General and optimal decay rates for a system of wave equations with damping and a coupled source term

1.   Introduction

In this paper, we study the following system of nonlinear viscoelastic equations:

Here, $\Omega$ is a bounded domain of $R^n\, (n\geq1)$ with a smooth boundary $\partial\Omega$, $g_1$ and $g_2$ denote the kernel of the memory term and the index number $m, r\geq 2$, $u$ and $v$ represent the transverse displacements of waves, and $f_1$ and $f_2$ are given functions to be specified later.

To motivate our work, we recall some results related to our work. For a single viscoelastic wave equation of the form

Xu and Lian ^[1] studied the above initial boundary value problem at three different initial energy levels, they proved the local existence of a weak solution, and in the framework of a potential well, they showed the global existence and energy decay of the solution with sub-critical initial energy. Then, through a scaling technique, they parallelly extended all the results for the subcritical case to the critical case. A similar result was also obtained in ^[2,3]. In ^[4], Xu et al. considered the following initial boundary value problem

they obtained the invariant sets, and proved the existence and nonexistence of a global solution. Moreover, they got a finite time blow-up result for certain solutions under a high energy case. In ^[5,6], Messaoudi proved, by using the perturbed energy method and under the supposition that $g'(t)\leq-\xi(t)g(t)$, that the solution energy was a general decay, not necessarily of exponential or polynomial type. {Under the same assumptions on a relaxation function, Liu ^[7] studied a viscoelastic wave equation with the dispersion term and proved that the decay rate of the solution energy was similar to that of the relaxation function. Furthermore, Messaoudi and Al-Khulaifi ^[8] established a general and optimal decay of the solution energy where the relaxation function satisfies $g'(t)\leq-\xi(t)g^\theta(t)$, $1\leq \theta < \frac{3}{2}$. In ^[9], the author established the optimal explicit and general energy decay results under the condition $g'(t)\leq-\xi(t)H(g(t))$, where $H$ is an increasing and convex function near the origin and $\xi$ is a non-increasing function.

Subsequently, the above methods used in the single viscoelastic wave equation were extended to coupled wave systems. For example, Li ^[10] considered the following wave system:

where $\Omega\subset R^n$ $(n\geq1)$ is a bounded domain with a smooth boundary $\partial\Omega$, $0 < \rho$, $m, r\geq1$. Under the following assumptions on the relaxation functions: $g_i(t)\leq-\xi_i(t)g_i(t), t\geq0, i = 1, 2$. Based on the potential well method and the perturbed energy method, the author obtained a general decay result of the solution. Furthermore, in ^[11], the author got some sufficient conditions on initial data such that the solution blows up in finite time at arbitrarily high initial energy. In the same direction and in the case of $\rho = 0$, Liu et al. ^[12] proved that the solution energy was an exponential decay or polynomial decay. Later, this result was furthered by Said-Houari ^[13], who demonstrated that the solution energy was a general decay. In the same nature, Liu ^[14] considered the following system:

where $\gamma_1, \gamma_2\geq 0$ are constans, $\rho$ is a real number, $0 < \rho$, and $\Omega\subset R^n$ $(n\geq1)$ is a bounded domain with a smooth boundary $\partial\Omega$. By exploiting the perturbed energy method, the author proved that the solution energy was an exponential and polynomial decay.

As far as we know, the decay property for the coupled system (1.1) has not been considered. Inspired by the literature ^[8], our aim in this paper is to extend some existing results for a single equation to the case of a couple viscoelastic wave system (1.1), while handling the additional difficulty caused by the nonlinear weak damping term and coupled source term. In this article, the energy decay result is new and the assumptions on the relaxation functions are weak. Then, the purpose of this paper is to study the decay property of solution energy for the system (1.1) by modifying the method used in the above literatures. Namely, under a 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 the system (1.1). Moreover, without restrictive conditions, we also obtain the optimal polynomial decay rate which seldom appears in previous literatures.

This article is organized as follows: In Section 2, some materials needed for our work are presented. In Section 3, we show the global existence of a solution and establish the general and optimal decay rates of the solution for the system (1.1).

2.   Preliminaries

In this part, some theorems and lemmas needed for our work are given. First, we make the following assumptions:

(A1) $g_i:(0, +\infty)\rightarrow (0, +\infty) (i = 1, 2)$ are non-increasing $C^1$ functions satisfying

(A2) There exists two positive differentiable functions $\xi_i(t): [0, +\infty)\rightarrow (0, +\infty)$ such that

and the function $\xi_i(t)$ satisfies

(A3) For the functions $f_1$and$f_2$, we assume that

It is easy to verify that

where

(A4) For the nonlinearity, we assume that

For our work, we introduce the following functionals:

where

The following result is concerned with local existence and uniqueness of weak solutions to the system (1.1). We can easily obtain it by using the Faedo-Galerkin approximation methods and the Banach contraction mapping principle, which is similar to ^[15] with slight modification, and the process of the proof is standard, so we omit it here.

