Asymptotic Behavior of the Solution of the Initial Boundary Value Problem for a Boussı̇nesq Analog Equation

Sherif Amirov¹, Hale Koyuncu², Orhan Amirov³

¹Mathematics Department, Science Faculty, Karabuk University, Karabük, Türkiye.
²Faculty of Science Department of Mathematics, Karabuk University, Karabük, Türkiye.
³Faculty of Science Department of Mathematics, Ataturk Universty, Erzurum, Türkiye.

DOI: 10.4236/jamp.2025.134052                

Abstract

In this paper, the asymptotics of the solution of the initial boundary value problem for a sixth-order nonlinear partial differential equation of Boussinesq type is studied. First, the energy function is obtained. Then, apriori evaluations for this function are obtained. Then, by imposing some conditions on the function on the right-hand side of the equation and using appropriate inequalities, it is shown that the solution is asymptotically damped.

Keywords

Initial Boundary Value Problem, Boussinesq Equation, Asymptotic Behavior of the Solution

1. Introduction

[1] As I mentioned in my paper, the following equation:

$\left({\sigma }^2\Delta -1\right)\frac{\partial u^2}{\partial t^2}+{\gamma }^2\Delta u=F$

It is called the Boussinesq equation. ∆ where is the Laplace operator,$\sigma ,\gamma$ fixed numbers $F$ if it is $u$ is a linear or nonlinear function depending on derivatives of order less than the left-hand side derivatives of the solution function. This equation and its analogs appear in the study of long-wave motions in plasmas and liquids [1]-[35] They examined the solvability of initial boundary value problems and the stability of the solutions.

In this paper we investigate the asymptotic stability of the solution of the initial and boundary value problem for an equation which is an analog of the sixth-order Boussinesq equation with respect to space variables.

2. Discussion

In this paper, we have investigated the asymptotic behavior of the strong solution of the initial boundary-value problem for the sixth-order nonlinear Boussinsq type equation with a dispersion term.

3. Methods

By using the multiplier method and the integral estimate methods, we prove that the strong solutions of the problem decay to zero exponentially as the time tends to infinite, under weaker conditions regarding the nonlinear term. And Gromwells lemma.

4. Results

Now let’s move on to the content of the article.

In this study $x\in \left(0,1\right),t\ge 0$ of the initial boundary value problem for a sixth-order nonlinear partial differential equation of Boussinesq type given below

$u_{tt}-\text{Δ}u-\text{Δ}{u}_t-\text{Δ}{u}_{tt}-{\text{Δ}}^{3}u=f\left(u\right),x\in \left(0,1\right),t\ge 0$ (1)

$u\left(x,0\right)=u_0\left(x\right),u_t\left(x,0\right)=u_1\left(x\right),x\in \left[0,1\right]$ (2)

$u\left(0,t\right)=u\left(1,t\right)=0,t\ge 0$ (3)

We will study the asymptotic behavior of the solution satisfying the initial conditions (2) and boundary conditions (3). Here $u\left(x,t\right)$ is the sought solution function, $f\left(s\right)$ is a nonlinear function and $u_0\left(x\right),u_1\left(x\right)$ are the initial data.

Theorem 1. Suppose,

$0\le -F\left(u\right)\le -f\left(u\right)u\forall u\in R,F\left(u\right)={\displaystyle {\int }_0^{u}f\left(s\right)\text{d}s}.$

Let u be the solution of the problem (1) - (3). Then for positive constants $C$ and $\lambda$

$E\left(t\right)\le C\cdot E\left(0\right){\text{e}}^{-\lambda t},0\le t<+\infty$ (4)

inequality is true.

Here

$E\left(t\right)=\frac{{\Vert u_t\Vert }^2}{2}+\frac{{\Vert \nabla u\Vert }^2}{2}+\frac{{\Vert \nabla u_t\Vert }^2}{2}+\frac{{\Vert {\nabla }^{3}u\Vert }^2}{2}-\underset{0}{\overset{1}{{\displaystyle \int }}}F\left(u\right)\text{d}x$ (5)

It is a function of energy.

