1.1 Problem setting

There is an important advance in control theory to understand how the damping mechanism acts in the energy of systems governed by a partial differential equation. In particular, exponential stability for dispersive equations related to water waves posed on bounded domains has been intensively studied. For example, it is well known that the KdV equation [Reference Menzala, Vasconcellos and Zuazua23], Boussinesq system of KdV-KdV type [Reference Pazoto and Rosier24], Kawahara equation [Reference Araruna, Capistrano–Filho and Doronin1], and others are exponentially stable using the Compactness-Uniqueness developed by J.L. Lions [Reference Lions22]. Other results, such as those presented in [Reference Capistrano–Filho, Cerpa and Gallego5] and [Reference Capistrano–Filho and Gallego7] are obtained by using Urquiza's and Backstepping approach. All these results use damping mechanisms in the equation or the boundary as a control.

Recently, in [Reference Capistrano–Filho and Gonzalez Martinez8, Reference Chentouf9], the authors obtained exponential decay for a fifth-order KdV type equation via the Compactness-Uniqueness argument and Lyapunov approach. Additionally to that, in [Reference Gomes and Panthee11] and [Reference de Moura, Nascimento and Santos10], exponential decay for the KP-II and K-KP-II was shownFootnote ². In both works, the authors can prove regularity and well-posedness for these equations and show that the energy associated with these equations decays exponentially in the presence of a damping term acting on the equation.

As we can see in these articles, in the mathematical context, there is interest in studying the asymptotic behavior of solutions of equation (1.4). Additionally, as pointed out, the model under consideration in this article has importance in the context of the dispersive equations as well as, physical motivation. So, motivated by [Reference Capistrano–Filho and Gonzalez Martinez8–Reference Gomes and Panthee11] we will analyse the qualitative properties of the initial-boundary value problem for the K-KP-II equation posed on a bounded domain $\Omega =(0,\,L)\times (0,\,L)\subset \mathbb {R}^2$ with localized damping and delay terms

(1.5)$$\begin{align} \begin{cases} \begin{aligned} & \partial_tu(x,y,t) + \alpha \partial^3_xu(x,y,t) + \beta \partial^5_xu(x,y,t)\\ & + \gamma\partial_x^{{-}1}\partial^2_y u(x,y,t) + \frac{1}{2}\partial_x(u^2(x,y,t))\\ & + a(x,y)u(x,y,t) + b(x,y)u(x,y,t-h)=0, \end{aligned} & (x,y)\in \Omega,\ t>0,\\ u(0,y,t)=u(L,y,t)=0 , & y\in(0,L),\ t\in(0,T), \\ \partial_xu(L,y,t) = \partial_xu(0,y,t)=\partial^2_xu(L,y,t)=0, & y\in(0,L),\ t\in(0,T), \\ u(x,L,t)=u(x,0,t)=0, & x\in(0,L),\ t\in(0,T),\\ u(x,y,0)=u_0(x,y), \quad u(x,y,t)= z_0(x,y,t), & (x,y)\in\Omega,\ t\in({-}h,0). \end{cases} \end{align}$$

Here $h>0$ is the time delay, $\alpha >0$, $\gamma >0$ and $\beta <0$ are real constants. Additionally, define the operator $\partial _x^{-1}:=\partial _x^{-1} \varphi (x,\,y,\,t)= \psi (x,\,y,\,t)$ such that $\psi (L,\,y,\,t)=0$ and $\partial _x\psi (x,\,y,\,t)=\varphi (x,\,y,\,t)$Footnote ³ and, for our purpose, let us consider the following assumption.

Our propose here is to present, for the first time, the K-KP-II system not only with a damping mechanism $a(x,\,y)u$, which plays the role of a feedback-damping mechanism (see e.g. [Reference de Moura, Nascimento and Santos10]) but also with an anti-damping, that is, some feedback such that our system does not have decreasing energy. In this context, we would like to prove that the energy associated with the solutions of system (1.5)

(1.6)$$\begin{align} \begin{aligned} E_{u}(t) & =\frac{1}{2} \int_0^L\int_0^L u^2(x,y,t) \,{\rm d}x\,{\rm d}y \\ & \quad+\frac{h}{2} \int_0^L\int_0^L\int_0^1 b(x,y)u^2(x,y,t-\rho h) \,{\rm d}\rho\,{\rm d}x\,{\rm d}y. \end{aligned} \end{align}$$

decays exponentially. Precisely, we want to answer the following question:

