General and optimal decay rates for a viscoelastic wave equation with strong damping

1.   Introduction

In this article, we study the following nonlinear viscoelastic problem

Here $\Omega$ is a bounded domain of $R^n\, (n\geq1)$ with a smooth boundary $\partial\Omega$, $g$ denote the kernel of memory term and the index number $p$ and $\rho$ satisfy the following conditions

This model of equation in (1.1) arise in the theory of viscoelasticity and physics and represent the propagation of several materials which possess a capacity to storage and dissipate mechanical energy. In the last half century, the existence and stability properties of solutions have been considered by many mathematicians, to motivate our work, we recall some literature related to our work.

In the absence of viscoelastic term and $\rho = 0$, Xu and Lian ^[18] studied the following initial boundary value problem at three different initial energy levels,

they proved the local existence of weak solution, and in the framework of potential well, they showed the global existence, energy decay of the solution with sub-critical initial energy, then by scaling technique, parallelly extended all the results for the subcritical case to the critical case. A similar result was also obtained by ^[2,20].

In the case when $\rho = 0$ and the strong damping term $-\triangle u_t$ is replaced by the damping mechanism $u_t|u_t|^{m-2}$, the form of the classical equation as follows,

where $a, b$ are two positive constant, the index number $m\geq1, p\geq2$, $\Omega$ is a bounded domain. Messaoudi ^[17] discussed the interaction between the damping term and the source term, which was first considered by Levine ^[8,10] for $m = 1$. He obtained, under suitable conditions on $g$ and initial data, that the solutions exist globally for any initial data if $m\geq p$ and blow up in finite time with negative initial energy if $p > m$. On the other hand, Messaoudi ^[15,16] considered equation (1.3) for $a = b = 0$ or $a = 0$, by using the perturbed energy method and under the supposition that $g'(t)\leq-\xi(t)g(t)$, proved that the solution energy is general decay not necessarily of exponential or polynomial type.

In the case when $\rho\neq0$, as the following class of quasilinear viscoelastic equations,

defined in a bounded domain and the index number $m\geq 1$, $p > 2$. This equation can model some materials whose density rely on the velocity $u_t$. The asymptotic behavior of solutions for equation (1.4) has been studied by many authors. For example, when $a = b = 0$, Cavalcanti et al. ^[1] proved that there exist the global results for $\gamma\geq 0$ and exponential decay for $\gamma > 0$. When $a = 0$, by using the potential well method, Messaoudi et al. ^[14] obtained a global existence and an exponential decay result. When $a = \gamma = 0$, Liu ^[11], by constructing a suitable Lyapunov function and using the perturbed energy method, proved that the solution energy is general decay. Furthermore, when $a = b = \gamma = 0$, Messaoudi and Al-Khulaifi ^[13] establish a general and optimal decay of solution energy with the relaxation function satisfies $g'(t)\leq-\xi(t)g^\theta(t)$, $1\leq \theta < \frac{3}{2}$. Also, in ^[12], under the condition $g'(t)\leq-\varepsilon(t)\chi(g(t))$, where $\chi$ is increasing and convex without any additional constraints, Mustafa established energy decay results that address both the optimality and generality by using the multiplier method and some properties of convex functions. For results of the same nature, we refer the readers to ^[4,6,7,19] and the references therein.

As far as we know, the decay property for equation (1.1) has not been considered. In this article, we consider the decay property of solution energy for problem (1.1), we obtain the following result: under certain class of relaxation functions and initial data, by using some inequalities and constructing a suitable Lyapunov function, we establish a general decay result for problem (1.1). Moreover, without restrictive conditions, we also obtain the optimal polynomial decay which seldom appear in previous literature.

This article is organized as follows. In section 2, we present some material needed for our work. In section 3, we show the global existence of solution and establish the general decay result.

2.   Preliminaries

In this part, we give some theorems and lemmas needed in the proof of our results. Firstly, we make the following assumptions.

(A1) $g(t):[0, +\infty)\rightarrow (0, +\infty)$ is a non-increasing $C^1$ function such that

(A2) There exists a positive differentiable function $\xi(t): [0, +\infty)\rightarrow (0, +\infty)$ such that

and $\xi(t)$ satisfies

For our work, we introduce the following functionals:

where

Then, we state a local existence theorem to the problem (1.1) that can be proved by combining arguments of ^[1,3,9].

Theorem 2.1 (^[5], Theorem 2.1). Suppose that $(1.2)$ and $(A1)$ hold and initial data $(u_0, u_1)\in H_0^1(\Omega)\times L^2(\Omega)$ is given.Then problem (1.1) has a unique local solution

Lemma 2.2. Assume the assumption $(1.2)$ and $(A1)$ hold. Let $u$ be a solution of $(1.1)$. Then $E(t)$ is non-increasing. In addition, we get the following energy inequality

Proof. Multiplying (1.1) by $u_{t}$ and integrating over $\Omega$, we can get

Now, we estimate the last term in the left-hand side of equation $(2.2)$ as follows

Insert $(2.3)$ into $(2.2)$, we obtain

This complete the proof.

The following Lemma is used for studying the decay of solution.

Lemma 2.3 (^[13], Corollary 2.1). Assume that $g(t)$ satisfies $(A1)$ and $(A2)$ and $u$ is the solution of $(1.1)$ then we have

3.   The decay result

In this section, we state and prove the decay result for global solutions. Firstly, we establish the global existence theorem.

