Existence and General Decay of Solutions for a Weakly Coupled System of Viscoelastic Kirchhoff Plate and Wave Equations

Abstract

In this paper, a weakly coupled system (by the displacement of symmetric type) consisting of a viscoelastic Kirchhoff plate equation involving free boundary conditions and the viscoelastic wave equation with Dirichlet boundary conditions in a bounded domain is considered. Under the assumptions on a more general type of relaxation functions, an explicit and general decay rate result is established by using the multiplier method and some properties of the convex functions.

1. Introduction

In this paper, we consider the following weakly coupled system of Kirchhoff plate and wave equations:

$$\left\{\begin{array}{ccc}{\displaystyle u_{tt}-\gamma \Delta u_{tt}+{\Delta }^{2}u-{\int }_0^t{g}_1(t-s){\Delta }^{2}u\left(s\right)ds+\alpha v=0}\hfill & \mathrm{in}\hfill & \Omega \times (0,\infty )\hfill \\ {\displaystyle v_{tt}-\Delta v+{\int }_0^t{g}_2(t-s)\Delta v\left(s\right)ds+\alpha u=0}\hfill & \mathrm{in}\hfill & \Omega \times (0,\infty )\hfill \\ u={\partial }_{\nu }u=0\hfill & \mathrm{on}\hfill & {\Gamma }_0\times (0,\infty )\hfill \\ {\mathbf{B}}_{1}u-{\mathbf{B}}_1\left\{{\displaystyle {\int }_0^t{g}_1(t-s)u\left(s\right)ds}\right\}=0\hfill & \mathrm{on}\hfill & {\Gamma }_1\times (0,\infty )\hfill \\ {\mathbf{B}}_{2}u-\gamma {\partial }_{\nu }{u}_{tt}-{\mathbf{B}}_2\left\{{\displaystyle {\int }_0^t{g}_1(t-s)u\left(s\right)ds}\right\}=0\hfill & \mathrm{on}\hfill & {\Gamma }_1\times (0,\infty )\hfill \\ v=0\hfill & \mathrm{on}\hfill & \Gamma \times (0,\infty )\hfill \\ u\left(0\right)=u^0,u_t\left(0\right)=u^1,v\left(0\right)=v^0,v_t\left(0\right)=v^1\hfill & \mathrm{in}\hfill & \Omega ,\hfill \end{array}\right.$$

(1)

where $\Omega$ is an open set of ${\mathbb{R}}^2$ with regular boundary $\Gamma =\partial \Omega ={\Gamma }_0\cup {\Gamma }_1$ (class $C^4$ will be enough), such that ${\overline{\Gamma }}_0\cap {\overline{\Gamma }}_1=\mathrm{\varnothing }$, the initial data $u^0$, $u^1$, $v^0$ and $v^1$ lie in an appropriate Hilbert space; the constant $\gamma >0$ is the rotational inertia of the plate; and the constant $0<\mu <\frac{1}{2}$ is the Poisson coefficient. The boundary operators ${\mathbf{B}}_1$, ${\mathbf{B}}_2$ are defined by

$${\mathbf{B}}_{1}u=\Delta u+(1-\mu )B_{1}u,$$

$${\mathbf{B}}_{2}u={\partial }_{\nu }\Delta u+(1-\mu )B_{2}u,$$

and

$$B_{1}u=2{\nu }_1{\nu }_2{u}_{x_1{x}_2}-{\nu }_1^2{u}_{x_2{x}_2}-{\nu }_2^2{u}_{x_1{x}_1},$$

$$B_{2}u={\partial }_{\tau }\left(({\nu }_1^2-{\nu }_2^2)u_{x_1{x}_2}+{\nu }_1{\nu }_2(u_{x_2{x}_2}-u_{x_1{x}_1})\right),$$

where $\nu =({\nu }_1,{\nu }_2)$ is the unit outer normal vector to $\Gamma$, and $\tau =(-{\nu }_2,{\nu }_1)$ is a unit tangent vector.

The coupling parameter $\alpha$ is assumed to satisfy:

$$\left|\alpha \right|<{\lambda }_0\eta ,$$

where ${\lambda }_0^2$ is the first eigenvalue of the operator ‘$-\Delta$’ with Dirichlet boundary conditions, and ${\eta }^2$ is the coercivity constant of the operator $\mathring{A}={\Delta }^2$, defined as follows:

$$\mathring{A}:D\left(\mathring{A}\right)\subset L^2(\Omega )\to L^2(\Omega ),$$

with domain

$$\begin{array}{c}D\left(\mathring{A}\right)=\{u\in H^4(\Omega )\cap H_{{\Gamma }_0}^2(\Omega ):\Delta u+(1-\mu )B_{1}u={\partial }_{\nu }\Delta u+(1-\mu )B_{2}u=0\mathrm{on}{\Gamma }_1\},\hfill \end{array}$$

with $H_{{\Gamma }_0}^2=V=\{u\in H^2(\Omega ):u={\partial }_{\nu }u=0\mathrm{on}{\Gamma }_0\}$.

It is clear that $\mathring{A}$ is positive definite and self-adjoint. We also define

$$H_{{\Gamma }_0}^1(\Omega )=\{u\in H^1(\Omega ):u=0\mathrm{on}{\Gamma }_0\}.$$

We have, for all $u,v\in V$ (see [1]):

$${⟨{\mathring{A}}^{\frac{1}{2}}u,{\mathring{A}}^{\frac{1}{2}}v⟩}_{L^2(\Omega )}={\left(\mathring{A}u,v\right)}_{V^{\prime }\times V}=a(u,v),$$

where $a:V\times V\to \mathbb{R}$ is a symmetric bilinear form defined by

$$a(u,v)={\int }_{\Omega }\left\{u_{x_1{x}_1}{v}_{x_1{x}_1}+u_{x_2{x}_2}{v}_{x_2{x}_2}+2(1-\mu )u_{x_1{x}_2}{v}_{x_1{x}_2}+\mu \left(u_{x_1{x}_1}{v}_{x_2{x}_2}+u_{x_2{x}_2}{v}_{x_1{x}_1}\right)\right\}dx.$$

We first recall the following Green’s formula (see [2]):

$$a(u,v)={\int }_{\Omega }{\Delta }^{2}uvdx+{\int }_{\Gamma }\left({\mathcal{B}}_{1}u{\partial }_{\nu }v-{\mathcal{B}}_{2}uv\right)d\Gamma ,\forall u\in H^4(\Omega ),v\in H^2(\Omega ).$$

For further purposes, we need a weaker version of the above. Indeed, as $\mathcal{D}\left(\overline{\Omega }\right)$ (the space of all functions defined in $\Omega$, which are restrictions to $\Omega$ of $C^{\infty }$ functions with compact support in ${\mathbb{R}}^2$) is dense in $E({\Delta }^2,L^2(\Omega )):=\left\{u\in H^2(\Omega )|{\Delta }^{2}u\in L^2(\Omega )\right\}$ equipped with its natural norm, we deduce that $u\in E({\Delta }^2,L^2(\Omega ))$ (see Theorem 5.6 in [3]) satisfies ${\mathcal{B}}_{1}u\in H^{-\frac{1}{2}}(\Gamma )$ and ${\mathcal{B}}_{2}u\in H^{-\frac{3}{2}}(\Gamma )$ with

$$a(u,v)={\int }_{\Omega }{\Delta }^{2}uvdx+{⟨{\mathcal{B}}_{1}u,{\partial }_{\nu }v⟩}_{H^{-\frac{1}{2}}(\Gamma ),H^{\frac{1}{2}}(\Gamma )}-{⟨{\mathcal{B}}_{2}u,v⟩}_{H^{-\frac{3}{2}}(\Gamma ),H^{\frac{3}{2}}(\Gamma )},\forall v\in H^2(\Omega ).$$

Now, with the parameter $\gamma >0$, we define a space $W=H_{{\Gamma }_0,\gamma }^1(\Omega )$ equivalent to $H_{{\Gamma }_0}^1(\Omega )$, with its inner product being

$${⟨u_1,u_2⟩}_{H_{{\Gamma }_0,\gamma }^1(\Omega )}\equiv {⟨u_1,u_2⟩}_{L^2(\Omega )}+\gamma {⟨\nabla u_1,\nabla u_2⟩}_{L^2(\Omega )}\forall u_1,u_2\in H_{{\Gamma }_0}^1(\Omega ),$$

and with its dual (pivotal with respect to $L_2$ inner product) denoted as $H_{{\Gamma }_0,\gamma }^{-1}(\Omega )$.

When $g_1=g_2=\alpha =0$, the first equation, in system (1), is well known as the Kirchhoff plate equation, while the second equation represents the classical wave equation. We study, in the present paper, a weak coupling of a symmetric type of these two equations (with the presence of memory terms), which means that the equations are coupled by displacements.

Model (1) describes the interaction of a viscoelastic Kirchhoff plate with rotational forces, and a viscoelastic membrane. The plate is clamped along ${\Gamma }_0$, and without bending and twisting moments on ${\Gamma }_1$.

We first recall some results for a single-wave equation and Kirchhoff plate equation. For a viscoelastic wave equation, we refer to [4,5,6,7,8] and references therein, in which the authors proved that the energy decays exponentially if the relaxation function g decays exponentially, and polynomially if g decays polynomially. In [9], Cavalcanti et al. considered the following wave equation:

$${\displaystyle u_{tt}-{\kappa }_0\Delta u+{\int }_0^{t}div[a\left(x\right)g(t-s)\nabla u\left(s\right)]ds+f\left(u\right)+b\left(x\right)h\left(u_t\right)=0,}$$

where frictional damping was also considered. They proved an exponential stability result for g decaying exponentially and h having linear and polynomial stability result for g decaying polynomially and h having a polynomial growth near zero. We mention, in the case where ${\kappa }_0=1$ and $f=h=0$, that the uniform decay of solutions was obtained in [10]. For the viscoelastic Kirchhoff plate equation, in [11], the authors showed the exponential and polynomial decay of the solutions to the viscoelastic plate equation. They considered a relaxation function satisfying

$$-d_{0}g\left(t\right)\le g^{\prime }\left(t\right)\le -d_{1}g\left(t\right),0\le g{}^{″}\left(t\right)\le d_{2}g\left(t\right).$$

For some positive constant $d_i,i=0,1,2$. Park et al. [12] obtained a general decay for weak viscoelastic Kirchhoff plate equations with delay boundary conditions. Motivated by the work of Lasiecka and Tatar [13], where a wave equation with frictional damping was considered, another step forward was taken by considering relaxation functions satisfying

$$g^{\prime }\left(t\right)\le -H\left(g\left(t\right)\right),$$

where the function $H>0$ satisfies $H\left(0\right)=H^{\prime }\left(0\right)=0$, and is strictly increasing and strictly convex near the origin. This condition was first introduced by Alabau-Boussouira and Cannarsa [14]. It turned out that the convexity properties can be explored for a general class of dissipative systems [15,16]. We also point out that the importance of the works [15,16] in which simple sharp optimal or quasi-optimal upper energy decay rates have been established.