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

1.2 Notation and main results

Before presenting answers to this 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

\[ X^k(\Omega):= \left\{ \begin{array} {@{}l@{}} \varphi \in H^k(\Omega) \colon {\partial_x^{{-}1}} \varphi(x,y)= \psi(x,y)\in H^k(\Omega)\text{ such that, }\\ \psi(L,y)=0\text{ and }\partial_x\psi(x,y)=\varphi(x,y) \end{array} \!\right\} \]

endowed with the norm $\left \lVert \varphi \right \rVert _{X^k(\Omega )}^2 = \left \lVert \varphi \right \rVert _{H^k(\Omega )}^2 + \left \lVert {\partial _x^{-1}} \varphi \right \rVert _{H^k(\Omega )}^2.$ We also define the normed space $H_x^k(\Omega )$,

\[ H_x^k(\Omega):= \left\lbrace \varphi \colon \partial_x^j \varphi \in L^2(\Omega),\ \text{for } 0\leq j \leq k \right\rbrace \]

with the norm $\left \lVert \varphi \right \rVert _{H_x^k(\Omega )}^2 =\sum _{j=0}^k \left \lVert \partial _x^j \varphi \right \rVert _{L^2(\Omega )}^2$ and the space

\[ X_{x}^{k}(\Omega):=\left\{ \begin{array}{@{}l@{}} \varphi \in H_x^k(\Omega) \colon {\partial_x^{{-}1}} \varphi(x,y)= \psi(x,y)\in H_x^k(\Omega)\text{ such that }\\ \psi(L,y)=0\text{ and }\partial_x\psi(x,y)=\varphi(x,y) \end{array} \!\right\} \]

with $\left \lVert \varphi \right \rVert _{X_x^k(\Omega )}^2 = \left \lVert \varphi \right \rVert _{H_x^k(\Omega )}^2 + \left \lVert {\partial _x^{-1}} \varphi \right \rVert _{H_x^k(\Omega )}^2.$ Finally, $H_{x0}^k(\Omega )$ will denote the closure of $C_0^\infty (\Omega )$ in $H_x^k(\Omega )$.

The next result will be used repeatedly throughout the article:

Theorem 1.2 [Reference Besov, Il'in and Nikol'skii2, Theorem 15.7]

Let $\beta$ and $\alpha ^{(j)},$ for $j = 1,\,\ldots,\, N,$ 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

\[ \sum_{ j = 1 }^N \mu_j = 1, \quad \frac{1}{q} \leq \sum_{j = 1}^N \frac{\mu_j}{p^{(j)}},\quad \text{and} \quad \beta - \frac{1}{q} = \sum_{j=1}^N \mu_j\left(\alpha^{(j)} - \frac{1}{p^{(j)}}\right). \]

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 $D_{x_i}^{\beta _i} = \frac {\partial ^{\beta _i}}{\partial x_i^{k_i}}$

From now on, the constants of system (1.5) satisfy $\alpha >0,\, \beta <0$, and $\gamma ^2=1$. With this, the first result of this work ensures that without a restrictive assumption on the length $L$ of the domain and with the weight of the delayed feedback small enough energy (1.6) associated with the solution of system (1.5) is locally stable.

Theorem 1.3 Optimal local stabilization

Assume that the functions $a(x,\,y), b(x,\,y)$ satisfy the conditions given in assumption 1.1. Let $L>0,$ $\xi >1,$ $0<\mu <1$ and $T_0$ given by

(1.7)$$\begin{align} T_0 = \frac{1}{2\theta} \ln \left( \frac{2\xi \kappa }{\mu}\right) +1, \end{align}$$

with $\theta = \frac {3\alpha \eta }{(1+2\eta L)L^2},\, \kappa = 1+\max \left \lbrace {2\eta L,\, \frac {\sigma }{\xi }}\right \rbrace$ and $\eta \in (0,\, \frac {\xi -1}{2L(1+2\xi )})$ satisfying

\[ \frac{2\alpha \eta}{(2+2\eta L)L^2}=\frac{\sigma}{2h(\xi+\sigma)} \]

where $\sigma =\xi -1 -2L\eta (1+2\xi )$. Let $T_{\min }>0$ given by

\[ T_{\min}:={-}\frac{1}{\nu} \ln\left( \frac{\mu}{2} \right) +\left( \frac{2\lVert b \rVert_\infty}{\nu}+1 \right)T_0, \hbox{ with } \nu=\frac{1}{T_0}\ln\left(\frac{1}{(\mu+\varepsilon)}\right). \]