Lemma 3.1. Assume the assumption $(A1)$, $(A2)$ and $(1.2)$ hold and the initial data $(u_0, u_1)\in H^1_0(\Omega)\times L^2(\Omega)$ such that

then $I(u(t)) > 0$ for $\forall \; t > 0$.

Proof. Due to $I(u_{0}) > 0$, then there exists a time $T_{m} < T$ such that $I\big(u(t)\big)\geq0$ for $\forall t \in[0, T_{m})$. So, we have

By applying $(A1)$, $(2.1)$ and $(3.2)$, we deduce

Then, combine $(A1)$, $(3.1)$ and $(3.3)$, we arrive at

Therefore,

for $\forall t\in [0, T_{m})$. Repeating the process and using the fact that

$T_{m}$ is extended to $T$.

Theorem 3.2. Assume that $(A1)$, $(A2)$ and $(1.2)$ hold, and the initial data $(u_0, u_1)\in H^1_0(\Omega)\times L^2(\Omega)$ and satisfies $(3.1)$, then the solution is global and bounded.

Proof. Our objective is to prove $||\nabla u(t)||^2_2+||u_t(t)||^{\rho+2}_{\rho+2}$ is bounded independently of t. For the aim, we apply the Lemma $3.1$ to obtain

for $I(u(t))\geq0$ and $(g\circ \nabla u)(t)$ is positive. Therefore, we have

Then we complete the proof. For obtaining the general decay rate estimate, let us consider the following functionals

where $M$ and $\varepsilon$ are positive constants and

Lemma 3.3. For $M$ large enough while $\varepsilon$ is small enough, then we have the following relation

holds, where $\alpha_1$ and $\alpha_2$ are two positive constants.

Proof. Applying Hölder inequality, Young inequality, Sobolev embedding theorem and (3.3) we have

and

When $M$ is large enough and $\varepsilon$ is small enough, we arrive at

Analogously, we have

Lemma 3.4. Under the assumption $(A1)$ and $(A2)$, let $u$ be the solution of (1.1), then the functional

satisfies

Proof. Taking a time derivative of $(3.7)$ and applying equation $(1.1)$, we can deduce

We now estimate the third term on the right-hand side of $(3.9)$, yields

at present, estimate the second term in the right-hand side of $(3.10)$, for $\forall\; \eta_1 > 0$, we can arrive at

Inserting (3.11) into (3.10), we get

For the forth term in the right-hand side of $(3.9)$, for $\forall\; \eta_2 > 0$, we can get

Inserting (3.12) and (3.13) into (3.9), and choosing $\eta_1 = l/(1-l)$, we can deduce

Lemma 3.5. Under the assumption $(A1)$ and $(A2)$, and let $u$ be the solution of (1.1), then the functional

satisfies

Proof. Taking a derivative of $\zeta(t)$, we have

Inserting equation (1.1) into (3.17), we get

We now estimate the first term in the right-hand side of (3.18), for $\forall\; \delta > 0$, we can deduce

For the second term

Similarly, we have

and

Combining (3.18)–(3.23), we can deduce

Theorem 3.6. Let initial data $(u_0, u_1)\in H^1_0(\Omega)\times L^2(\Omega)$ be given and satisfy (3.1). Assume that (1.2), (A1) and (A2) hold. Then, for each $t_0 > 0$, there exist strictly positive constants K and k such that the solution of (1.1) satisfies, for all $t\geq t_0$,

Proof. Taking a derivative of (3.5), we can obtain

Since $g$ is continuous and $g(0) > 0$, then there exists $t_0 > 0$ such that

By using (2.1), (3.8), (3.16) and (3.27), we arrive at

At this point, we choose $\varepsilon < g_0$ such that

then take $\delta$ and $\eta_2$ small enough such that

Once $\delta$ and $\eta_2$ are fixed, we choose M sufficiently large such that

and

Hence, there exist two positive constants $k_1$ and $k_2$ such that

Multiplying both sides of (3.34) by $\xi(t)$, we obtain

In the case when $r = 1$, by using (A2) and (2.1), then from (3.35) we can infer that

We let $F(t): = \xi(t)L(t)+k_2E(t)$, which is equivalent to $E(t)$, then from (3.36) we can arrive at

A simple integration of (3.37) leads to

In the case when $1 < r < \frac{3}{2}$, we again consider (3.35) and use Lemma 2.2 to get

Multiplying both sides of (3.39) by $\xi^\nu E^\nu (t)$, where $\nu = 2r-2$, then applying Young inequality, we can infer that

We choose $\varepsilon < \frac{k_1}{k_2C}$ such that $k_3: = k_1-k_2C\varepsilon > 0$ and thanks to $\xi'(t)\leq 0$ and $E'(t)\leq 0$, we can deduce that

Then we have

Let $G: = \xi^{\nu+1}E^\nu L+k_2CC_\varepsilon E\sim E$. Then we deduce

By integrating over $(t_0, t)$ and applying the condition that $G\sim E$, we arrive at

This completes the proof.

4.   Conclusions

In this paper, we consider a viscoelastic wave equation with strong damping, by constructing a suitable Lyapunov function, we establish a general decay result, Moreover, without restrictive conditions, we also obtain the optimal polynomial decay result.

Acknowledgments

The authors would like to thank the referees for many valuable comments and suggestions. This work was supported by the Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi(2021L516).

Conflict of interest

The authors declare no conflict of interest.