Proof: Let us scalar multiply Equation (1) by $u_t$ and integrate it from 0 to 1:

$\underset{0}{\overset{1}{{\displaystyle \int }}}{u}_{tt}{u}_t\text{d}x-\underset{0}{\overset{1}{{\displaystyle \int }}}{u}_{xx}{u}_t\text{d}x-\underset{0}{\overset{1}{{\displaystyle \int }}}{u}_{xxt}{u}_t\text{d}x-\underset{0}{\overset{1}{{\displaystyle \int }}}{u}_{xxtt}{u}_t\text{d}x-\underset{0}{\overset{1}{{\displaystyle \int }}}{u}_{xxxxxx}{u}_t\text{d}x=\underset{0}{\overset{1}{{\displaystyle \int }}}f\left(u\right)u_t\text{d}x$

Now if we use the partial integration formula

$\frac{\text{d}}{2\text{d}t}{\Vert u_t\Vert }^2+\frac{\text{d}}{2\text{d}t}{\Vert u_x\Vert }^2+{\Vert u_{xt}\Vert }^2+\frac{\text{d}}{2\text{d}t}{\Vert u_{xt}\Vert }^2+\frac{\text{d}}{2\text{d}t}{\Vert {\nabla }^{3}u\Vert }^2-\frac{\text{d}}{\text{d}t}\underset{0}{\overset{1}{{\displaystyle \int }}}F\left(u\right)\text{d}x=0$

$\frac{\text{d}}{\text{d}t}\left(\frac{{\Vert u_t\Vert }^2}{2}+\frac{{\Vert \nabla u\Vert }^2}{2}+\frac{{\Vert \nabla u_t\Vert }^2}{2}+\frac{{\Vert {\nabla }^{3}u\Vert }^2}{2}-\underset{0}{\overset{1}{{\displaystyle \int }}}F\left(u\right)\text{d}x\right)+{\Vert u_{xt}\Vert }^2=0$ (6)

we obtain equality.

If we substitute (6) in (5):

$\frac{\text{d}}{\text{d}t}E\left(t\right)+{\Vert \nabla u_t\Vert }^2=0$ (7)

we get.

Let us multiply the inequality (7) by ${\text{e}}^{\delta t}$ provided that $\delta >0$ :

${\text{e}}^{\delta t}\frac{\text{d}}{\text{d}t}E\left(t\right)+{\text{e}}^{\delta t}{\Vert \nabla u_t\Vert }^2=0$

We can write this equation as follows.

$\frac{\text{d}}{\text{d}t}\left({\text{e}}^{\delta t}E\left(t\right)\right)+{\text{e}}^{\delta t}{\Vert \nabla u_t\Vert }^2=\delta {\text{e}}^{\delta t}E\left(t\right)$ (8)

If we integrate Equation (8) from 0 to t:

$\underset{0}{\overset{t}{{\displaystyle \int }}}\frac{\text{d}}{\text{d}\tau }\left({\text{e}}^{\delta \tau }E\left(\tau \right)\right)\text{d}\tau +\underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }{\Vert \nabla u_{\tau }\Vert }^2\text{d}\tau =\delta \underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }E\left(\tau \right)\text{d}\tau$

${\text{e}}^{\delta t}E\left(t\right)-E\left(0\right)+\underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }{\Vert \nabla u_{\tau }\Vert }^2\text{d}\tau =\delta \underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }E\left(\tau \right)\text{d}\tau$

${\text{e}}^{\delta t}E\left(t\right)+\underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }{\Vert \nabla u_{\tau }\Vert }^2\text{d}\tau =E\left(0\right)+\delta \underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }E\left(\tau \right)\text{d}\tau$

we obtain. In the integral on the right side of the equation $E\left(t\right)$ if we use (5) instead:

$\begin{array}{l}{\text{e}}^{\delta t}E\left(t\right)+\underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }{\Vert \nabla u_{\tau }\Vert }^2\text{d}\tau \\ =E\left(0\right)+\delta \underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }\left(\frac{{\Vert u_{\tau }\Vert }^2}{2}+\frac{{\Vert \nabla u\Vert }^2}{2}+\frac{{\Vert \nabla u_{\tau }\Vert }^2}{2}+\frac{{\Vert {\nabla }^{3}u\Vert }^2}{2}-\underset{0}{\overset{1}{{\displaystyle \int }}}F\left(u\right)\text{d}x\right)\text{d}\tau \\ =E\left(0\right)+\delta \underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }\left(\frac{{\Vert u_{\tau }\Vert }^2}{2}+\frac{{\Vert \nabla u\Vert }^2}{2}+\frac{{\Vert \nabla u_{\tau }\Vert }^2}{2}\right)\text{d}\tau +\delta \underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }\left(\frac{{\Vert {\nabla }^{3}u\Vert }^2}{2}-\underset{0}{\overset{1}{{\displaystyle \int }}}F\left(u\right)\text{d}x\right)\text{d}\tau \end{array}$ (9)

we get.