For a coupled wave system, a general model on coupled wave equations with weak damping is given by:

$$\left\{\begin{array}{c}{\displaystyle u_{tt}-\Delta u+{\int }_0^t{g}_1(t-s)\Delta u\left(s\right)ds+h_1\left(u_t\right)=f_1(u,v),}\hfill \\ {\displaystyle v_{tt}-\Delta v+{\int }_0^t{g}_2(t-s)\Delta v\left(s\right)ds+h_2\left(v_t\right)=f_2(u,v).}\hfill \end{array}\right.$$

In [17], Han and Wang established several results related to local existence, global existence and finite time blow-up (the initial energy $E\left(0\right)<0$), by taking $h_1\left(u_t\right)={\left|u_t\right|}^{m-1}{u}_t$ and $h_2\left(v_t\right)={\left|v_t\right|}^{r-1}{v}_t$. Later on, Houari et al. [18] improved the last results by considering a larger class of initial data for which the initial energy can take positive values. Messaoudi and Tatar [13] considered a coupled system only with viscoelastic terms, and proved exponential decay and polynomial decay results. Al-Gharabli and Kafini considered the system in [13] and established a more general decay result; see [19]. Mustafa [20] considered the following problem

$$\left\{\begin{array}{c}{\displaystyle u_{tt}-\Delta u+{\int }_0^t{g}_1(t-s)\Delta u\left(s\right)ds+f_1(u,v)=0,}\hfill \\ {\displaystyle v_{tt}-\Delta v+{\int }_0^t{g}_2(t-s)\Delta v\left(s\right)ds+f_2(u,v)=0,}\hfill \end{array}\right.$$

and proved the well-posedness and energy decay result. The decay result was improved by Messaoudi and Hassan in their recent paper [21], where they established a new general decay result for a wider class of relaxation functions. We also mention the work [22], in which the authors proved the global existence and decay rate estimates of solutions for a system of viscoelastic wave equations of the Kirchhoff type with logarithmic nonlinearity.

For indirect stabilization, Alabau et al. [23] considered the stabilization of an abstract system of two coupled second-order evolution equations, wherein only one of the equations is stabilized and showed that the energy decays polynomially. Recently, Hajej et al. [24] studied the indirect stabilization (only one equation of the coupled system is damped) of a coupled wave equation and Kirchhoff plate equation without viscoelastic terms ($g_1=g_2=0$), and with frictional damping, the polynomial decay was derived. Motivated by these works, in this paper, we study the stability of this coupled system but only with the presence of viscoelastic terms in the two equations with a wider class of relaxation functions. We establish a very general energy decay result of the system by the general approach in [14].

Hereinafter, we assume that

(A1):

$g_i:[0,+\infty )\to (0,+\infty )$ (for $i=1,2$) are two non-increasing $C^1$ functions such that:

$$1-{\int }_0^{\infty }{g}_i\left(\tau \right)d\tau =l_i>0.$$

(A2):

There exists a positive $C^1$ function $Q:(0,+\infty )\to (0,+\infty )$, where Q is linear or a strictly increasing and strictly convex $C^2$ function on $(0,r],(r<1)$, with $Q\left(0\right)=Q^{\prime }\left(0\right)=0$, such that

$$g_i^{\prime }\left(t\right)\le -{\xi }_i\left(t\right)Q\left(g_i\left(t\right)\right),\forall t\ge 0,$$

where ${\xi }_1$ and ${\xi }_2$ are positive non-increasing differentiable functions.

Remark 1. 

The function Q, defined in assumption (A2), was introduced by Alabau-Boussouira and Cannarsa [14].

To simplify calculations in our analysis, we introduce the following notations:

$${\displaystyle (g_1\square u)\left(t\right)={\int }_0^t{g}_1(t-s)a\left(u\left(t\right)-u\left(s\right),u\left(t\right)-u\left(s\right)\right)ds,}$$

$${\displaystyle (g_2\circ v)\left(t\right)={\int }_0^t{g}_2(t-s){\parallel v\left(t\right)-v\left(s\right)\parallel }^{2}ds.}$$

We will use C and c, throughout this paper, to denote generic positive constants.

The paper is organized as follows. The well-posedness of the problem, that is, the existence of a global weak solution, is proved in Section 2. In Section 3, we state and establish the general decay result of the energy by using the perturbed energy method, which introduces a new Lyapunov function.

2. Global Existence

This section deals with the existence and uniqueness of a global weak solution. In fact, we start by proving the existence and uniqueness of a unique local weak solution by using the Faedo–Galerkin approach, and afterward, show that this solution is global. This means that our system is well-posed.

We start this section by presenting the definition of a weak solution of the problem (1).

Definition 1. 

Let $T>0$. A pair of functions $(u,v)$ such that

$$u\in C([0,T],V)\cap C^1([0,T],W),v\in C([0,T],H_0^1(\Omega ))\cap C^1([0,T],L^2(\Omega )),$$

is called a weak solution of the problem (1) if

$$\begin{array}{ccc}& & {\displaystyle {\int }_{\Omega }{u}_{tt}wdx+\gamma {\int }_{\Omega }\nabla u_{tt}\nabla wdx+a(u,w)-{\int }_0^t{g}_1(t-s)a(u\left(s\right),w)ds+\alpha {\int }_{\Omega }vwdx=0,}\hfill \\ & & u(x,0)=u_0\left(x\right),u_t(x,0)=u_1\left(x\right)\hfill \end{array}$$

and

$$\begin{array}{ccc}& & {\displaystyle {\int }_{\Omega }{v}_{tt}ydx+{\int }_{\Omega }\nabla v\nabla ydx-{\int }_0^t{g}_2(t-s){\int }_{\Omega }\nabla v\left(s\right)\nabla ydxds+\alpha {\int }_{\Omega }uydx=0,}\hfill \\ & & v(x,0)=v_0\left(x\right),v_t(x,0)=v_1\left(x\right)\hfill \end{array}$$

for all test functions $w\in V,y\in H_0^1$ and almost all $t\in [0,T].$

Now, we state the local existence theorem.

Theorem 1. 

Suppose (A1) holds and let $(u_0,u_1)\in V\times W$ and $(v_0,v_1)\in H_0^1(\Omega )\times L^2(\Omega )$. Then, problem (1) has a unique local weak solution on $[0,T]$, for any $T>0$.

Proof. 

The existence is proven using the Faedo–Galerkin method. In order to do so, let ${\left\{w_j\right\}}_{j=1}^{\infty }$ and ${\left\{y_j\right\}}_{j=1}^{\infty }$ be a basis of V and $H_0^1$, respectively. Define $V_m=\mathrm{s}pan\{w_1,w_2,\cdots ,w_m\}$ and $Y_m=\mathrm{s}pan\{y_1,y_2,\cdots ,y_m\}$. The projection of the initial data on the finite dimensional subspaces $V_m$ and $Y_m$ is given by

$$u_0^m\left(x\right)=\sum _{j=1}^m{a}_j{w}_j,u_1^m\left(x\right)=\sum _{j=1}^m{b}_j{w}_j,v_0^m\left(x\right)=\sum _{j=1}^m{c}_j{y}_j,v_1^m\left(x\right)=\sum _{j=1}^m{d}_j{y}_j,$$

such that

$$(u_0^m,v_0^m)\to (u_0,v_0)\mathrm{i}nV\times H_0^1(\Omega ),\mathrm{a}nd(u_1^m,v{}_1^m)\to (u_1,v_1)\mathrm{i}nW\times L^2(\Omega ).$$

(2)

We search a solution of the form

$$u^m(x,t)=\sum _{j=1}^m{h}_j\left(t\right)w_j\left(x\right),v^m(x,t)=\sum _{j=1}^m{k}_j\left(t\right)y_j\left(x\right),$$

which satisfy the approximate problem in $V_m$ and $Y_m$, respectively:

$$\begin{array}{ccc}\hfill & & {\displaystyle {\int }_{\Omega }{u}_{tt}^{m}wdx+\gamma {\int }_{\Omega }\nabla u_{tt}^m\nabla wdx+a(u^m,w)-{\int }_0^t{g}_1(t-s)a(u^m\left(s\right),w)ds+\alpha {\int }_{\Omega }{v}^{m}wdx=0,}\hfill \\ & & {\displaystyle {\int }_{\Omega }{v}_{tt}^{m}ydx+{\int }_{\Omega }\nabla v^m\nabla ydx-{\int }_0^t{g}_2(t-s){\int }_{\Omega }\nabla v^m\left(s\right)\nabla ydxds+\alpha {\int }_{\Omega }{u}^{m}ydx=0,}\hfill \\ & & u^m\left(0\right)=u_0^m,u_t^m\left(0\right)=u_1^m,v^m\left(0\right)=v_0^m,v_t^m\left(0\right)=v_1^m.\hfill \end{array}$$

(3)

This system leads to a system of ODEs for unknown functions $h_j\left(t\right)$ and $k_j\left(t\right)$. Based on the standard existence theory for ODE, one can conclude the existence of a solution $(u^m,v^m)$ of (3) on a maximal interval $[0,t_m),0<t_m\le T$ for each $m\ge 1$. In fact, $t_m=T$, and the local solution is uniformly bounded independent of m and t. To show this, we take $w=u_t^m$ in the first equation of (3) and $y=v_t^m$ in the second one. By summing up the resulting equations and integrating by parts over $\Omega$, we obtain

$$\begin{array}{c}\hfill \frac{d}{dt}{E}^m\left(t\right)=\frac{1}{2}(g_1{}^{\prime }\square u^m)\left(t\right)-\frac{1}{2}{g}_1\left(t\right)a(u^m,u^m)+\frac{1}{2}(g_2^{\prime }\circ \nabla v^m)\left(t\right)-\frac{1}{2}{g}_2\left(t\right){\parallel \nabla v^m\left(t\right)\parallel }^2,\end{array}$$

(4)

where

$$\begin{array}{ccc}\hfill E^m\left(t\right)& =& \frac{1}{2}\left({\displaystyle 1-{\int }_0^t{g}_1\left(s\right)ds}\right)a(u^m,u^m)+\frac{1}{2}\parallel u_t^m{\parallel }^2+\frac{\gamma }{2}{\parallel \nabla u_t^m\parallel }^2+\frac{1}{2}(g_1\square u^m)\left(t\right)\hfill \\ & +& {\displaystyle \frac{1}{2}\left({\displaystyle 1-{\int }_0^t{g}_2\left(s\right)ds}\right)\parallel \nabla v^m{\parallel }^2+\frac{1}{2}(g_2\circ \nabla v^m)\left(t\right)+\frac{1}{2}{\parallel v_t^m\parallel }^2+\alpha {\int }_{\Omega }{u}^m{v}^{m}dx.}\hfill \end{array}$$

Notably, by (2), that $(u_0^m,v_m^0)$ and $(u_1^m,v_1^m)$ are bounded, respectively, in $V\times H_0^1(\Omega )$ and $W\times L^2(\Omega )$, we integrate (4) over $(0,t),0<t<t_m$, to obtain

$$E^m\left(t\right)\le E^m\left(0\right)\le M,$$

where M is a positive constant independent of t and m. Thus, we can extend $t_m$ to T and, in addition, we have

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\left(u^m\right)\mathrm{is}\mathrm{a}\mathrm{bounded}\mathrm{sequence}\mathrm{in}{L}^{\infty }(0,T;V),\hfill \\ \\ \left(u_t^m\right)\mathrm{is}\mathrm{a}\mathrm{bounded}\mathrm{sequence}\mathrm{in}{L}^{\infty }(0,T;W),\hfill \\ \\ \left(v^m\right)\mathrm{is}\mathrm{a}\mathrm{bounded}\mathrm{sequence}\mathrm{in}{L}^{\infty }(0,T;H_0^1(\Omega )),\hfill \\ \\ \left(v_t^m\right)\mathrm{is}\mathrm{a}\mathrm{bounded}\mathrm{sequence}\mathrm{in}{L}^{\infty }(0,T;L^2(\Omega )).\hfill \end{array}\right.\end{array}$$