Then, there exists $\delta >0$, $r>0$, $C>0$ and $\gamma$, depending on $T_{\min },\, \xi,\, L,\, h$, such that if $\lVert b \rVert _\infty \leq \delta$, then for every $(u_0,\,z_0) \in \mathcal {H} = L^2(\Omega )\times L^2(\Omega \times (0,\,1))$ satisfying $\lVert (u_0,\, z_0) \rVert _{\mathcal {H}}\leq r$, the energy of system (1.5) satisfies

\[ E_u(t)\leq Ce^{-\gamma t}E_u(0), \hbox{ for all } t>T_{\min}. \]

Next, following the ideas presented in [Reference Capistrano–Filho and Gonzalez Martinez8], we obtain some stability properties about the next system, called $\mu _i$–system. Note that if we choose $a(x,\,y)= \mu _1 a(x,\,y)$ and $b(x,\,y)=\mu _2 a(x,\,y)$ in (1.5), where $\mu _1$ and $\mu _2$ are real constants we obtain the system

(1.8)$$\begin{align} \begin{cases} \begin{aligned} & \partial_tu(x,y,t) + \alpha \partial_{x}^3u(x,y,t) + \beta\partial_{x}^5u(x,y,t) \\ & + \gamma\partial_x^{{-}1} \partial^2_yu(x,y,t)+ \frac{1}{2}\partial_x(u^2(x,y,t)) \\ & + a(x,y)\left(\mu_1u(x,y,t) + \mu_2 u(x,y,t-h)\right)=0, \end{aligned} & (x,y,t)\in \Omega\times\mathbb{R}^+,\\ u(0,y,t)=u(L,y,t)=0, & y\in(0,L), \ t\in(0,T), \\ \partial_x u(L,y,t) = \partial_xu(0,y,t)=\partial^2_xu(L,y,t)=0, & y\in(0,L), \ t\in(0,T), \\ u(x,L,t)=u(x,0,t)=0, & x\in(0,L), \ t\in(0,T),\\ u(x,y,0)=u_0(x,y),\quad u(x,y,t)= z_0(x,y,t), & (x,y)\in \Omega, \ t\in({-}h,0). \end{cases} \end{align}$$

Here, $\mu _1>\mu _2$ are positive real number and $a(x,\,y)$ satisfies assumption 1.1. We define the total energy associated to (1.8)

(1.9)$$\begin{align} \begin{aligned} E_u(t) & =\frac{1}{2}\int_0^L\int_0^L u^2(x,y,t)\,{\rm d}x\,{\rm d}y \\ & \quad+\frac{\xi}{2}\int_0^L\int_0^L \int_0^1 a(x,y) u^2(x,y,t-\rho h)\,{\rm d}\rho\,{\rm d}x\,{\rm d}y, \end{aligned} \end{align}$$

where $\xi >0$ satisfies

(1.10)$$\begin{align} h\mu_2<\xi< h(2\mu_1-\mu_2). \end{align}$$

Note that the derivative of energy (1.9) satisfies

$$\begin{align*} \frac{{\rm d}}{{\rm d}t} E_u(t) & \leq{-}C \left( \int_0^L(\partial^2_xu(0,y,t))^2\,{\rm d}y+ \int_0^L \left( {\partial_x^{{-}1}} \partial_yu(0,y,t) \right)^2\,{\rm d}y \right. \\ & \quad\left. +\int_0^L\int_0^L a(x,y)u^2(x,y,t-h)\,{\rm d}x\,{\rm d}y \right) \end{align*}$$

for $C:= C(\mu _1,\,\mu _2,\,\xi,\,h)\geq 0$. This indicates that the function $a(x,\,y)$ plays the role of a feedback-damping mechanism, at least for the linearized system. Therefore, for system (1.8) we split the behavior of the solutions into two parts. Employing Lyapunov's method, it can be deduced that the energy $E_u(t)$ goes exponentially to zero as $t \rightarrow \infty$, however, the initial data needs to be sufficiently small in this case. Precisely, the second local result can be read as follows:

Theorem 1.4 Local stabilization

Let $L>0$. Assume that $a(x,\,y)\in L^\infty (\Omega )$ is a non-negative function, that relation (1.10) holds and $\beta < -\frac {1}{30}$. Then, there exists

\[ { 0< r< \frac{\sqrt[4]{216\alpha^3}}{CL^{\frac{5}{2}}} } \]

such that for every $(u_0,\,z_0(\cdot,\,\cdot,\,-h(\cdot )))\in \mathcal {H}$ satisfying $\left \lVert {(u_0,\,z_0(\cdot,\,\cdot,\,-h(\cdot )))}\right \rVert _{\mathcal {H}} \leq r$, the energy defined in (1.9) decays exponentially. More precisely, there exists two positives constants $\theta$ and $\kappa$ such that $E_u(t) \leq \kappa E_u(0)e^{-2\theta t}$ for all $t>0.$ Here,

