed in a bounded domain, Journal of Mathematical Analysis and Applications 385:2, 743–756 (2012).
  * [2] O. V. Besov, V. P. Il’in and S. M. Nikol’skii,  Integral Representations of Functions and Imbedding Theorems, Vol. I., New York-Toronto, Ont.-London, 1978.
  * [3] J. Bourgain, On the compactness of the support of solutions of dispersive equations, Int. Math. Res. Notices, 9 (1997), 437-447.
  * [4] H. Brezis, Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York, 2011.
  * [5] R. A. Capistrano–Filho, E. Cerpa, and F. A. Gallego, Rapid exponential stabilization of a Boussinesq system of KdV–KdV Type, Communications in Contemporary Mathematics, Vol. 25, No. 03, 2150111 (2023).
  * [6] R. A. Capistrano–Filho and F. A. Gallego, Asymptotic behavior of Boussinesq system of KdV–KdV type, Journal of Differential Equations 265:6, 2341–2374 (2018).
  * [7] R. A. Capistrano–Filho, V. H. Gonzalez and J.R, Muñoz, Stabilization of the Kawahara – Kadomtsev Petviashvili equation with time-delayed feedback, Proceedings of the Royal Society of Edinburgh,pag 1-27 DOI:10.1017/prm.2023.92
  * [8] R. P. de Moura and A. C. Nascimento and G. N. Santos, _O_ n the stabilization for the high-order Kadomtsev-Petviashvili and the Zakharov-Kuznetsov equations with localized damping, Evolution Equations and Control Theory, 11, 711–727 (2022).
  * [9] D. A. Gomes and M. Panthee, Exponential Energy Decay for the Kadomtsev-Petviashvili (KP-II) equation, São Paulo Journal of Mathematical Sciences 5:2, 135–148 (2011).
  * [10] C. A. Jones and P. H. Roberts, Motions in a Bose condensate, IV: Axisymmetric solitary waves, J. Phys. A Math. Gen. 15 (1982), 2599 – 2619. 353
  * [11] J.-L. Lions, _E_ xact controllability, stabilization and perturbations for distributed systems. SIAM Rev;30:1, 1–68 (1988).
  * [12] J.-L. Lions, Controlabilite exacte, perturbations et stabilisation de systemes distribues. Tome 1, in: Recherches en Mathematiques Appliquees [Research in Applied Mathematics], vol. 8, Masson, Paris, 1988, Controlabilite exacte. [Exact controllability], With appendices by E. Zuazua, C. Bardos, G. Lebeau and J. Rauch.
  * [13] J.-L. Lions, Controlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués. Vol. 22. Elsevier-Masson, 1990.
  * [14] G. Menzala, C. Vasconcellos and E. Zuazua, Stabilization of the Korteweg-De Vries equation with localized damping., Quarterly of Applied Mathematics LX, 111–129.
  * [15] A. F. Pazoto and L. Rosier, Stabilization of a Boussinesq system of KdV–KdV type, Systems & Control Letters 57:8, 595–601 (2008).
  * [16] M. Panthee. Unique continuation property for the Kadomtsev-Petviashvili (KP-II) equation. Electronic Journal of Differential Equations (EJDE), 2005, vol. 2005, p. Paper No. 59, 12.
  * [17] A. Pazy, Semigroups os Linear Operators and Applications to Partial Differential Equations, Springer Verlag, New York, USA, 1983.
  * [18] L. Rosier, Exact boundary controllability for the Korteweg–de Vries equation on a bounded domain, ESAIM Control Optim. Calc. Var. 2 (1997) 33–55 (electronic).
  * [19] B. B. Kadomtsev and V. I. Petviashvili, On the stability of solitary waves in weakly dispersive media, Sov. Phys. Dokl., 15, 539–549 (1970).