Therefore, there exists a subsequence of $\left(u^m\right)$ and $\left(v^m\right)$, still denoted by $\left(u^m\right)$ and $\left(v^m\right)$, respectively, such that

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{u}^m\rightharpoonup u\mathrm{weakly}\mathrm{star}\mathrm{in}{L}^{\infty }(0,T;V)\mathrm{and}\mathrm{weakly}\mathrm{in}{L}^2(0,T;V),\hfill \\ \\ u_t^m\rightharpoonup u_t\mathrm{weakly}\mathrm{star}\mathrm{in}{L}^{\infty }(0,T;W)\mathrm{and}\mathrm{weakly}\mathrm{in}{L}^2(0,T;W),\hfill \\ \\ v^m\rightharpoonup v\mathrm{weakly}\mathrm{star}\mathrm{in}{L}^{\infty }(0,T;H_0^1(\Omega ))\mathrm{and}\mathrm{weakly}\mathrm{in}{L}^2(0,T;H_0^1(\Omega )),\hfill \\ \\ v_t^m\rightharpoonup v_t\mathrm{weakly}\mathrm{star}\mathrm{in}{L}^{\infty }(0,T;L^2(\Omega ))\mathrm{and}\mathrm{weakly}\mathrm{in}{L}^2(0,T;L^2(\Omega )).\hfill \end{array}\right.\end{array}$$

(5)

Now, integrate (3) over $(0,t)$ to obtain

$$\begin{array}{ccc}\hfill & & {\displaystyle {\int }_{\Omega }{u}_t^{m}wdx+\gamma {\int }_{\Omega }\nabla u_t^m\nabla wdx+{\int }_0^{t}a(u^m,w)ds-{\int }_0^t{\int }_0^s{g}_1(s-\zeta )a(u^m\left(\zeta \right),w)d\zeta ds}\hfill \\ & & {\displaystyle +\alpha {\int }_0^t{\int }_{\Omega }{v}^{m}wdxds={\int }_{\Omega }{u}_1^{m}wdx+\gamma {\int }_{\Omega }\nabla u_1^m\nabla wdx.}\hfill \\ & & {\displaystyle {\int }_{\Omega }{v}_t^{m}ydx+{\int }_0^t{\int }_{\Omega }\nabla v^m\nabla ydxds-{\int }_0^t{\int }_0^s{g}_2(s-\zeta ){\int }_{\Omega }\nabla v^m\left(\zeta \right)\nabla ydxd\zeta ds}\hfill \\ & & {\displaystyle +\alpha {\int }_0^t{\int }_{\Omega }{u}^{m}ydxds={\int }_{\Omega }{v}_1^{m}ydx.}\hfill \end{array}$$

Using (5) and letting $m\to \infty$, we obtain for all $w\in V$ and $y\in H_0^1$

$$\begin{array}{ccc}\hfill & & {\displaystyle {\int }_{\Omega }{u}_{t}wdx+\gamma {\int }_{\Omega }\nabla u_t\nabla wdx-{\int }_{\Omega }{u}_{1}wdx-\gamma {\int }_{\Omega }\nabla u_1\nabla wdx}\hfill \\ & & {\displaystyle =-{\int }_0^{t}a(u,w)ds+{\int }_0^t{\int }_0^s{g}_1(s-\zeta )a(u^m\left(\zeta \right),w)d\zeta ds-\alpha {\int }_0^t{\int }_{\Omega }vwdxds}\hfill \\ & & {\displaystyle {\int }_{\Omega }{v}_{t}ydx-{\int }_{\Omega }{v}_{1}ydx=-{\int }_0^t{\int }_{\Omega }\nabla v\nabla ydxds}\hfill \\ & & {\displaystyle +{\int }_0^t{\int }_0^s{g}_2(s-\zeta ){\int }_{\Omega }\nabla v\left(\zeta \right)\nabla ydxd\zeta ds-\alpha {\int }_0^t{\int }_{\Omega }uydxds.}\hfill \end{array}$$

(6)

Using the fact that the right-hand side of the first equation and the second one in (6) is an absolutely continuous function—hence, it is differentiable almost everywhere—we obtain

$$\begin{array}{ccc}& & {\displaystyle {\int }_{\Omega }{u}_{tt}wdx+\gamma {\int }_{\Omega }\nabla u_{tt}\nabla wdx+a(u,w)-{\int }_0^t{g}_1(t-s)a(u\left(s\right),w)ds+\alpha {\int }_{\Omega }vwdx=0,\forall w\in V,}\hfill \\ & & {\displaystyle {\int }_{\Omega }{v}_{tt}ydx+{\int }_{\Omega }\nabla v\nabla ydx-{\int }_0^t{g}_2(t-s){\int }_{\Omega }\nabla v\left(s\right)\nabla ydxds+\alpha {\int }_{\Omega }uydx=0,\forall y\in H_0^1(\Omega ).}\hfill \end{array}$$

Regarding the initial conditions, we can also use (6) to verify that

$$u\left(0\right)=u_0,u_t\left(0\right)=u_1,v\left(0\right)=v_0,v_t\left(0\right)=v_0.$$

For uniqueness, let us assume that $(u_1,v_1),(u_2,v_2)$ are two weak solutions of (1). Then, $(p,q)=(u_1-u_2,v_1-v_2)$ satisfies

$$\begin{array}{ccc}\hfill & & {\displaystyle p_{tt}-\gamma \Delta p_{tt}+{\Delta }^{2}p-{\int }_0^t{g}_1(t-s){\Delta }^{2}p\left(s\right)ds+\alpha q=0,\mathrm{in}{L}^2(0,T;V^{\prime }),}\hfill \\ & & {\displaystyle q_{tt}-\Delta q+{\int }_0^t{g}_2(t-s)\Delta q\left(s\right)ds+\alpha p=0,\mathrm{in}{L}^2(0,T;H^{-1}(\Omega )),}\hfill \\ & & p\left(0\right)=p_t\left(0\right)=q\left(0\right)=q_t\left(0\right)=0.\hfill \end{array}$$

(7)

We shall use the Visik–Ladyzenskaya method. We consider, for each $s\in [0,T]$, the following functions:

$$\psi \left(t\right)=\left\{\begin{array}{cc}{\displaystyle -{\int }_t^{s}p\left(\zeta \right)d\zeta ,}\hfill & \hfill 0\le t\le s,\\ 0,\hfill & \hfill s\le t\le T,\end{array}\right.\mathrm{and}\phi \left(t\right)=\left\{\begin{array}{cc}{\displaystyle -{\int }_t^{s}q\left(\zeta \right)d\zeta ,}\hfill & \hfill 0\le t\le s,\\ 0,\hfill & \hfill s\le t\le T.\end{array}\right.$$

The derivatives (in the distributions sense) of $\psi$ and $\phi$ are given by

$${\psi }^{\prime }\left(t\right)=\left\{\begin{array}{cc}p\left(t\right),\hfill & \hfill 0\le t\le s,\\ 0,\hfill & \hfill s\le t\le T,\end{array}\right.and{\phi }^{\prime }\left(t\right)=\left\{\begin{array}{cc}q\left(t\right),\hfill & \hfill 0\le t\le s,\\ 0,\hfill & \hfill s\le t\le T.\end{array}\right.$$

It is clear that

$$\psi ,{\psi }^{\prime }\in L^{\infty }(0,T;V)and\phi ,{\phi }^{\prime }\in L^{\infty }(0,T;H_0^1(\Omega )),$$

which implies that

$$\psi \in C^0([0,T];V)and\phi \in C^0([0,T];H_0^1(\Omega )).$$

By composing the first equation in (7) using $\psi$ and the second equation using $\phi$, we obtain

$$\begin{array}{ccc}& & {\displaystyle {\int }_0^s{((I-\gamma \Delta )p_{tt},\psi \left(t\right))}_{V^{\prime }\times V}dt+{\int }_0^s{({\Delta }^{2}p,\psi \left(t\right))}_{V^{\prime }\times V}dt-{\int }_0^s{\int }_0^t{g}_1(t-\zeta ){({\Delta }^{2}p\left(\zeta \right),\psi \left(t\right))}_{V^{\prime }\times V}d\zeta dt}\hfill \\ & & {\displaystyle +\alpha {\int }_0^s{(q,\psi \left(t\right))}_{V^{\prime }\times V}dt=0,}\hfill \\ & & {\displaystyle {\int }_0^s{(q_{tt},\phi \left(t\right))}_{H^{-1}(\Omega )\times H_0^1(\Omega )}dt-{\int }_0^s{(\Delta q,\phi \left(t\right))}_{H^{-1}(\Omega )\times H_0^1(\Omega )}dt}\hfill \\ & & {\displaystyle +{\int }_0^s{\int }_0^t{g}_2(t-\zeta ){(\Delta q\left(\zeta \right),\phi \left(t\right))}_{H^{-1}(\Omega )\times H_0^1(\Omega )}d\zeta dt+\alpha {\int }_0^s{(p,\phi \left(t\right))}_{H^{-1}(\Omega )\times H_0^1(\Omega )}dt=0.}\hfill \end{array}$$

Using the fact that $\psi \left(s\right)=\phi \left(s\right)=p_t\left(0\right)=q_t\left(0\right)=0$, ${\psi }^{\prime }\left(t\right)=p\left(t\right)$ and ${\phi }^{\prime }\left(t\right)=q\left(t\right)$ in $[0,s]$, we integrate by parts and add the resulting equations to obtain

$$\begin{array}{ccc}\hfill & & {\displaystyle -{\int }_0^s{(p_t,p\left(t\right))}_{W^{\prime }\times W}dt+{\int }_0^{s}a({\psi }^{\prime }\left(t\right),\psi )dt-{\int }_0^s{\int }_0^t{g}_1(t-\zeta )a({\psi }^{\prime }\left(\zeta \right),\psi \left(t\right))d\zeta dt}\hfill \\ & & {\displaystyle -{\int }_0^s{(q_t,q\left(t\right))}_{H^{-1}(\Omega )\times H_0^1(\Omega )}dt+{\int }_0^s{({\phi }^{\prime }\left(\zeta \right),\phi )}_{H_0^1(\Omega )}dt}\hfill \\ & & {\displaystyle -{\int }_0^s{\int }_0^t{g}_2(t-\zeta ){({\phi }^{\prime }\left(\zeta \right),\phi \left(t\right))}_{H_0^1(\Omega )}d\zeta dt+\alpha {\int }_0^s\left({(q\left(t\right),\psi \left(t\right))}_{L^2(\Omega )}+{(p\left(t\right),\phi \left(t\right))}_{L^2(\Omega )}\right)dt=0,}\hfill \end{array}$$

which, by using (11), results in: and (12)

$$\begin{array}{ccc}& & {\displaystyle \frac{1}{2}\frac{d}{dt}\{-{\int }_0^s{\parallel p\parallel }_W^{2}dt+{\int }_0^s\left({\displaystyle 1-{\int }_0^t{g}_1\left(\zeta \right)d\zeta }\right)a(\psi ,\psi )dt+{\int }_0^s(g_1\square \psi )\left(t\right)dt-{\int }_0^s{\parallel q\parallel }_{H_0^1(\Omega )}^{2}dt}\hfill \\ & & {\displaystyle +{\int }_0^s\left({\displaystyle 1-{\int }_0^t{g}_2\left(\zeta \right)d\zeta }\right){\parallel \phi \left(t\right)\parallel }_{H_0^1(\Omega )}^{2}dt\}}\hfill \\ & & {\displaystyle +\frac{1}{2}{\int }_0^s{g}_1\left(t\right)a(\psi ,\psi )dt-\frac{1}{2}{\int }_0^s(g_1^{\prime }\square \psi )\left(t\right)dt+\frac{1}{2}{\int }_0^s{g}_1\left(t\right){\parallel \phi \parallel }_{H_0^1(\Omega )}^{2}dt-\frac{1}{2}{\int }_0^s(g_2^{\prime }\circ \nabla \phi )\left(t\right)dt}\hfill \\ & & {\displaystyle =-\alpha {\int }_0^s\left({(q\left(t\right),\psi \left(t\right))}_{L^2(\Omega )}+{(p\left(t\right),\phi \left(t\right))}_{L^2(\Omega )}\right)dt.}\hfill \end{array}$$