Considering the conditions of Theorem 1, let us evaluate the last integral in (9):

$\frac{{\Vert {\nabla }^{3}u\Vert }^2}{2}-\underset{0}{\overset{1}{{\displaystyle \int }}}F\left(u\right)\text{d}x\le \frac{{\Vert {\nabla }^{3}u\Vert }^2}{2}-\underset{0}{\overset{1}{{\displaystyle \int }}}f\left(u\right)u\text{d}x$

$\delta \underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }\left(\frac{{\Vert {\nabla }^{3}u\Vert }^2}{2}-\underset{0}{\overset{1}{{\displaystyle \int }}}F\left(u\right)\text{d}x\right)\text{d}\tau \le \delta \underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }\left(\frac{{\Vert {\nabla }^{3}u\Vert }^2}{2}-\underset{0}{\overset{1}{{\displaystyle \int }}}f\left(u\right)u\text{d}x\right)\text{d}\tau$ (10)

As seen in this inequality, if we substitute the left side of Equation (1) for $f\left(u\right)$ in the expression ${\displaystyle {\int }_0^{1}f\left(u\right)u\text{d}x}$ in the integral on the right side of the inequality

$\begin{array}{l}\underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }\left(\frac{{\Vert {\nabla }^{3}u\Vert }^2}{2}-\underset{0}{\overset{1}{{\displaystyle \int }}}F\left(u\right)\text{d}x\right)\text{d}\tau \\ \le \underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }\left(\frac{{\Vert {\nabla }^{3}u\Vert }^2}{2}-{\Vert {\nabla }^{3}u\Vert }^2-\left(u_{\tau \tau }u\right)-{\Vert \nabla u\Vert }^2-\frac{\text{d}}{2\text{d}\tau }{\Vert \nabla u\Vert }^2-\left(\nabla u_{\tau \tau }\nabla u\right)\right)\text{d}\tau \\ =\underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }\left(-\frac{{\Vert {\nabla }^{3}u\Vert }^2}{2}-\left(u_{\tau \tau }u\right)-{\Vert \nabla u\Vert }^2-\frac{\text{d}}{2\text{d}\tau }{\Vert \nabla u\Vert }^2-\left(\nabla u_{\tau \tau }\nabla u\right)\right)\text{d}\tau \end{array}$ (11)

we get.

From the right hand side of (11)

$-\frac{{\Vert {\nabla }^{3}u\Vert }^2}{2}\text{and}-{\Vert \nabla u\Vert }^2$

If we drop terms that are strictly negative, such as (11), then (11) becomes the following inequality.

$\begin{array}{l}\delta \underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }\left(\frac{{\Vert {\nabla }^{3}u\Vert }^2}{2}-\underset{0}{\overset{1}{{\displaystyle \int }}}F\left(u\right)\text{d}x\right)\text{d}\tau \\ \le \delta \underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }\left(-\left(u_{\tau \tau }u\right)-\frac{\text{d}}{2\text{d}\tau }{\Vert \nabla u\Vert }^2-\left(\nabla u_{\tau \tau }u\right)\right)\text{d}\tau \end{array}$ (12)

Let us consider the terms on the right hand side of inequality (12) respectively:

before

$-\delta {\displaystyle {\int }_0^t{\text{e}}^{\delta \tau }\left(u_{\tau \tau }u\right)\text{d}\tau }$

Let’s look at the integral:

For this, let’s examine the following expression

$-\frac{\text{d}}{\text{d}t}\left(\underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }\left(u_{\tau }u\right)\text{d}\tau \right)=-\delta \underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }\left(u_{\tau }u\right)\text{d}\tau -\underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }\left(u_{\tau \tau }u\right)\text{d}\tau -\underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }{\Vert u_{\tau }\Vert }^2\text{d}\tau$

$-{\text{e}}^{\delta \tau }\left(u_{t}u\right)+\left(u_1{u}_0\right)=-\delta \underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }\left(u_{\tau }u\right)\text{d}\tau -\underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }\left(u_{\tau \tau }u\right)\text{d}\tau -\underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }{\Vert u_{\tau }\Vert }^2\text{d}\tau$

Now multiply both sides of the equation by $\delta$ , $\left(\delta >0\right)$ and leave the desired integral alone

$-\delta \underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }\left(u_{\tau \tau }u\right)\text{d}\tau =-\delta {\text{e}}^{\delta \tau }\left(u_{t}u\right)+\delta \left(u_1{u}_0\right)+{\delta }^2\underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }\left(u_{\tau }u\right)\text{d}\tau +\delta \underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }{\Vert u_{\tau }\Vert }^2\text{d}\tau$

we obtain equality.

If we apply Young’s inequality to the remaining terms in this equation except the last term, we obtain the following inequality.