Theorem 2.1 (^[15], Theorem 2.1). Suppose that $(A1)$, $(A2)$, and $(A4)$ hold and initial data $(u_0, u_1)\in H_0^1(\Omega)\times L^2(\Omega)$, $(v_0, v_1)\in H_0^1(\Omega)\times L^2(\Omega)$ are given. Then there exists a unique local weak solution $(u, v)$ of problem (1.1) defined in $[0, T]$ for some $T > 0$ small enough.

Lemma 2.2. Assume that $(A1)$–$(A4)$ hold. Let $(u, v)$ be the solution of the system $(1.1)$. Then the energy functional $E(t)$ is non-increasing. In addition, we get the following energy inequality:

Proof. Multiplying the first two equations in system (1.1) by $u_{t}$ and $v_{t}$, respectively, integrating over $\Omega$, and then adding them up, we can get

where

Similarly, we have

Substituting (2.3) and (2.4) into (2.2) yields

Then, we have

This completes the proof. □

Now, we establish the global existence theorem.

Lemma 2.3. Assume the assumptions $(A1)$–$(A4)$ hold and the initial data $(u_0, u_1)\in H^1_0(\Omega)\times L^2(\Omega)$, $(v_0, v_1)\in H^1_0(\Omega)\times L^2(\Omega)$ such that

and then we have

Proof. Due to $I(u_0) > 0$, then by continuity of $I(t)$ about variable $t$, there exists a maximal time $0 < T^* < T$ such that

which implies that $\forall t\in [0, T^*]$,

By (A1) and (2.1), we see that

By (A3), (2.25), and (2.6), we can deduce that

From (2.6) and (2.10), we infer that

Therefore,

By repeating these steps, $T^*$ is extended to $T_m$. □

Theorem 2.4. Assume that $(A1)$–$(A4)$ hold, the initial data $(u_0, u_1)\in H^1_0(\Omega)\times L^2(\Omega)$, $(v_0, v_1)\in H^1_0(\Omega)\times L^2(\Omega)$, and satisfies $(2.6)$. Then the solution is bounded and global in time.

Proof. Applying Lemmas $2.2$ and $2.3$, we can obtain

which implies that the solution of system $(1.1)$ is global and bounded. □

Lemma 2.5 (^[16]). Suppose that $g\in C[0, \infty]$, $\omega\in L^1_{loc}(0, \infty)$, and $0\leq\theta\leq1$; and then we have that

Lemma 2.6 (^[8]). Assume that $g_i$ satisfies $(A1)$ and $(A2)$, for $i = 1, 2$, and then

Proof. From $(A1)$ and $(A2)$, we can obtain

Integrating (2.14) over $(0, +\infty)$ and using the condition $0\leq\theta < 2-\gamma$, we can deduce that

□

Lemma 2.7 (^[17]). Suppose that the conditions $(A1)$–$(A4)$, (2.6), and (2.11) hold, $u\in L^{\infty}(0, T; H^1_0(\Omega))$, and $g$ is a continuous function. For $0 < \theta < 1$, there exists $C > 0$ such that

Proof. From the hypothesis on $u$ and Lemma 2.7, we can obtain

Now, for $0 < \theta < 1$ and the conditions (2.6) and (2.11), we have

The proof is now complete. □

By taking $\theta = \frac{1}{2}$, we have

Similarly, we have

The following lemmas are crucial for studying the decay of a solution.

Lemma 2.8 (^[8]). Assume that $g_i(t)$ satisfies $(A1)$ and $(A2)$, and $(u, v)$ is the solution of $(1.1)$, and then we obtain

Proof. Multiplying both sides of (2.19) by $\xi_1(t)$ and using $(A2)$, (2.1), and (2.13), we can deduce that

Similarly, we have

□

Lemma 2.9 (^[18], Lemma 4.2). There exist two positive constants $\gamma_1$ and $\gamma_2$ such that

Lemma 2.10 (^[18], Lemma 3.2). Assume that (A4) holds. Then there exists $\eta > 0$ such that, for any $(u, v)\in H^1_0(\Omega)\times H^1_0(\Omega)$, we obtain

3.   The decay result

In this section, we state and prove the decay result for the global solutions. For obtaining the general decay rate estimate, let us consider the following functionals:

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

Lemma 3.1. For $M$ large enough and $\varepsilon$ small enough, we have that the following relation

holds, where $\nu_1$ and $\nu_2$ are two positive constants.