Now, using the fact that $g_i,-g_i^{\prime },a(\psi ,\psi )\ge 0$ for $i=1,2$, and $W,H_0^1\subset L^2(\Omega )$, we obtain the existence of a positive constant C such that

$$\begin{array}{ccc}\hfill & & \frac{1}{2}{\parallel p\left(s\right)\parallel }^2+\frac{1}{2}{\parallel \psi \left(0\right)\parallel }^2+\frac{1}{2}{\parallel q\left(s\right)\parallel }^2+\frac{1}{2}{\parallel \phi \left(0\right)\parallel }^2\hfill \\ & & {\displaystyle \le C{\int }_0^s(\parallel q\left(t\right)\parallel \parallel \psi \left(t\right)\parallel +\parallel p\left(t\right)\parallel \parallel \phi \left(t\right)\parallel )dt.}\hfill \end{array}$$

(8)

Finally, let ${\displaystyle p_1\left(t\right)={\int }_0^{t}p\left(\zeta \right)d\zeta }$ and ${\displaystyle q_1\left(t\right)={\int }_0^{t}q\left(\zeta \right)d\zeta }$. We have, for all $t\in [0,s]$

$$\psi \left(t\right)=p_1\left(t\right)-p_1\left(s\right),\psi \left(0\right)=-p_1\left(s\right),$$

and

$$\phi \left(t\right)=q_1\left(t\right)-q_1\left(s\right),\phi \left(0\right)=-q_1\left(s\right).$$

Consequently, (8) becomes

$$\begin{array}{ccc}& & \frac{1}{2}{\parallel p\left(s\right)\parallel }^2+\frac{1}{2}\parallel p_1{\left(s\right)\parallel }^2+\frac{1}{2}{\parallel q\left(s\right)\parallel }^2+\frac{1}{2}{\parallel q_1\left(s\right)\parallel }^2\hfill \\ & & {\displaystyle \le C{\int }_0^s(\parallel q\left(t\right)\parallel \parallel p_1\left(t\right)-p_1\left(s\right)\parallel +\parallel p\left(t\right)\parallel \parallel q_1\left(t\right)-q_1\left(s\right)\parallel )dt}\hfill \\ & & {\displaystyle \le C\{{\int }_0^s\parallel q\left(t\right)\parallel \parallel p_1\left(t\right)\parallel dt+{\int }_0^s\parallel q\left(t\right)\parallel \parallel p_1\left(s\right)\parallel dt}\hfill \\ & & {\displaystyle +{\int }_0^s\parallel p\left(t\right)\parallel \parallel q_1\left(t\right)\parallel dt+{\int }_0^s\parallel p\left(t\right)\parallel \parallel q_1\left(s\right)\parallel dt\}}\hfill \\ & & {\displaystyle \le C\{{\int }_0^s\parallel q\left(t\right)\parallel \parallel p_1\left(t\right)\parallel dt+{\int }_0^s\sqrt{2Cs}\parallel q\left(t\right)\parallel \frac{1}{\sqrt{2Cs}}\parallel p_1\left(s\right)\parallel dt}\hfill \\ & & {\displaystyle +{\int }_0^s\parallel p\left(t\right)\parallel \parallel q_1\left(t\right)\parallel dt+{\int }_0^s\sqrt{2Cs}\parallel p\left(t\right)\parallel \frac{1}{\sqrt{2Cs}}\parallel q_1\left(s\right)\parallel dt\}}\hfill \\ & & {\displaystyle \le \frac{C}{2}{\int }_0^s{\parallel p\left(t\right)\parallel }^{2}dt+\frac{C}{2}{\int }_0^s\parallel p_1{\left(t\right)\parallel }^{2}dt+T{C}^2{\int }_0^s\parallel p^2{\left(t\right)\parallel }^{2}dt+\frac{1}{4}{\parallel p_1\left(s\right)\parallel }^2}\hfill \\ & & {\displaystyle +\frac{C}{2}{\int }_0^s{\parallel q\left(t\right)\parallel }^{2}dt+\frac{C}{2}{\int }_0^s\parallel q_1{\left(t\right)\parallel }^{2}dt+T{C}^2{\int }_0^s\parallel q^2{\left(t\right)\parallel }^{2}dt+\frac{1}{4}{\parallel q_1\left(s\right)\parallel }^2,}\hfill \end{array}$$

which implies that

$$\begin{array}{ccc}& & \frac{1}{4}{\parallel p\left(s\right)\parallel }^2+\frac{1}{4}\parallel p_1{\left(s\right)\parallel }^2+\frac{1}{4}{\parallel q\left(s\right)\parallel }^2+\frac{1}{4}{\parallel q_1\left(s\right)\parallel }^2\hfill \\ & & {\displaystyle \le C{\int }_0^s\left({\parallel p\left(t\right)\parallel }^2+\parallel p_1{\left(t\right)\parallel }^2+{\parallel q\left(t\right)\parallel }^2+{\parallel q_1\left(t\right)\parallel }^2\right)dt.}\hfill \end{array}$$

By using Gronwall’s Lemma, we deduce that

$$\frac{1}{4}{\parallel p\left(s\right)\parallel }^2+\frac{1}{4}\parallel p_1{\left(s\right)\parallel }^2+\frac{1}{4}{\parallel q\left(s\right)\parallel }^2+\frac{1}{4}{\parallel q_1\left(s\right)\parallel }^2\le 0.$$

Then, we can determine that

$$p\left(s\right)=q\left(s\right)=0,\mathrm{in}{L}^2(\Omega ),\forall s\in (0,T),$$

and since $p\left(0\right)=q\left(0\right)=0$, we obtain

$$p\left(s\right)=q\left(s\right)=0,\mathrm{in}{L}^2(\Omega ),\forall s\in [0,T],$$

which means that $(u_1,v_1)=(u_2,v_2)$.

Consequently, the proof of the local existence of a weak solution is complete. Furthermore, it is easy to see that

$$l_{1}a(u,u)+\parallel u_t{\parallel }^2+\gamma \parallel \nabla u_t{\parallel }^2+l_2{\parallel \nabla v\parallel }^2+{\parallel v_t\parallel }^2\le 2E\left(t\right)\le 2E\left(0\right),$$

which shows that the solution is bounded and global in time.

This completes the proof. □

We also need the following regularity result. Indeed, in some parts of the paper, we multiply the first equation by $u_t$ and the second one by $v_t$. This is only possible if we are working with regular solutions. For this reason, we will introduce a theorem for regular solutions as well. Thus, it is enough to work with regular solutions all time. The decay rate estimates for weak solutions are obtained using standard density arguments. But, before performing this, we present the definition of regular solutions in our case, which was introduced in Definition 2 [11].

Definition 2. 

We previously stated that $(u_0,u_1)$ is 2-regular if

$$u_j\in H^{4-j}(\Omega )\cap V,j=0,\cdots ,2;u_3\in W,$$

where $u_j$ is obtained by the following recursive formulas:

$$\begin{array}{ccc}& & u_{j+2}-\gamma \Delta u_{j+2}={\Delta }^2{u}_j,\hfill \\ & & u_j={\partial }_{\nu }{u}_j=0,\mathrm{on}{\Gamma }_0,\hfill \\ & & {\mathbf{B}}_1{u}_j=0,\mathrm{on}{\Gamma }_1,\forall j=1,2.\hfill \\ & & {\mathbf{B}}_2{u}_j=\gamma {\partial }_{\nu }{u}_{j+2},\mathrm{on}{\Gamma }_1.\hfill \end{array}$$

Now, we present our regularity result.

Theorem 2. 

Suppose (A1) holds and suppose that $(u_0,u_1)$ is 2-regular and $(v_0,v_1)\in \left(H^2(\Omega )\cap \right.\left.H_0^1(\Omega )\right)\times H_0^1(\Omega )$. Then, the solution of problem (1) satisfies

$$u\in C([0,T],V\cap H^4(\Omega ))\cap C^1([0,T],W\cap H^3(\Omega )),$$

$$v\in L^{\infty }([0,T],H^2(\Omega )\cap H_0^1(\Omega ))\cap W^{1,\infty }([0,T],H_0^1(\Omega ))\cap W^{2,\infty }([0,T],L^2(\Omega )).$$

Proof. 

The proof can be performed by combining the arguments used, for example, in [9,11]. □

3. General Decay

In this section, we will present and establish our principal theorem of this paper, which states the general decay of the energy of our system. This will be conducted by the help of the perturbed energy method. First, we introduce the energy functional by

$$\begin{array}{ccc}\hfill & & E\left(t\right)=\frac{1}{2}\left({\displaystyle 1-{\int }_0^t{g}_1\left(s\right)ds}\right)a(u,u)+\frac{1}{2}\parallel u_t{\parallel }^2+\frac{\gamma }{2}\parallel \nabla u_t{\parallel }^2+\frac{1}{2}(g_1\square u)\left(t\right)+\frac{1}{2}\left({\displaystyle 1-{\int }_0^t{g}_2\left(s\right)ds}\right){\parallel \nabla v\parallel }^2\hfill \\ & & {\displaystyle +\frac{1}{2}(g_2\circ \nabla v)\left(t\right)+\frac{1}{2}{\parallel v_t\parallel }^2+\alpha {\int }_{\Omega }uvdx,}\hfill \end{array}$$

which satisfies the following dissipation identity:

Proposition 1. 

Under the hypothesis of Theorem 2, the following identity holds:

$$E^{\prime }\left(t\right)=\frac{1}{2}(g_1^{\prime }\square u)\left(t\right)-\frac{1}{2}{g}_1\left(t\right)a(u,u)+\frac{1}{2}(g_2^{\prime }\circ \nabla v)\left(t\right)-\frac{1}{2}{g}_2\left(t\right){\parallel \nabla v\left(t\right)\parallel }^2\le 0.$$

(9)

Proof. 

In (1), upon multiplying the first equation by $u_t$ and the second one by $v_t$, add the resulting equations and integrate by parts over $\Omega$ to obtain

$$\begin{array}{ccc}\hfill & & \frac{d}{dt}\left\{{\displaystyle \frac{1}{2}\parallel u_t{\parallel }^2+\frac{\gamma }{2}\parallel \nabla u_t{\parallel }^2+\frac{1}{2}a(u,u)+\frac{1}{2}\parallel v_t{\parallel }^2+\frac{1}{2}{\parallel \nabla v\parallel }^2+\alpha {\int }_{\Omega }uvdxdy}\right\}\hfill \\ & & {\displaystyle -{\int }_0^t{g}_1(t-s)a\left(u\left(s\right),u_t\right)ds-{\int }_0^t{g}_2(t-s){\int }_{\Omega }\nabla v\left(s\right)\nabla v_t\left(t\right)dxds=0}\hfill \end{array}$$

(10)

By the virtue of Lemma 2.1 in [11], we have

$$\begin{array}{ccc}\hfill a\left({\displaystyle {\int }_0^t{g}_1(t-s)u\left(s\right)ds,u_t}\right)& =& -\frac{1}{2}{g}_1\left(t\right)a(u,u)-\frac{1}{2}\frac{d}{dt}\left\{(g_1\square u)\left(t\right)-\left({\displaystyle {\int }_0^t{g}_1\left(s\right)ds}\right)a(u,u)\right\}\hfill \\ & +& \frac{1}{2}(g_1^{\prime }\square u)\left(t\right),\hfill \end{array}$$

(11)

For any $u\in C^1(0,T;H^2(\Omega ))$.