$\begin{array}{c}-\delta \underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }\left(u_{\tau \tau }u\right)\text{d}\tau \le \frac{\delta }{2}{\text{e}}^{\delta \tau }\left({\Vert u_t\Vert }^2+{\Vert u\Vert }^2\right)+\frac{\delta }{2}\left({\Vert u_1\Vert }^2+{\Vert u_0\Vert }^2\right)\\ +\frac{{\delta }^2}{2}\underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }\left({\Vert u_{\tau }\Vert }^2+{\Vert u\Vert }^2\right)\text{d}\tau +\delta \underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }{\Vert u_{\tau }\Vert }^2\text{d}\tau \end{array}$ (13)

Now let’s look at the following integrand

$-\delta \underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }\frac{\text{d}}{2\text{d}\tau }{\Vert \nabla u\Vert }^2\text{d}\tau$

For this, let us examine the following expression:

$-\underset{0}{\overset{t}{{\displaystyle \int }}}\frac{\text{d}}{2\text{d}t}\left({\text{e}}^{\delta \tau }{\Vert \nabla u\Vert }^2\right)\text{d}\tau =-\frac{\delta }{2}\underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }{\Vert \nabla u\Vert }^2\text{d}\tau -\underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }\frac{\text{d}}{2\text{d}t}{\Vert \nabla u\Vert }^2\text{d}\tau$

$-\frac{1}{2}{\text{e}}^{\delta \tau }{\Vert \nabla u\Vert }^2+\frac{1}{2}{\Vert \nabla u_0\Vert }^2=-\frac{\delta }{2}\underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }{\Vert \nabla u\Vert }^2\text{d}\tau -\underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }\frac{\text{d}}{2\text{d}t}{\Vert \nabla u\Vert }^2\text{d}\tau$

Now multiply both sides of the equation by $\delta$ , $\left(\delta >0\right)$ and leave the desired integral alone

$-\delta \underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }\frac{\text{d}}{2\text{d}\tau }{\Vert \nabla u\Vert }^2\text{d}\tau =-\frac{\delta }{2}{\text{e}}^{\delta t}{\Vert \nabla u\Vert }^2+\frac{\delta }{2}{\Vert \nabla u_0\Vert }^2+\frac{{\delta }^2}{2}\underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }{\Vert \nabla u\Vert }^2\text{d}\tau$

we obtain equality.

If we drop the strictly negative term in the above equation

$-\delta \underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }\frac{\text{d}}{2\text{d}\tau }{\Vert \nabla u\Vert }^2\text{d}\tau \le \frac{\delta }{2}{\Vert \nabla u_0\Vert }^2+\frac{{\delta }^2}{2}\underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }{\Vert \nabla u\Vert }^2\text{d}\tau$ (14)

we obtain the inequality. Finally, let us look at the following integrand:

$-\delta \underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }\left(\nabla u_{\tau \tau }\nabla u\right)\text{d}\tau$

For this, let us examine the following expression:

$-\frac{\text{d}}{\text{d}t}\left(\underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }\left(\nabla u_{\tau }\nabla u\right)\text{d}\tau \right)=-\delta \underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }\left(\nabla u_{\tau }\nabla u\right)\text{d}\tau -\underset{0}{\overset{t}{{\displaystyle \int }}}\left(\nabla u_{\tau \tau }\nabla u\right)\text{d}\tau -\underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }{\Vert \nabla u_{\tau }\Vert }^2\text{d}\tau$

$-{\text{e}}^{\delta \tau }\left(\nabla u_{\tau }\nabla u\right)+\left(\nabla u_1\nabla u_0\right)=-\delta \underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }\left(\nabla u_{\tau }\nabla u\right)\text{d}\tau -\underset{0}{\overset{t}{{\displaystyle \int }}}\left(\nabla u_{\tau \tau }\nabla u\right)\text{d}\tau -\underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }{\Vert \nabla u_{\tau }\Vert }^2\text{d}\tau$

Now multiply both sides of the equation by $\delta$ , $\left(\delta >0\right)$ and leave the desired integral alone

$\begin{array}{c}-\delta \underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }\left(\nabla u_{\tau \tau }\nabla u\right)\text{d}\tau =-\delta {\text{e}}^{\delta \tau }\left(\nabla u_t\nabla u\right)+\delta \left(\nabla u_1\nabla u_0\right)\\ +{\delta }^2\underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }\left(\nabla u_{\tau }\nabla u\right)\text{d}\tau +\delta \underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }{\Vert \nabla u_{\tau }\Vert }^2\text{d}\tau \end{array}$

we obtain equality.

If we apply Young’s inequality to the remaining terms in this equation except the last term, we obtain the following inequality.