Proof. Applying the Hölder inequality, Young inequality, Sobolev embedding theorem, and (2.9), for $\forall \delta > 0$, we have

Similarly, we have

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

Analogously, we have

□

Lemma 3.2. Under the assumptions $(A1)$–$(A4)$, let $(u, v)$ be the solution of system (1.1). Then the functional

satisfies

Proof. Taking a time derivative of $(3.5)$ and applying Eq $(1.1)$, we can deduce

We now estimate the third term on the right-hand side of $(3.7)$, yielding

at present. We estimate the second term in the right-hand side of $(3.8)$, for $\forall\; \eta_1 > 0$, and we arrive at

Inserting (3.9) into (3.8), we get

Similarly, for $\forall\; \eta_2 > 0$, we have

For the fourth term in the right-hand side of $(3.7)$, for $\forall \alpha > 0$, we can get

Similarly, we can get

Inserting (3.10)–(3.13) into (3.7), and choosing

we can deduce

□

Lemma 3.3. Under the assumptions $(A1)$–$(A4)$, letting $(u, v)$ be the solution of (1.1), then the functional

satisfies

Proof. Taking the derivative of $\zeta(t)$ with respect to variable $t$, we have

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

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

For the fourth term, it follows from Lemma $2.9$ that

Similarly, we have

Combining (3.18)–(3.30), we can deduce that

□

Theorem 3.4. Let initial data $(u_0, u_1), (v_0, v_1)\in H^1_0(\Omega)\times L^2(\Omega)$ be given and satisfy (2.6). Assume that $(A1)$–$(A4)$ hold. Then, for each $t_0 > 0$, there exist strictly positive constants $k_1, k_2$, and $k_3$ such that the solution of system (1.1) satisfies, for all $t\geq t_0$,

where $\xi(t): = \min\{\xi_1(t), \xi_2(t)\}$.

Proof. Taking a derivative of (3.1), we can obtain

Since $g_1$ and $g_2$ are continuous and $g_i(0) > 0$, then there exists $t\geq t_0 > 0$ such that

and

By using (2.1), (3.6), (3.16), and (3.34), we arrive at

At this point, we choose $\varepsilon$ and $\delta$ small enough such that

Once $\varepsilon$ and $\delta$ are fixed, we choose ital $"M"$ sufficiently large such that

Hence, the inequality (3.37) can be changed to

Then, there exist two positive constants $m_1$ and $m_2$ such that

We let $\xi(t): = \min\{\xi_1(t), \xi_2(t)\}$, since $\xi_1(t)$ and $\xi_2(t)$ are non-increasing and non-negative functions, and then the function $\xi(t)$ is a non-increasing and non-negative function. In the case when $\gamma = 1$, by multiplying both sides of (3.39) by $\xi(t)$ and using $(A2)$ and (2.1), we can deduce that

We let $F(t): = \xi(t)L(t)+m_2E(t)$, which is equivalent to $E(t)$, since the function $L(t)$ is also equivalent to $E(t)$ and $E(t)\geq0$, $\xi'(t)\leq 0$. Then we can deduce that $\xi'(t)L(t)\leq0$. Then from (3.40), for some $m_3, m_4 > 0$, we can arrive at

A simple integration of (3.41) leads to

In the case when $1 < \gamma < \frac{3}{2}$, we again consider (3.40) and use Lemma 2.8 to get, for $C_1 > 0$,

Multiplying both sides of (3.43) by $\xi^\sigma(t) E^\sigma (t)$, where $\sigma = 2\gamma-2$, then applying the Young inequality, we can infer that

We choose $\varepsilon < \frac{m_1}{C_1}$ such that $m_5: = m_1-C_1\varepsilon > 0$ and thanks to $\xi'(t)\leq 0$ and $E'(t)\leq 0$, we can deduce that

Then we have

Let $G: = \xi^{\sigma+1}E^\sigma L+C_1C_\varepsilon E\sim E$. Then we deduce that

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 system of two viscoelastic wave equations with Dirichlet boundary conditions. By constructing a suitable Lyapunov function, we establish a general decay result. Moreover, without restrictive conditions, we also obtain the optimal polynomial decay result.

Author contributions

Qian Li: Conceptualization, methodology, writing original draft, writing and editing, formal analysis, funding acquisition; Yanyuan Xing: methodology, writing and editing, formal analysis supervision, funding acquisition. Both authors have read and approved the final version of the manuscript for publication.

Acknowledgments

The authors would like to thank the referees for many valuable comments and suggestions. This work is supported by the Fundamental Science Research Projects of Shanxi Province (No. 202203021222332, 202103021223379), China.

Conflict of interest

The authors declare no conflicts of interest.