Besides, a direct computation shows that

$$\begin{array}{ccc}\hfill {\displaystyle {\int }_0^t{g}_2(t-s){\int }_{\Omega }\nabla v\left(s\right)\nabla v_t\left(t\right)dxds}& =& \frac{1}{2}(g_2^{\prime }\circ \nabla v)\left(t\right)-\frac{1}{2}{g}_2\left(t\right){\parallel \nabla v\left(t\right)\parallel }^2\hfill \\ & -& \frac{1}{2}\frac{d}{dt}\left\{(g_2\circ \nabla v)\left(t\right)-\left({\displaystyle {\int }_0^t{g}_2\left(s\right)ds}\right){\parallel \nabla v\left(t\right)\parallel }^2\right\}.\hfill \end{array}$$

(12)

□

By replacing (11) and (12) in (10), we obtain the desired result.

The main result of this paper reads as follows.

Theorem 3. 

Suppose that $(u_0,u_1)$ is 2-regular and $(v_0,v_1)\in \left(H^2(\Omega )\cap H_0^1(\Omega )\right)\times H_0^1(\Omega )$. Assume that (A1) and (A2) hold. Then, there exist positive constants ${\beta }_1$ and ${\beta }_2$, such that the energy $E\left(t\right)$ satisfies for any $t>g^{-1}\left(r\right)$

$$E\left(t\right)\le {\beta }_2{Q}_1^{-1}\left({\beta }_1{\int }_{g^{-1}\left(r\right)}^t\xi \left(s\right)ds\right),$$

(13)

where $\xi \left(t\right)=min\{{\xi }_1\left(t\right),{\xi }_2\left(t\right)\}$, $g\left(t\right)=max\{g_1\left(t\right),g_2\left(t\right)\}$ and ${\displaystyle Q_1\left(t\right)={\int }_t^r\frac{1}{s{Q}^{\prime }\left(s\right)}ds}$.

Remark 2 

([25]).

1. 

The following Jensen’s inequality is critical to prove our main result. Let G be a convex increasing function on $[a,b],h:\Omega \to [a,b]$ and m is the integrable function on Ω, such that $m\left(x\right)\ge 0$ and ${\displaystyle {\int }_{\Omega }m\left(x\right)dx=k>0}$, then Jensen’s inequality states that

$${\displaystyle G\left[\frac{1}{k}{\int }_{\Omega }h\left(x\right)m\left(x\right)dx\right]\le \frac{1}{k}{\int }_{\Omega }G\left[h\left(x\right)\right]m\left(x\right)dx.}$$

2. 

From (A2), we infer that ${\displaystyle \underset{t\to +\infty }{lim}{g}_i\left(t\right)=0}$. Then, there exists some large enough $t_1\ge 0$, such that

$$\begin{array}{c}\hfill g_i\left(t_1\right)=r\Rightarrow g_i\left(t\right)\le r,\forall t\ge t_1.\end{array}$$

(14)

Since Q is a positive continuous function and $g_i,{\xi }_i$ are positive non-increasing continuous functions, we can obtain for every $t\in [0,t_1]$,

$$0<g_i\left(t_1\right)\le g_i\left(t\right)\le g_i\left(0\right)and0<{\xi }_i\left(t_1\right)\le {\xi }_i\left(t\right)\le {\xi }_i\left(0\right),i=1,2,$$

which implies for some positive constants $a_i$ and $b_i$:

$$a_i\le {\xi }_i\left(t\right)Q\left(g_i\left(t\right)\right)\le b_i,i=1,2.$$

This shows that for every $t\in [0,t_1]$,

$$\begin{array}{c}\hfill g_i^{\prime }\left(t\right)\le -{\xi }_i\left(t\right)Q\left(g_i\left(t\right)\right)\le -\frac{a_i}{g_i\left(0\right)}{g}_i\left(0\right)\le -\frac{a_i}{g_i\left(0\right)}{g}_i\left(t\right),i=1,2.\end{array}$$

(15)

3. 

If different functions $Q_1$ and $Q_2$ have the properties mentioned in (A2), such that $g_1^{\prime }\left(t\right)\le -Q_1\left(g_1\left(t\right)\right)$ and $g_2^{\prime }\left(t\right)\le -Q_2\left(g_2\left(t\right)\right)$, then there exists $r<min\{r_1,r_2\}$ small enough so that, e.g., $Q_1\left(t\right)\le Q_2\left(t\right)$ on the interval $(0,r]$. Thus, the function $Q\left(t\right)=Q_1\left(t\right)$ satisfies (A2) for both functions $g_1$ and $g_2$ for all $t\ge t_1$.

We will work with regular solutions; by standard density arguments, the decay result remains valid for weak solutions as well. In order to prove the main Theorem (3), we need to introduce several lemmas. To this end, let us introduce the functionals

$$\begin{array}{c}\hfill {\displaystyle I\left(t\right)={\int }_{\Omega }\left(u{u}_t+\gamma \nabla u_t\nabla u\right)dx+{\int }_{\Omega }v{v}_{t}dx,}\end{array}$$

(16)

and

$$\begin{array}{ccc}\hfill K\left(t\right)& =& {\displaystyle -{\int }_{\Omega }{u}_t{\int }_0^t{g}_1(t-s)(u\left(t\right)-u\left(s\right))dsdx-\gamma {\int }_{\Omega }\nabla u_t{\int }_0^t{g}_1(t-s)\nabla (u\left(t\right)-u\left(s\right))dsdx}\hfill \\ & -& {\int }_{\Omega }{v}_t{\int }_0^t{g}_2(t-s)(v\left(t\right)-v\left(s\right))dsdx.\hfill \end{array}$$

(17)

Lemma 1. 

Assume that (A1) and (A2) hold. Then, the functional $I\left(t\right)$ introduced in (16) satisfies (along the solution) the estimate

$$\begin{array}{ccc}\hfill I^{\prime }\left(t\right)& \le & {\displaystyle {\int }_{\Omega }|u_t{|}^{2}dx+\gamma {\int }_{\Omega }{|\nabla u_t|}^{2}dx-\frac{l_1}{2}a(u,u)+\frac{1-l_1}{2l_1}(g_1\square u)\left(t\right)}\hfill \\ & +& {\displaystyle {\int }_{\Omega }|v_t{|}^{2}dx-\frac{l_2}{2}{\int }_{\Omega }{|\nabla v|}^{2}dx+\frac{1-l_2}{2l_2}(g_2\circ \nabla v)\left(t\right)-2\alpha {\int }_{\Omega }uvdx.}\hfill \end{array}$$

(18)

Proof. 

Direct differentiation of I, using (1), yields

$$\begin{array}{ccc}\hfill I^{\prime }\left(t\right)& =& {\displaystyle {\int }_{\Omega }|u_t{|}^{2}dx+{\int }_{\Omega }u{u}_{tt}dx+\gamma {\int }_{\Omega }|\nabla u_t{|}^{2}dx+\gamma {\int }_{\Omega }\nabla u\nabla u_{tt}dx+{\int }_{\Omega }{|v_t|}^{2}dx+{\int }_{\Omega }v{v}_{tt}dx}\hfill \\ & =& {\displaystyle {\int }_{\Omega }|u_t{|}^{2}dx+\gamma {\int }_{\Omega }{|\nabla u_t|}^{2}dx-a(u,u)+{\int }_0^t{g}_1(t-s)a(u\left(s\right),u\left(t\right))ds}\hfill \\ & & {\displaystyle +{\int }_{\Omega }|v_t{|}^{2}dx-{\int }_{\Omega }{|\nabla v|}^{2}dx+{\int }_0^t{g}_2(t-s){\int }_{\Omega }\nabla v\left(s\right)\nabla v\left(t\right)dxds-2\alpha {\int }_{\Omega }uvdx.}\hfill \end{array}$$

(19)

By using the Cauchy–Schwarz inequality, Young’s inequality and the fact that ${\displaystyle {\int }_0^t{g}_1\left(s\right)ds\le {\int }_0^{+\infty }{g}_1\left(s\right)ds=1-l_1}$, we obtain

$$\begin{array}{ccc}\hfill & & {\int }_0^t{g}_1(t-s)a(u\left(t\right),u\left(s\right))ds\hfill \\ & & ={\int }_0^t{g}_1(t-s)a(u\left(s\right)-u\left(t\right),u\left(t\right))ds+{\int }_0^t{g}_1(t-s)a(u\left(t\right),u\left(t\right))ds\hfill \\ & & \le {\int }_0^t{g}_1(t-s){\left\{a\left(u\right(s)-u(t),u(s)-u(t\left)\right)\right\}}^{\frac{1}{2}}{\left\{a\left(u\right(t),u(t\left)\right)\right\}}^{\frac{1}{2}}ds+\left({\int }_0^t{g}_1\left(s\right)ds\right)a(u\left(t\right),u\left(t\right))\hfill \\ & & \le \frac{l_1}{2}a(u\left(t\right),u\left(t\right))+\frac{1}{2l_1}{\left({\int }_0^t\sqrt{g_1(t-s)}{\left\{g_1(t-s)a(u\left(s\right)-u\left(t\right),u\left(s\right)-u\left(t\right))\right\}}^{\frac{1}{2}}ds\right)}^2\hfill \\ & & +(1-l_1)a(u\left(t\right),u\left(t\right))\hfill \\ & & \le \left(1-\frac{l_1}{2}\right)a(u\left(t\right),u\left(t\right))+\frac{1-l_1}{2l_1}(g_1\square u)\left(t\right).\hfill \end{array}$$

(20)

Furthermore, we have (see, for example, [5]) that

$$\begin{array}{ccc}\hfill & & {\displaystyle {\int }_0^t{g}_2(t-s){\int }_{\Omega }\nabla v\left(s\right)\nabla v\left(t\right)dxds}\hfill \\ & & {\displaystyle \le (1-\frac{l_2}{2}){\int }_{\Omega }{|\nabla v|}^{2}dx+\frac{1-l_2}{2l_2}(g_2\circ \nabla v)\left(t\right).}\hfill \end{array}$$

(21)

Inserting (20) and (21) in (19), the assertion of the lemma is established. □

Lemma 2. 