$\begin{array}{c}-\delta \underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }\left(\nabla u_{\tau \tau }\nabla u\right)\text{d}\tau \le \frac{\delta }{2}{\text{e}}^{\delta t}\left({\Vert \nabla u_t\Vert }^2+{\Vert \nabla u\Vert }^2\right)+\frac{\delta }{2}\left({\Vert \nabla u_1\Vert }^2+{\Vert \nabla u_0\Vert }^2\right)\\ +\frac{{\delta }^2}{2}\underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }\left({\Vert \nabla u_{\tau }\Vert }^2+{\Vert \nabla u\Vert }^2\right)\text{d}\tau +\delta \underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }{\Vert \nabla u_{\tau }\Vert }^2\text{d}\tau \end{array}$ (15)

If we consider inequalities (13), (14) and (15) in (9) and (12)

$\begin{array}{l}{\text{e}}^{\delta t}E\left(t\right)+\underset{0}{\overset{1}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }{\Vert \nabla u_{\tau }\Vert }^2\text{d}\tau \\ \le E\left(0\right)+\delta \underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }\left(\frac{{\Vert u_{\tau }\Vert }^2}{2}+\frac{{\Vert \nabla u\Vert }^2}{2}+\frac{{\Vert \nabla u_{\tau }\Vert }^2}{2}\right)\text{d}\tau +\frac{\delta }{2}{\text{e}}^{\delta t}\left({\Vert \nabla u_t\Vert }^2+{\Vert \nabla u\Vert }^2\right)\\ +\frac{\delta }{2}\left({\Vert \nabla u_1\Vert }^2+{\Vert \nabla u_0\Vert }^2\right)+\frac{{\delta }^2}{2}\underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }\left({\Vert \nabla u_{\tau }\Vert }^2+{\Vert \nabla u\Vert }^2\right)\text{d}\tau +\delta \underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }{\Vert \nabla u_{\tau }\Vert }^2\text{d}\tau \\ +\frac{\delta }{2}{\Vert \nabla u_0\Vert }^2+\frac{{\delta }^2}{2}\underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }{\Vert \nabla u\Vert }^2\text{d}\tau +\frac{\delta }{2}{e}^{\delta t}\left({\Vert \nabla u_t\Vert }^2+{\Vert \nabla u\Vert }^2\right)\\ +\frac{\delta }{2}\left({\Vert \nabla u_1\Vert }^2+{\Vert \nabla u_0\Vert }^2\right)+\frac{{\delta }^2}{2}\underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }\left({\Vert \nabla u_{\tau }\Vert }^2+{\Vert \nabla u\Vert }^2\right)\text{d}\tau +\delta \underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }{\Vert \nabla u_{\tau }\Vert }^2\text{d}\tau \end{array}$ (16)

inequality is obtained.

(16) in the inequality

$\frac{\delta }{2}\underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }{\Vert \nabla u_{\tau }\Vert }^2\text{d}\tau ,\frac{{\delta }^2}{2}\underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }{\Vert \nabla u\Vert }^2\text{d}\tau ,\delta \underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }{\Vert \nabla u_{\tau }\Vert }^2\text{d}\tau ,\frac{\delta }{2}{\text{e}}^{\delta t}{\Vert \nabla u\Vert }^2$

Using the Sobolev-Poincare inequality $\forall u\in H_0^1\left(\Omega \right)$ , ${\Vert u\Vert }^2\le {\lambda }_0{\Vert \nabla u\Vert }^2$ we obtain the following inequalities.

$\delta \underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }{\Vert u_{\tau }\Vert }^2\text{d}\tau \le \delta {\lambda }_0\underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }{\Vert \nabla u_{\tau }\Vert }^2\text{d}\tau$

$\frac{\delta }{2}\underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }{\Vert u_{\tau }\Vert }^2\text{d}\tau \le {\lambda }_0\frac{\delta }{2}\underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }{\Vert \nabla u_{\tau }\Vert }^2\text{d}\tau$

$\frac{\delta }{2}{\text{e}}^{\delta t}{\Vert u\Vert }^2\le {\lambda }_0\frac{\delta }{2}{\text{e}}^{\delta t}{\Vert \nabla u\Vert }^2$

$\frac{{\delta }^2}{2}\underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }{\Vert u\Vert }^2\text{d}\tau \le {\lambda }_0\frac{{\delta }^2}{2}\underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }{\Vert \nabla u\Vert }^2\text{d}\tau$

If we substitute these inequalities in (16)