\[ \theta < \min\left\lbrace \frac{\eta}{(1+2\eta L)L^2} \left[ 3\alpha - \frac{1}{2} C^{\frac{4}{3}} r^{\frac{4}{3}} L^{\frac{10}{3}} \right], \frac{\xi\sigma}{2h(\xi+\sigma\xi)} \right\rbrace, \quad \kappa = 1+\max\lbrace{2\eta L, \sigma}\rbrace \]

and $\eta$ and $\sigma$ are positive constants such that

$$\begin{align*} \sigma & < \frac{2h}{\xi}\left( \mu_1-\frac{\mu_2}{2}-\frac{\xi}{2h} \right)\\ \eta & <\min\left\lbrace \frac{1}{2L\mu_2} \left[ \frac{\xi}{h}-\mu_2 \right], \frac{1}{2L\mu_1+L\mu_2} \left[ \mu_1 - \frac{\mu_2}{2} - \frac{\xi}{2h}(1+\sigma) \right] \right\rbrace. \end{align*}$$

The last result of the manuscript, still related to the system (1.8), removes the hypothesis of the initial data being small. To do that, we use, as mentioned before, the compactness-uniqueness argument due to J.-L. Lions [Reference Lions21], which reduces our problem to prove an observability inequality for the nonlinear system (1.8) and removes the hypotheses that the initial data are small enough.

1.3 Novelty and outline of the article

We finish the introduction by highlighting some facts about our problem in comparison with the works previously mentioned, as well as the organization of the paper.

1. a. Observe that the absence of drift term $u_x$, in comparison with Kawahara equation in [Reference Capistrano–Filho and Gonzalez Martinez8, Reference Chentouf9], leads to get stabilization results without restriction in the length of the spatial domain. This term is not important in our analysis, the term only plays an important role in the problems where the control (damping or delay) is acting in the boundary conditionFootnote ⁴.

2. b. As stated earlier, we introduce an anti-damping together with the damping mechanism to show that the energy of system (1.5) decays exponentially. Compared with the known result [Reference de Moura, Nascimento and Santos10], the novelty of this paper is twofold:

   1. (i) Our work gives the precise decay rate, see theorems 1.3 and 1.4.

   2. (ii) Lyapunov's method shows an optimal decay rate in terms of $\theta$ in theorem 1.3. Observe that the value of $\theta$ can be optimized as a function of $\eta$, that is, we can choose

      (1.11)$$\begin{align} \eta \in \left(0,\frac{\xi-1}{2L(1+2\xi)}\right) \end{align}$$
      such that the value of $\theta$ is the largest possible, which implies that the decay rate $\theta$ obtained in this way, is the best one. It can be shown defining functions $f,\, g : [0,\,\frac {\xi -1}{2L(1+2\xi )}] \longrightarrow \mathbb {R}$ by
      \[ f(\eta)= \frac{3\alpha\eta }{L^2(1+2\eta L)}, \quad g(\eta)= \frac{\xi-1-2L\eta(1+2\xi)}{2h(2\xi-1-2\eta L(1+2\xi))}, \]
      and considering $\gamma (\eta )=\min \{f(\eta ),\,g(\eta )\}$. So, the function $f$ is increasing in the interval $[0,\,\frac {\xi -1}{2L(1+2\xi )})$ while the function $g$ is decreasing in this same interval. In fact, note that
      \[ f(\eta)= \frac{3\alpha}{2L^3}\left(1-\frac{1}{1+2\eta L}\right) \]
      and
      \[ g(\eta)= \frac{1}{2h}-\left(\frac{\xi}{4hL(1+2\xi)}\right)\left(\frac{1}{\frac{\xi}{2L(1+2\xi)}+\frac{\xi-1}{2L(1+2\xi)}-\eta}\right). \]
      If $-\frac {1}{2L}<\eta$, then
      \[[ f'(\eta)= \frac{3\alpha}{2L^3}\frac{2L}{(1+2L\eta)^2}>0. \]
      In particular, $f'(\eta )>0$ when
      \[ \eta \in \left[0,\frac{\xi-1}{2L(1+2\xi)}\right). \]
      Analogously,
      \[ g'(\eta)={-}\left(\frac{\xi}{4hL(1+2\xi)}\right)\frac{1}{\left(\frac{\xi}{2L(1+2\xi)}+\frac{\xi-1}{2L(1+2\xi)}-\eta\right)^2}<0, \]
      since $\xi >1$ and $\eta <\frac {\xi -1}{2L(1+2\xi )},$ showing our claim. Now, we claim that there exists only one point satisfying (1.11) such that $f(\eta )=g(\eta )$. To show the existence of this point, it is sufficient to note that $f(0)=0$, $g(\frac {\xi -1}{2L(1+2\xi )})=0$ and
      \[ f\left(\frac{\xi-1}{2L(1+2\xi)}\right)=\frac{3\alpha}{2L^3} \left(\frac{3\xi-1}{3\xi}\right)>0, \quad g(0)=\frac{1}{2h}\left(1-\frac{\xi}{2\xi-1}\right)>0. \]
      The uniqueness follows from the fact that $f$ is increasing while $g$ is decreasing in this interval.