Assume that (A1) and (A2) hold. Then, the functional $K\left(t\right)$ introduced in (17) satisfies, along the solution, the estimate

$$\begin{array}{ccc}\hfill K^{\prime }\left(t\right)& & \le \left(\delta -{\int }_0^t{g}_1\left(s\right)ds\right)\left({\int }_{\Omega }{\left|u_t\right|}^{2}dx\right)\hfill \\ & & +\left\{\delta (C+1)+\delta (1-l_1)(\delta +2(1-l_1))-{\int }_0^t{g}_1\left(s\right)ds\right\}a(u,u)\hfill \\ & & +\gamma \left(\delta -{\int }_0^t{g}_1\left(s\right)ds\right)\left({\int }_{\Omega }{|\nabla u_t|}^{2}dx\right)\hfill \\ & & +\left\{\frac{(1-l_1)(C+2)}{4\delta }+\delta (1-l_1)+{(1-l_1)}^2\right\}(g_1\square u)\left(t\right)\hfill \\ & & -\frac{C{g}_1\left(0\right)}{2\delta }(g_1^{\prime }\square u)\left(t\right)+\left(\delta -{\int }_0^t{g}_2\left(s\right)ds\right)\left({\int }_{\Omega }{\left|v_t\right|}^{2}dx\right)\hfill \\ & & +\delta \left(C+1+2{(1-l_2)}^2\right)\left({\int }_{\Omega }{|\nabla v|}^{2}dx\right)\hfill \\ & & +\left\{\frac{(1-l_2)(C+1)}{4\delta }+(2\delta +\frac{1}{4\delta })(1-l_2)\right\}(g_2\circ \nabla v)\left(t\right)\hfill \\ & & -\frac{C{g}_2\left(0\right)}{4\delta }(g_2^{\prime }\circ \nabla v)\left(t\right),\forall \delta >0.\hfill \end{array}$$

(22)

Proof. 

By exploiting equation (1) and integrating by parts, we have

$$\begin{array}{ccc}\hfill K^{\prime }\left(t\right)& =& \left(-{\int }_0^t{g}_1\left(s\right)ds\right)\left({\int }_{\Omega }{\left|u_t\right|}^{2}dx\right)-{\int }_{\Omega }{u}_t{\int }_0^t{g}_1^{\prime }(t-s)(u\left(t\right)-u\left(s\right))dsdx\hfill \\ & & -{\int }_{\Omega }{u}_{tt}{\int }_0^t{g}_1(t-s)(u\left(t\right)-u\left(s\right))dsdx\hfill \\ & & -\gamma {\int }_{\Omega }\nabla u_{tt}{\int }_0^t{g}_1(t-s)\nabla (u\left(t\right)-u\left(s\right))dsdx\hfill \\ & & -\gamma {\int }_{\Omega }\nabla u_t{\int }_0^t{g}_1^{\prime }(t-s)\nabla (u\left(t\right)-u\left(s\right))dsdx-\gamma \left({\int }_0^t{g}_1\left(s\right)ds\right)\left({\int }_{\Omega }{|\nabla u_t|}^{2}dx\right)\hfill \\ & & -{\int }_{\Omega }{v}_{tt}{\int }_0^t{g}_2(t-s)(v\left(t\right)-v\left(s\right)dsdx\hfill \\ & & -{\int }_{\Omega }{v}_t{\int }_0^t{g}_2^{\prime }(t-s)(u\left(t\right)-u\left(s\right))dsdx-\left({\int }_0^t{g}_2\left(s\right)ds\right)\left({\int }_{\Omega }{\left|v_t\right|}^{2}dx\right)\hfill \\ & =& \left(-{\int }_0^t{g}_1\left(s\right)ds\right)\left({\int }_{\Omega }{\left|u_t\right|}^{2}dx\right)-{\int }_{\Omega }{u}_t{\int }_0^t{g}_1^{\prime }(t-s)(u\left(t\right)-u\left(s\right))dsdx\hfill \\ & & +a\left(u,{\int }_0^t{g}_1(t-s)(u\left(t\right)-u\left(s\right))ds\right)\hfill \\ & & -{\int }_0^t{g}_1(t-s)a\left(u\left(s\right),{\int }_0^t{g}_1(t-s)(u\left(t\right)-u\left(s\right))ds\right)ds\hfill \\ & & +\alpha {\int }_{\Omega }v{\int }_0^t{g}_1(t-s)(u\left(t\right)-u\left(s\right))dsdx\hfill \end{array}$$

$$\begin{array}{ccc}& & -\gamma {\int }_{\Omega }\nabla u_t{\int }_0^t{g}_1^{\prime }(t-s)\nabla (u\left(t\right)-u\left(s\right))dsdx-\gamma \left({\int }_0^t{g}_1\left(s\right)ds\right)\left({\int }_{\Omega }{|\nabla u_t|}^{2}dx\right)\hfill \\ & & +{\int }_{\Omega }\nabla v{\int }_0^t{g}_2(t-s)\nabla (v\left(t\right)-v\left(s\right))dsdx\hfill \\ & & -{\int }_{\Omega }\left({\int }_0^t{g}_2(t-s)\nabla v\left(s\right)ds\right)\left({\int }_0^t{g}_2(t-s)\nabla (u\left(t\right)-u\left(s\right))ds\right)dx\hfill \\ & & +\alpha {\int }_{\Omega }u{\int }_0^t{g}_2(t-s)(v\left(t\right)-v\left(s\right))dsdx-\left({\int }_0^t{g}_2\left(s\right)ds\right)\left({\int }_{\Omega }{\left|v_t\right|}^{2}dx\right)\hfill \\ & & -{\int }_{\Omega }{v}_t{\int }_0^t{g}_2^{\prime }(t-s)(u\left(t\right)-u\left(s\right))dsdx.\hfill \end{array}$$

(23)

Using Young’s inequality and Cauchy–Schwarz’s inequality, we obtain for any $\delta >0$

$$\begin{array}{ccc}\hfill -{\int }_{\Omega }{u}_t{\int }_0^t{g}_1^{\prime }(t-s)(u\left(t\right)-u\left(s\right))dsdx& \le & \delta {\int }_{\Omega }{\left|u_t\right|}^2+\frac{1}{4\delta }{\int }_{\Omega }{\left({\int }_0^t{g}_1^{\prime }(t-s)(u\left(t\right)-u\left(s\right))ds\right)}^{2}dx\hfill \\ & \le & \delta {\int }_{\Omega }|u_t{|}^2-\frac{g_1\left(0\right)}{4\delta }{\int }_0^t{g}_1^{\prime }(t-s){\int }_{\Omega }{|u\left(t\right)-u\left(s\right)|}^{2}dxds\hfill \\ & \le & \delta {\int }_{\Omega }{\left|u_t\right|}^2-\frac{C{g}_1\left(0\right)}{4\delta }(g_1^{\prime }\square u)\left(t\right),\hfill \end{array}$$

(24)

and

$$\begin{array}{ccc}\hfill a\left(u,{\int }_0^t{g}_1(t-s)(u\left(t\right)-u\left(s\right))ds\right)& =& {\int }_0^t{g}_1(t-s)a\left(u\left(t\right),u\left(t\right)-u\left(s\right)\right)ds\hfill \\ & \le & {\int }_0^t{g}_1(t-s){\left[a\left(u\right(t),u(t\left)\right)\right]}^{\frac{1}{2}}{\left[a\left(u\left(t\right)-u\left(s\right),u\left(t\right)-u\left(s\right)\right)\right]}^{\frac{1}{2}}ds\hfill \\ & \le & \delta a(u,u)+\frac{1}{4\delta }{\left\{{\int }_0^t{g}_1(t-s){\left[a\left(u\left(t\right)-u\left(s\right),u\left(t\right)-u\left(s\right)\right)\right]}^{\frac{1}{2}}ds\right\}}^2\hfill \\ & \le & \delta a(u,u)+\frac{1-l_1}{4\delta }(g_1\square u)\left(t\right).\hfill \end{array}$$

(25)

Furthermore, we have

$$\begin{array}{ccc}\hfill & -& {\int }_0^t{g}_1(t-s)a\left(u\left(s\right),{\int }_0^t{g}_1(t-s)(u\left(t\right)-u\left(s\right))ds\right)ds\hfill \\ & \le & \left({\int }_0^t{g}_1(t-s){\left[a\left(u\right(s),u(s\left)\right)\right]}^{\frac{1}{2}}ds\right)\left({\int }_0^t{g}_1(t-s){\left[a\left(u\left(t\right)-u\left(s\right),u\left(t\right)-u\left(s\right)\right)\right]}^{\frac{1}{2}}ds\right)\hfill \\ & \le & \delta {\left({\int }_0^t{g}_1(t-s){\left[a\left(u\right(s),u(s\left)\right)\right]}^{\frac{1}{2}}ds\right)}^2+\frac{1}{4\delta }{\left({\int }_0^t{g}_1(t-s){\left[a\left(u\left(t\right)-u\left(s\right),u\left(t\right)-u\left(s\right)\right)\right]}^{\frac{1}{2}}ds\right)}^2\hfill \\ & \le & \delta {\left({\int }_0^t{g}_1(t-s){\left[a\left(u\right(s),u(s\left)\right)\right]}^{\frac{1}{2}}ds\right)}^2+\frac{1-l_1}{4\delta }(g_1\square u)\left(t\right).\hfill \end{array}$$

(26)

Now, we will estimate the term ${\left({\displaystyle {\int }_0^t{g}_1(t-s){\left[a\left(u\right(s),u(s\left)\right)\right]}^{\frac{1}{2}}ds}\right)}^2$. We have

$$\begin{array}{ccc}\hfill & & {\left({\int }_0^t{g}_1(t-s){\left[a\left(u\right(s),u(s\left)\right)\right]}^{\frac{1}{2}}ds\right)}^2\hfill \\ & & \le (1-l_1){\int }_0^t{g}_1(t-s)a(u\left(s\right),u\left(s\right))ds\hfill \\ & & =(1-l_1){\int }_0^t{g}_1(t-s)\left(a\left(u\right(t)-u(s),u(t)-u(s\left)\right)+2a\left(u\right(t),u(s\left)\right)-a\left(u\right(t),u(t\left)\right)\right)ds\hfill \\ & & =(1-l_1)(g_1\square u)\left(t\right)-(1-l_1)\left({\int }_0^t{g}_1\left(s\right)ds\right)a(u,u)+2(1-l_1){\int }_0^t{g}_1(t-s)a(u\left(t\right),u\left(s\right))ds\hfill \\ & & \le (1-l_1)(g_1\square u)\left(t\right)-(1-l_1)\left({\int }_0^t{g}_1\left(s\right)ds\right)a(u,u)+\delta (1-l_1)a(u,u)\hfill \\ & & +\frac{{(1-l_1)}^2}{\delta }(g_1\square u)\left(t\right)+2{(1-l_1)}^{2}a(u,u)\hfill \\ & & =(1-l_1)\left(\delta +2(1-l_1)-{\int }_0^t{g}_1\left(s\right)ds\right)a(u,u)+\left(1-l_1+\frac{{(1-l_1)}^2}{\delta }\right)(g_1\square u)\left(t\right).\hfill \end{array}$$

(27)

Inserting (27) in (26), we obtain

$$\begin{array}{ccc}\hfill & & -{\int }_0^t{g}_1(t-s)a\left(u\left(s\right),{\int }_0^t{g}_1(t-s)(u\left(t\right)-u\left(s\right))ds\right)ds\hfill \\ & & \le \delta (1-l_1)\left(\delta +2(1-l_1)-{\int }_0^t{g}_1\left(s\right)ds\right)a(u,u)\hfill \\ & & +\left(\delta (1-l_1)+{(1-l_1)}^2+\frac{(1-l_1)}{4\delta }\right)(g_1\square u)\left(t\right).\hfill \end{array}$$

(28)

Next, we have

$$\begin{array}{ccc}\hfill & & \alpha {\int }_{\Omega }v{\int }_0^t{g}_1(t-s)(u\left(t\right)-u\left(s\right))dsdx\hfill \\ & & \le \delta {\int }_{\Omega }{\left|v\right|}^{2}dx+\frac{C(1-l_1)}{4\delta }(g_1\square u)\left(t\right)\hfill \\ & & \le C\delta {\int }_{\Omega }{|\nabla v|}^{2}dx+\frac{C(1-l_1)}{4\delta }(g_1\square u)\left(t\right).\hfill \end{array}$$