$\begin{array}{l}{\text{e}}^{\delta t}E\left(t\right)+\underset{0}{\overset{1}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }{\Vert \nabla u_{\tau }\Vert }^2\text{d}\tau \\ \le E\left(0\right)+\delta \underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }\left(\frac{{\lambda }_0{\Vert u_{\tau }\Vert }^2}{2}+\frac{{\Vert \nabla u\Vert }^2}{2}+\frac{{\Vert \nabla u_{\tau }\Vert }^2}{2}\right)\text{d}\tau \\ +\frac{\delta }{2}\left({\Vert \nabla u_t\Vert }^2+{\lambda }_0{\Vert \nabla u\Vert }^2\right)+\frac{\delta }{2}\left({\Vert \nabla u_1\Vert }^2+{\Vert \nabla u_0\Vert }^2\right)\\ +\frac{{\delta }^2}{2}\underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }\left({\Vert \nabla u_{\tau }\Vert }^2+{\lambda }_0{\Vert \nabla u\Vert }^2\right)\text{d}\tau +\delta {\lambda }_0\underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }{\Vert \nabla u_{\tau }\Vert }^2\text{d}\tau \\ +\frac{\delta }{2}{\Vert \nabla u_0\Vert }^2+\frac{{\delta }^2}{2}\underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }{\Vert \nabla u\Vert }^2\text{d}\tau +\frac{\delta }{2}{\text{e}}^{\delta t}\left({\Vert \nabla u_t\Vert }^2+{\Vert \nabla u\Vert }^2\right)\\ +\frac{\delta }{2}\left({\Vert \nabla u_1\Vert }^2+{\Vert \nabla u_0\Vert }^2\right)+\frac{{\delta }^2}{2}\underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }\left({\Vert \nabla u_{\tau }\Vert }^2+{\Vert \nabla u\Vert }^2\right)\text{d}\tau \\ +\delta \underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }{\Vert \nabla u_{\tau }\Vert }^2\text{d}\tau \end{array}$ (17)

we obtain the inequality.

We can rearrange the terms on the right-hand side of inequality (17) as follows.

$\frac{3}{2}\delta {\lambda }_0\underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }{\Vert \nabla u_{\tau }\Vert }^2\text{d}\tau +\frac{3}{2}\delta \underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }{\Vert \nabla u_{\tau }\Vert }^2\text{d}\tau =\frac{3}{2}\delta \left(1+{\lambda }_0\right)\underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }{\Vert \nabla u_{\tau }\Vert }^2\text{d}\tau$

$\delta {\text{e}}^{\delta t}\left(\frac{{\Vert u_t\Vert }^2}{2}+\frac{{\Vert \nabla u\Vert }^2}{2}+\frac{{\Vert \nabla u_t\Vert }^2}{2}+\frac{{\Vert {\nabla }^{3}u\Vert }^2}{2}-\underset{0}{\overset{1}{{\displaystyle \int }}}F\left(u\right)\text{d}x\right)\le c_1\delta {\text{e}}^{\delta t}E\left(t\right)$

${\delta }^2\underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }\left(\frac{{\Vert u_{\tau }\Vert }^2}{2}+\frac{{\Vert \nabla u\Vert }^2}{2}+\frac{{\Vert \nabla u_{\tau }\Vert }^2}{2}+\frac{{\Vert {\nabla }^{3}u\Vert }^2}{2}-\underset{0}{\overset{1}{{\displaystyle \int }}}F\left(u\right)\text{d}x\right)\text{d}\tau \le c_2{\delta }^2\underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }E\left(\tau \right)\text{d}\tau$

If we substitute these expressions in (17)

$\begin{array}{l}{\text{e}}^{\delta t}E\left(t\right)+\underset{0}{\overset{1}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }{\Vert \nabla u_{\tau }\Vert }^2\text{d}\tau \\ \le E\left(0\right)+\frac{3}{2}\delta \left(1+{\lambda }_0\right)\underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }{\Vert \nabla u_{\tau }\Vert }^2\text{d}\tau +c_1\delta {\text{e}}^{\delta \tau }E\left(t\right)+c_2{\delta }^2\underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }E\left(\tau \right)\text{d}\tau \end{array}$ (18)

we get.

From (18), substitute $\delta$ for $0<\text{min}\left\{\frac{2}{3\left(1+{\lambda }_0\right)},\frac{1}{2c_1}\right\}$ to get the following inequality

$\begin{array}{l}{\text{e}}^{\delta t}E\left(t\right)+\underset{0}{\overset{1}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }{\Vert \nabla u_{\tau }\Vert }^2\text{d}\tau \\ \le E\left(0\right)+\underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }{\Vert \nabla u_{\tau }\Vert }^2\text{d}\tau +\frac{{\text{e}}^{\delta \tau }}{2}E\left(t\right)+c_2{\delta }^2\underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }E\left(\tau \right)\text{d}\tau \end{array}$

${\text{e}}^{\delta t}E\left(t\right)\le E\left(0\right)+\frac{{\text{e}}^{\delta t}}{2}E\left(t\right)+c_2{\delta }^2\underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }E\left(\tau \right)\text{d}\tau$

$\frac{{\text{e}}^{\delta t}}{2}E\left(t\right)\le E\left(0\right)+c_2{\delta }^2\underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }E\left(\tau \right)\text{d}\tau$

${\text{e}}^{\delta t}E\left(t\right)\le 2E\left(0\right)+2c_2{\delta }^2\underset{0}{\overset{t}{{\displaystyle \int }}}{\text{e}}^{\delta \tau }E\left(\tau \right)\text{d}\tau$

we get.