3. c. Taking into account the above information about $f$ and $g$, the maximum value of the function must be reached at the point $\eta$ satisfying (1.11), where $f(\eta )=g(\eta )$. The figure 1 below shows, in a simple case, what was said earlier to the functions $f$ and $g$ when we consider some values, for example, $L=1$, $\xi =2.3$, $\alpha =0.5$ and $h=1.5$:

4. d. Still concerning the theorem 1.3, observe that we do not need to localize the solution of the transport equation in a small subset of $(0,\,L)$ as in [Reference Valein29, Section 4]. Moreover, we emphasize that we can take $a=0$ in theorem 1.3. Finally, it is important to mention that we do not know if the time $T_{\min }$ is optimal.

5. e. Aiming to present optimal decay results, note that for the nonlinear system, we obtain one stabilization result with no restriction in the length of the spatial domain but carries a restriction in one parameter of the system, see theorem 1.4. Once again, it is possible to waive one of the conditions (either the restriction on $L$ or a restriction in one system parameter). Observe that, using theorem 1.2 like as (2.7) below, we have

   (1.12)$$\begin{align} \begin{aligned} \int_0^L\int_0^L u^3(x,y,t)\,{\rm d}x\,{\rm d}y & \leq cL \left\lVert{u_{xx}}\right\rVert_{L^2(\Omega)}^{\frac{1}{2}} \left\lVert{u}\right\rVert_{L^2(\Omega)}^{\frac{5}{2}} \\ & \leq { \frac{1}{4}(CL)^{4} \left\lVert u_{xx} \right\rVert_{L^2(\Omega)}^2 +\frac{3}{4} r^{\frac{4}{3}} \left\lVert u \right\rVert_{L^2(\Omega)}^2.} \end{aligned} \end{align}$$
   This estimate allows obtaining, with an analogous argument another result for exponential stability without restriction in the parameter $\beta$ but with restriction in the length $L$ of the domain. Thus, in theorem 1.4, we can remove the hypothesis over $\beta$, however, a hypothesis over $L$ is necessary. The result is the following:

   Theorem 1.6 Local stabilization-bis

   Let $0< L< \sqrt [4]{\frac {-30\beta }{C}}.$ Assume that $a(x,\,y)\in L^\infty (\Omega )$ is a non-negative function and that the relation (1.10) holds. Then, there exists $0< r< \frac {\sqrt [4]{216\alpha ^3}}{CL^{\frac {5}{2}}}$ such that for every $(u_0,\,z_0(\cdot,\,\cdot,\,-h(\cdot )))\in \mathcal {H}$ satisfying $\left \lVert {(u_0,\,z_0(\cdot,\,\cdot,\,-h(\cdot )))}\right \rVert _{\mathcal {H}} \leq r$, the energy defined in (1.9) decays exponentially. More precisely, there exists two positives constants $\theta$ and $\kappa$ such that $E_u(t) \leq \kappa E_u(0)e^{-2\theta t}$ for all $t>0,$ where $\theta$, $\kappa$, $\eta$ and $\sigma$ are positive constants defined as in theorem 1.4.

   

   Figure 1. Maximum of $\gamma (\eta )=\min \{f(\eta ),\,g(\eta )\}$.

6. f. The results obtained here can be easily adapted for the KP-II system (1.2) with or without the drift term $u_x$, extending the results of [Reference de Moura, Nascimento and Santos10] and [Reference Gomes and Panthee11].

The work is organized as follows:

– § 2 is devoted to proving the first, and optimal, local stability result, that is, theorem 1.3.

– In § 3 we are able to prove the exponential stability, theorem 1.4, for the energy associated with the $\mu _i$–system (1.8).

– Additionally, to extend the local property to the global one, in § 3 we give the proof of theorem 1.5.

– For the sake of completeness, we present in Appendix A, at the end of the work, the well-posedness of the time-delayed K-KP-II system.