(29)

The term ${\displaystyle -\gamma {\int }_{\Omega }\nabla u_t{\int }_0^t{g}_1^{\prime }(t-s)\nabla (u\left(t\right)-u\left(s\right))dsdx}$ can be estimated as follows:

$$\begin{array}{ccc}\hfill & & -\gamma {\int }_{\Omega }\nabla u_t{\int }_0^t{g}_1^{\prime }(t-s)\nabla (u\left(t\right)-u\left(s\right))dsdx\hfill \\ & & \le \gamma \delta {\int }_{\Omega }{|\nabla u_t|}^2-\frac{Cg\left(0\right)}{4\delta }(g_1^{\prime }\square u)\left(t\right).\hfill \end{array}$$

(30)

Furthermore, we determine that

$$\begin{array}{ccc}\hfill & & \alpha {\int }_{\Omega }u{\int }_0^t{g}_2(t-s)(v\left(t\right)-v\left(s\right))dsdx\hfill \\ & & \le C\delta a(u,u)+\frac{C(1-l_2)}{4\delta }(g_2\circ \nabla v)\left(t\right).\hfill \end{array}$$

(31)

The remaining terms can be estimated as, for example, in [5] (see estimates (3.14)–(3.16) in the mentioned paper).

$$\begin{array}{ccc}\hfill & & {\int }_{\Omega }\nabla v{\int }_0^t{g}_2(t-s)\nabla (v\left(t\right)-v\left(s\right))dsdx\hfill \\ & & \le \delta {\int }_{\Omega }{|\nabla v|}^{2}dx+\frac{1-l_2}{4\delta }(g_2\circ \nabla v)\left(t\right),\hfill \end{array}$$

(32)

$$\begin{array}{ccc}\hfill & & -{\int }_{\Omega }\left({\int }_0^t{g}_2(t-s)\nabla v\left(s\right)ds\right)\left({\int }_0^t{g}_2(t-s)\nabla (v\left(t\right)-v\left(s\right))ds\right)dx\hfill \\ & & \le (2\delta +\frac{1}{4\delta })(1-l_2)(g_2\circ \nabla v)\left(t\right)+2\delta {(1-l_2)}^2{\int }_{\Omega }{|\nabla v|}^{2}dx,\hfill \end{array}$$

(33)

and

$$\begin{array}{ccc}\hfill & & -{\int }_{\Omega }{v}_t{\int }_0^t{g}_2^{\prime }(t-s)(u\left(t\right)-u\left(s\right))dsdx\hfill \\ & & \le \delta {\int }_{\Omega }{\left|v_t\right|}^{2}dx-\frac{C{g}_2\left(0\right)}{4\delta }(g_2^{\prime }\circ \nabla v)\left(t\right).\hfill \end{array}$$

(34)

By combining (23) and (34), we achieve the desired estimate. □

Now, we define the functional $F\left(t\right)$. The idea is to construct a new Lyapunow function, equivalent to the energy quantity, that will satisfy an “appropriate” inequality. Let

$$\begin{array}{c}\hfill F\left(t\right)=NE\left(t\right)+N_{1}I\left(t\right)+N_{2}K\left(t\right),\end{array}$$

where $N,N_1$ and $N_2$ are positive constants that will be chosen later. It is easy to verify that for a large enough N, we have $F\sim E$, i.e.,

$$c_{1}E\left(t\right)\le F\left(t\right)\le c_{2}E\left(t\right),$$

for some $c_1,c_2>0.$

Lemma 3. 

The functional F satisfies

$$\begin{array}{ccc}\hfill F^{\prime }\left(t\right)& \le & -\left({\int }_{\Omega }|u_t{|}^{2}dx+{\int }_{\Omega }|v_t{|}^{2}dx+\gamma {\int }_{\Omega }{|\nabla u_t|}^{2}dxdy\right)\hfill \\ & -& 4(1-l)\left(a(u,u)+{\int }_{\Omega }{|\nabla v|}^{2}dx\right)+c\left((g_1\square u)\left(t\right)+(g_2\circ \nabla v)\left(t\right)\right)-2\alpha N_1{\int }_{\Omega }uvdx,\forall t\ge t_1,\hfill \end{array}$$

(35)

where $t_1$ was introduced in (14) and $c>0$.

Proof. 

Let

$$g_0=min\left\{{\int }_0^{t_1}{g}_1\left(s\right)ds,{\int }_0^{t_1}{g}_2\left(s\right)ds\right\}>0,\mathrm{and}l=min\left\{l_1,l_2\right\}.$$

By using (9), (18) and (22), we obtain for any $t\ge t_1$

$$\begin{array}{ccc}\hfill F^{\prime }\left(t\right)& =& N{E}^{\prime }\left(t\right)+N_1{I}^{\prime }\left(t\right)+N_2{K}^{\prime }\left(t\right)\hfill \\ & \le & -\left(N_2(g_0-\delta )-N_1\right)\left({\int }_{\Omega }|u_t{|}^{2}dx+{\int }_{\Omega }|v_t{|}^{2}dx+\gamma {\int }_{\Omega }{|\nabla u_t|}^{2}dx\right)\hfill \\ & -& \left(\frac{N_{1}l}{2}-N_2\delta \left(C+1+2{(1-l)}^2\right)-N_2{\delta }^2(1-l)+N_2{g}_0\right)a(u,u)\hfill \\ & -& \left(\frac{N_{1}l}{2}-N_2\delta \left(C+1+2{(1-l)}^2\right)\right){\int }_{\Omega }{|\nabla v|}^{2}dx\hfill \\ & +& \left(\frac{N_1(1-l)}{2l}+N_2\left(\frac{(C+2)(1-l)}{4\delta }+\delta (1-l)+{(1-l)}^2\right)\right)(g_1\square u)\left(t\right)\hfill \\ & +& \left(\frac{N_1(1-l)}{2l}+N_2\left(\frac{(C+1)(1-l)}{4\delta }+(2\delta +\frac{1}{4\delta })(1-l)\right)\right)(g_2\circ \nabla v)\left(t\right)\hfill \\ & +& \left(\frac{N}{2}-\frac{C{N}_2{g}_0}{2\delta }\right)\left\{(g_1^{\prime }\square u)\left(t\right)+(g_2^{\prime }\circ \nabla v)\left(t\right)\right\}-2\alpha N_1{\int }_{\Omega }uvdx.\hfill \end{array}$$

(36)

Taking $\delta =\frac{l}{4N_2(C+1+2{(1-l)}^2)}$, (36) becomes

$$\begin{array}{ccc}\hfill F^{\prime }\left(t\right)& \le & -\left(N_2{g}_0-\frac{l}{4(C+1+2{(1-l)}^2)}-N_1\right)\left({\int }_{\Omega }|u_t{|}^{2}dx+{\int }_{\Omega }|v_t{|}^{2}dx+\gamma {\int }_{\Omega }{|\nabla u_t|}^{2}dx\right)\hfill \\ & -& \left(\frac{N_{1}l}{2}-\frac{l}{4}-\frac{l^2(1-l)}{16N_2{(C+1+2{(1-l)}^2)}^2}+N_2{g}_0\right)a(u,u)\hfill \\ & -& \left(\frac{N_{1}l}{2}-\frac{l}{4}\right){\int }_{\Omega }{|\nabla v|}^{2}dx\hfill \\ & +& (\frac{N_1(1-l)}{2l}+\frac{N_2^2(C+2)(1-l)(C+1+2{(1-l)}^2)}{l}\hfill \\ & +& \frac{l(1-l)}{4(C+1+2{(1-l)}^2)}+N_2{(1-l)}^2)(g_1\square u)\left(t\right)\hfill \\ & +& \left(\frac{N_1(1-l)}{2l}+\frac{N_2^2(C+2)(1-l)(C+1+2{(1-l)}^2)}{l}+\frac{l(1-l)}{2(C+1+2{(1-l)}^2)}\right)(g_2\circ \nabla v)\left(t\right)\hfill \\ & +& \left(\frac{N}{2}-\frac{2C{g}_0{N}_2^2(C+1+2{(1-l)}^2)}{l}\right)\left\{(g_1^{\prime }\square u)\left(t\right)+(g_2^{\prime }\circ \nabla v)\left(t\right)\right\}-2\alpha N_1{\int }_{\Omega }uvdx.\hfill \end{array}$$

At this point, we choose $N_1$ that is large enough, so that

$$\frac{N_{1}l}{2}-\frac{l}{4}>4(1-l),$$

and then $N_2$ that is large enough, such that

$$N_2{g}_0-\frac{l}{4(C+1+2{(1-l)}^2)}-N_1>1,$$

and

$$N_2{g}_0-\frac{l^2(1-l)}{16N_2{(C+1+2{(1-l)}^2)}^2}>0.$$

Now, we choose N that is large enough, such that

$$\frac{N}{2}-\frac{2C{g}_0{N}_2^2(C+1+2{(1-l)}^2)}{l}>0.$$

Thus, (35) is established. □

Now, we are in a position to prove our main result.

Proof of Theorem (3). 

Taking into account (9) and (15), we obtain that for any $t\ge t_1$

$$\begin{array}{ccc}& & {\int }_0^{t_1}{g}_1\left(s\right)a(u\left(t\right)-u(t-s),u\left(t\right)-u(t-s))ds\hfill \\ & & \le -\frac{g_1\left(0\right)}{a_1}{\int }_0^{t_1}{g}_1^{\prime }\left(s\right)a(u\left(t\right)-u(t-s),u\left(t\right)-u(t-s))ds\le -c{E}^{\prime }\left(t\right),\hfill \end{array}$$

and

$$\begin{array}{ccc}& & {\int }_0^{t_1}{g}_2\left(s\right){\int }_{\Omega }{|\nabla v\left(t\right)-\nabla v(t-s)|}^{2}dxds\hfill \\ & & \le -\frac{g_2\left(0\right)}{a_2}{\int }_0^{t_1}{g}_2^{\prime }\left(s\right){\int }_{\Omega }{|\nabla v\left(t\right)-\nabla v(t-s)|}^{2}dxds\le -c{E}^{\prime }\left(t\right).\hfill \end{array}$$

Therefore, (35) yields for some $m>0$ and all $t\ge t_1$,

$$\begin{array}{ccc}\hfill F^{\prime }\left(t\right)& \le & -mE\left(t\right)+c(g_1\square u)\left(t\right)+c(g_2\circ \nabla v)\left(t\right)\hfill \\ & \le & -mE\left(t\right)-c{E}^{\prime }\left(t\right)+c{\int }_{t_1}^t{g}_1\left(s\right)a(u\left(t\right)-u(t-s),u\left(t\right)-u(t-s))ds\hfill \\ & & +c{\int }_{t_1}^t{g}_2\left(s\right){\int }_{\Omega }{|\nabla v\left(t\right)-\nabla v(t-s)|}^{2}dxds.\hfill \end{array}$$

(37)

Denote $\mathcal{L}\left(t\right)=F\left(t\right)+cE\left(t\right)$. Clearly, $\mathcal{L}\left(t\right)$ is equivalent to $E\left(t\right)$. It follows from (37) that

$$\begin{array}{ccc}\hfill {\mathcal{L}}^{\prime }\left(t\right)& \le & -mE\left(t\right)+c{\int }_{t_1}^t{g}_1\left(s\right)a(u\left(t\right)-u(t-s),u\left(t\right)-u(t-s))ds\hfill \\ & & +c{\int }_{t_1}^t{g}_2\left(s\right){\int }_{\Omega }{|\nabla v\left(t\right)-\nabla v(t-s)|}^{2}dxds.\hfill \end{array}$$

(38)

Next, the following two cases are considered.