If we apply Gronwall’s lemma here

${\text{e}}^{\delta t}E\left(t\right)\le 2E\left(0\right){\text{e}}^{2c_2{\delta }^{2}t},0\le t<\infty$

and

$E\left(t\right)\le CE\left(0\right){\text{e}}^{-\lambda t}$

we get.

If we select $\delta$ from the following range

$0<\delta <\text{min}\left\{\frac{2}{3\left(1+{\lambda }_0\right)},\frac{1}{2c_1},\frac{1}{2c_2}\right\}$

(4). That is, we show that the solution of the problem converges to zero of exponential order when $t\to \infty$ under the condition $c_1,c_2>1$ .

5. Conclusion

This paper has investigated the asymptotic behavior of the strong solution to a class of sixth-order nonlinear evolutions equations with both dispersive terms.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

References

[1]	Amirov, S. and Kozhanov, A.I. (2016) Global Solvability of Initial Boundary-Value Problems for Nonlinear Analogs of the Boussinesq Equation. Mathematical Notes, 99, 183-191.[CrossRef]
[2]	Amirov, S. and Anutgan, M. (2017) Boundary Value Problem for the Nonlinear Analogues of the Boussinesq Equation: Numerical Solution and Its Stability. New Trends in Mathematical Science, 3, 245-252.[CrossRef]
[3]	Amirov, Sh. and Anutgan, M. (2017) Analytical Solitary Wave Solutions for the Nonlinear Analogues of the Boussinesq and Sixth-Order Modified Boussinesq Equations. Journal of Applied Analysis & Computation, 7, 1613-1623.[CrossRef]
[4]	Âmin, E. and Mohammadi, H.B. (2010) Asymptotic Behavior of Solutions for the Cauchy Problem of a Dissipative Boussinesq-Type Equation. Analysis of PDEs, 367, 15-364.
[5]	Bayraktar, S. and Gür, Ş. (2020) Continuous Dependence of Solutions for Damped Improved Boussinesq Equation. Turkish Journal of Mathematics, 44, 334-341.[CrossRef]
[6]	Webb, G.F. (1980) Existence and Asymptotic Behavior for a Strongly Damped Nonlinear Wave Equation. Canadian Journal of Mathematics, 32, 631-643.[CrossRef]
[7]	Güleç, İ. (2016) Continuous Dependence of Solutions to Fourth-Order Nonlinear Wave Equation. Hacettepe Journal of Mathematics and Statistics, 45, 367-371.[CrossRef]
[8]	Gür, Ş. and Güleç, İ. (2016) Structural Stability Analysis of Solutions to the Initial Boundary Value Problem for a Nonlinear Strongly Damped Wave Equation. Turkish Journal of Mathematics, 40, 1231-1236.[CrossRef]
[9]	Di, H. and Shang, Y. (2015) Global Existence and Asymptotic Behavior of Solutions for the Double Dispersive-Dissipative Wave Equation with Nonlinear Damping and Source Terms. Boundary Value Problems, 2015, Article No. 29.[CrossRef]
[10]	Yang, H. and Han, Y. (2010) Initial Boundary Value Problem for a Strongly Damped Wave Equation with a General Nonlinearity. School of Mathematics, Jilin University, 35L20, 35L71.
[11]	Nakao, M. and Ono, K. (1993) Existence of Global Solutions to the Cauchy Problem for the Semilinear Dissipative Wave Equations. Mathematische Zeitschrift, 214, 325-342.[CrossRef]
[12]	Polat, N. and Pişkin, E. (2012) Asymptotic Behavior of a Solution of the Cauchy Problem for the Generalized Damped Multidimensional Boussinesq Equation. Applied Mathematics Letters, 25, 1871-1874.[CrossRef]
[13]	Polat, N. and Piskin, E. (2012) Asymptotic Behavior of Solution of Cauchy Problem for the Generalized Damped Double Dispersion Equation. International Mathematical Forum, 3, 145-151.
[14]	Polat, N. and Kaya, D. (2009) Existence, Asymptotic Behaviour, and Blow up of Solutions for a Class of Nonlinear Wave Equations with Dissipative and Dispersive Terms. Zeitschrift für Naturforschung A, 64, 315-326.[CrossRef]
[15]	Pişkin, E. and Polat, N. (2014) Existence, Global Nonexistence, and Asymptotic Behavior of Solutionsfor the Cauchy Problem of a Multidimensional Generalized Damped Boussinesq-Type Equation. Turkish Journal of Mathematics, 38, 706-727.[CrossRef]
[16]	Pişkin, E. (2017) Sobolev Spaces. Seçkin Yayıncılık A.Ş., 27-39, 69-70, 96-97.
[17]	Xu, R., Zhao, X. and Shen, J. (2008) Asymptotic Behaviour of Solution for Fourth Order Wave Equation with Dispersive and Dissipative Terms. Applied Mathematics and Mechanics, 29, 259-262.[CrossRef]
[18]	Xu, R. and Liu, Y. (2008) Asymptotic Behavior of Solutions for Initial-Boundary Value Problems for Strongly Damped Nonlinear Wave Equations. Nonlinear Analysis: Theory, Methods & Applications, 69, 2492-2495.[CrossRef]
[19]	Xu, R. and Yang, Y. (2012) Finite Time Blow-Up for the Nonlinear Fourth-Order Dispersive-Dissipative Wave Equation at High Energy Level. International Journal of Mathematics, 23, Article ID: 1250060.[CrossRef]
[20]	Xu, R.Z. and Yang, Y.B. (2012) Global Existence and Asymptotic Behaviour of Solutions for a Class of Fourth Order Strongly Damped Nonlinear Wave Equations. Quarterly of Applied Mathematics, 71, 401-415.[CrossRef]
[21]	Wang, S. and Chen, G. (2002) The Cauchy Problem for the Generalized IMBq Equation in Ws,p(Rn). Journal of Mathematical Analysis and Applications, 266, 38-54. [Google Scholar] [CrossRef]
[22]	Gerbi, S. and Said-Houari, B. (2011) Asymptotic Stability and Blow up for a Semilinear Damped Wave Equation with Dynamic Boundary Conditions. Nonlinear Analysis: Theory, Methods & Applications, 74, 7137-7150.[CrossRef]
[23]	Chen, X. and Chen, G. (2008) Asymptotic Behavior and Blow-Up of Solutions to a Nonlinear Evolution Equation of Fourth Order. Nonlinear Analysis: Theory, Methods & Applications, 68, 892-904.[CrossRef]
[24]	Wang, Y. (2012) Existence and Asymptotic Behavior of Solutions to the Generalized Dumped Boussinesq Equation. Electronic Journal of Differential Equations, 96, 1-11.
[25]	Wang, Y. and Zhao, H. (2013) Global Existence and Asymptotic Behavior of Solutions to the Generalized Damped Boussinesq Equation. Advances in Mathematical Physics, 2013, Article ID: 364165.[CrossRef]
[26]	Ali Akbar, M., Akinyemi, L., Yao, S., Jhangeer, A., Rezazadeh, H., Khater, M.M.A., et al. (2021) Soliton Solutions to the Boussinesq Equation through Sine-Gordon Method and Kudryashov Method. Results in Physics, 25, Article ID: 104228.[CrossRef]
[27]	Dong, L. (2021) On Stability of Boussinesq Equations without Thermal Conduction. Zeitschrift für angewandte Mathematik und Physik, 72, Article No. 128.[CrossRef]
[28]	Dong, L. and Sun, Y. (2022) Asymptotic Stability of the 2D Boussinesq Equations without Thermal Conduction. Journal of Differential Equations, 337, 507-540.[CrossRef]
[29]	Dong, L., Sun, Y. and Zhu, F. (2023) On Asymptotic Stability of Boussinesq Equations without Heat Conduction. In: Bodnár, T., Galdi, G.P. and Nečasová, Š., Eds., Advances in Mathematical Fluid Mechanics, Springer International Publishing, 295-322.[CrossRef]
[30]	Dong, L. and Sun, Y. (2023) On Asymptotic Stability of the 3D Boussinesq Equations without Heat Conduction. Science China Mathematics, 67, 253-280.[CrossRef]
[31]	Linares, F. and Ponce, G. (2015) Asymptotic Behavior of Solutions for the k-gKdV Equations. In: Ponce, G. and Linares, F., Eds., Introduction to Nonlinear Dispersive Equations, Springer, 191-214.
[32]	Kawashima, S. and Wang, Y. (2015) Global Existence and Asymptotic Behavior of Solutions to the Generalized Cubic Double Dispersion Equation. Analysis and Applications, 13, 233-254.[CrossRef]
[33]	Xu, R. and Liu, Y. (2008) Asymptotic Behavior of Solutions for Initial-Boundary Value Problems for Strongly Damped Nonlinear Wave Equations. Nonlinear Analysis: Theory, Methods & Applications, 69, 2492-2495.[CrossRef]
[34]	Shang, Y.D. (2000) Initial Boundary Value Problem of Boussinesq Equation. Acta Mathematicae Applicatae Sinica, 23, 385-393.
[35]	Chen, X. and Chen, G. (2008) Asymptotic Behavior and Blow-Up of Solutions to a Nonlinear Evolution Equation of Fourth Order. Nonlinear Analysis: Theory, Methods & Applications, 68, 892-904.[CrossRef]