Case 1. The function $Q\left(t\right)$ is linear.

We multiply (38) by $\xi \left(t\right)$ and use Assumption (A2) and (9) to obtain

$$\begin{array}{ccc}\hfill \xi \left(t\right){\mathcal{L}}^{\prime }\left(t\right)& \le & -m\xi \left(t\right)E\left(t\right)+c\xi \left(t\right){\int }_{t_1}^t{g}_1\left(s\right)a(u\left(t\right)-u(t-s),u\left(t\right)-u(t-s))ds\hfill \\ & & +c\xi \left(t\right){\int }_{t_1}^t{g}_2\left(s\right){\int }_{\Omega }{|\nabla v\left(t\right)-\nabla v(t-s)|}^{2}dxds\hfill \\ & \le & -m\xi \left(t\right)E\left(t\right)+c{\int }_{t_1}^t{\xi }_1\left(s\right)g_1\left(s\right)a(u\left(t\right)-u(t-s),u\left(t\right)-u(t-s))ds\hfill \\ & & +c{\int }_{t_1}^t{\xi }_2\left(s\right)g_2\left(s\right){\int }_{\Omega }{|\nabla v\left(t\right)-\nabla v(t-s)|}^{2}dxds\hfill \\ & \le & -m\xi \left(t\right)E\left(t\right)-c{\int }_{t_1}^t{g}_1^{\prime }\left(s\right)a(u\left(t\right)-u(t-s),u\left(t\right)-u(t-s))ds\hfill \\ & & -c{\int }_{t_1}^t{g}_2^{\prime }\left(s\right){\int }_{\Omega }{|\nabla v\left(t\right)-\nabla v(t-s)|}^{2}dxds\hfill \\ & \le & -m\xi \left(t\right)E\left(t\right)-c{E}^{\prime }\left(t\right).\hfill \end{array}$$

(39)

Denote $\mathcal{F}\left(t\right)=\xi \left(t\right)\mathcal{L}\left(t\right)+cE\left(t\right)\sim E\left(t\right)$. Then, we have, from (39) and the fact that $\xi$ is non-increasing, that for any $t\ge t_1$:

$${\mathcal{F}}^{\prime }\left(t\right)\le -m\xi \left(t\right)E\left(t\right).$$

Using the fact that $\mathcal{F}\sim E$, we obtain

$${\mathcal{F}}^{\prime }\left(t\right)\le -c_1\mathcal{F}\left(t\right),$$

for some positive constant $c_1$. By applying Gronwall’s Lemma, we obtain the existence of a constant $c_2>0$ such that

$$\mathcal{F}\left(t\right)\le c_2{e}^{{\displaystyle -c_1{\int }_{t_1}^t\xi \left(s\right)ds}},$$

which yields to

$$E\left(t\right)\le c_3{e}^{{\displaystyle -c_1{\int }_{t_1}^t\xi \left(s\right)ds}},$$

for some constant $c_3>0$.

Case 2: Q is nonlinear. First, we define the following quantities

$$I_1\left(t\right)=\frac{\kappa }{t}{\int }_0^{t}a(u\left(t\right)-u(t-s),u\left(t\right)-u(t-s))ds,t>0$$

and

$$I_2\left(t\right)=\frac{\kappa }{t}{\int }_0^t{\int }_{\Omega }{|\nabla v\left(t\right)-\nabla v(t-s)|}^{2}dxds,t>0.$$

Then, we have

$$\begin{array}{ccc}\hfill I_1\left(t\right)& \le & \frac{2\kappa }{t}{\int }_0^t\left[a\left(u\right(t),u(t\left)\right)+a\left(u\right(t-s),u(t-s\left)\right)\right]ds\hfill \\ & \le & \frac{4\kappa }{lt}\left({\int }_0^t(E\left(t\right)+E(t-s))ds\right)\hfill \\ & \le & \frac{8\kappa }{lt}{\int }_0^{t}E\left(s\right)ds\hfill \\ & \le & \frac{8\kappa }{lt}{\int }_0^{t}E\left(0\right)ds=\frac{8\kappa }{l}E\left(0\right)<+\infty ,\hfill \end{array}$$

and likewise, we have

$$\begin{array}{c}\hfill I_2\left(t\right)\le \frac{8\kappa }{l}E\left(0\right)<+\infty .\end{array}$$

Thus, choosing $0<\kappa <1$ that is small enough so that, for all $t>0$:

$$I_i\left(t\right)<1,\mathrm{for}i=1,2.$$

(40)

Also, we define ${\lambda }_1\left(t\right)$ and ${\lambda }_2\left(t\right)$ by

$${\lambda }_1\left(t\right)=-{\int }_0^t{g}_1^{\prime }\left(s\right)a(u\left(t\right)-u(t-s),u\left(t\right)-u(t-s))ds,$$

and

$${\lambda }_2\left(t\right)=-{\int }_0^t{g}_2^{\prime }\left(s\right){\int }_{\Omega }{|\nabla v\left(t\right)-\nabla v(t-s)|}^{2}dxds.$$

It is obvious that ${\lambda }_i\left(t\right)\le -c{E}^{\prime }\left(t\right),i=1,2$.

Noting Q is strictly convex on $(0,r]$ and $Q\left(0\right)=0$, then $Q\left(\theta x\right)\le \theta Q\left(x\right),$ provided that $0\le \theta \le 1$ and $x\in (0,r].$ This, together with (A1), (40) and Jensen’s inequality, leads to

$$\begin{array}{ccc}\hfill {\lambda }_1\left(t\right)& =& \frac{1}{\kappa I_1\left(t\right)}{\int }_0^t{I}_1\left(t\right)(-g_1^{\prime }\left(s\right))\kappa a(u\left(t\right)-u(t-s),u\left(t\right)-u(t-s))ds\hfill \\ & \ge & \frac{1}{\kappa I_1\left(t\right)}{\int }_0^t{I}_1\left(t\right){\xi }_1\left(s\right)Q\left(g_1\left(s\right)\right)\kappa a(u\left(t\right)-u(t-s),u\left(t\right)-u(t-s))ds\hfill \\ & \ge & \frac{{\xi }_1\left(t\right)}{\kappa I_1\left(t\right)}{\int }_0^{t}Q\left(I_1\left(t\right)g_1\left(s\right)\right)\kappa a(u\left(t\right)-u(t-s),u\left(t\right)-u(t-s))ds\hfill \\ & \ge & \frac{{\xi }_1\left(t\right)}{\kappa }Q\left(\frac{1}{I_1\left(t\right)}{\int }_0^t{I}_1\left(t\right)g_1\left(s\right)\kappa a(u\left(t\right)-u(t-s),u\left(t\right)-u(t-s))ds\right)\hfill \\ & =& \frac{{\xi }_1\left(t\right)}{\kappa }Q\left(\kappa {\int }_0^t{g}_1\left(s\right)a(u\left(t\right)-u(t-s),u\left(t\right)-u(t-s))ds\right)\hfill \\ & =& \frac{{\xi }_1\left(t\right)}{\kappa }\overline{Q}\left(\kappa {\int }_0^t{g}_1\left(s\right)a(u\left(t\right)-u(t-s),u\left(t\right)-u(t-s))ds\right),\hfill \end{array}$$

where $\overline{Q}$ is an extension of Q such that $\overline{Q}$ is strictly increasing and a strictly convex $C^2$ function on $(0,+\infty )$. This implies that

$${\int }_0^t{g}_1\left(s\right)a(u\left(t\right)-u(t-s),u\left(t\right)-u(t-s))ds\le \frac{1}{\kappa }{\overline{Q}}^{-1}\left(\frac{\kappa {\lambda }_1\left(t\right)}{{\xi }_1\left(t\right)}\right).$$

Similarly, we have

$${\int }_0^t{g}_2\left(s\right){\int }_{\Omega }{|\nabla v\left(t\right)-\nabla v(t-s)|}^{2}dxds\le \frac{1}{\kappa }{\overline{Q}}^{-1}\left(\frac{\kappa {\lambda }_2\left(t\right)}{{\xi }_2\left(t\right)}\right).$$

We infer from (38) that for any $t\ge t_1$

$${\mathcal{L}}^{\prime }\left(t\right)\le -mE\left(t\right)+c{\overline{Q}}^{-1}\left(\frac{\kappa {\lambda }_1\left(t\right)}{{\xi }_1\left(t\right)}\right)+c{\overline{Q}}^{-1}\left(\frac{\kappa {\lambda }_2\left(t\right)}{{\xi }_2\left(t\right)}\right).$$

(41)

For ${\epsilon }_0<r$, using (41) and the fact that $E^{\prime }\le 0$, ${\overline{Q}}^{\prime }>0$, ${\overline{Q}}^{″}>0$, we find that the functional ${\mathcal{K}}_1$, defined by

$${\mathcal{K}}_1\left(t\right)={\overline{Q}}^{\prime }\left({\epsilon }_0\frac{E\left(t\right)}{E\left(0\right)}\right)\mathcal{L}\left(t\right)+E\left(t\right)$$

is equivalent to $E\left(t\right)$ and satisfies

$$\begin{array}{ccc}\hfill {\mathcal{K}}_1^{\prime }\left(t\right)& =& {\epsilon }_0\frac{E^{\prime }\left(t\right)}{E\left(0\right)}{\overline{Q}}^{″}\left({\epsilon }_0\frac{E\left(t\right)}{E\left(0\right)}\right)\mathcal{L}\left(t\right)+{\overline{Q}}^{\prime }\left({\epsilon }_0\frac{E\left(t\right)}{E\left(0\right)}\right){\mathcal{L}}^{\prime }\left(t\right)+E^{\prime }\left(t\right)\hfill \\ & \le & -mE\left(t\right){\overline{Q}}^{\prime }\left({\epsilon }_0\frac{E\left(t\right)}{E\left(0\right)}\right)+c{\overline{Q}}^{\prime }\left({\epsilon }_0\frac{E\left(t\right)}{E\left(0\right)}\right){\overline{Q}}^{-1}\left(\frac{\kappa {\lambda }_1\left(t\right)}{{\xi }_1\left(t\right)}\right)+c{\overline{Q}}^{\prime }\left({\epsilon }_0\frac{E\left(t\right)}{E\left(0\right)}\right){\overline{Q}}^{-1}\left(\frac{\kappa {\lambda }_2\left(t\right)}{{\xi }_2\left(t\right)}\right).\hfill \end{array}$$

(42)

Now, let ${\overline{Q}}^{*}$ be the convex conjugate of $\overline{Q}$ in the sense of Young (see [26]). Then,

$$\begin{array}{c}\hfill {\overline{Q}}^{*}\left(s\right)=s{\left({\overline{Q}}^{\prime }\right)}^{-1}\left(s\right)-\overline{Q}({({\overline{Q}}^{\prime })}^{-1}\left(s\right)),\end{array}$$

(43)

which satisfies

$$\begin{array}{c}\hfill A{B}_i\le {\overline{Q}}^{*}\left(A\right)+\overline{Q}\left(B_i\right),i=1,2,\end{array}$$

(44)

with $A={\overline{Q}}^{\prime }\left({\epsilon }_0\frac{E\left(t\right)}{E\left(0\right)}\right)$ and $B_i={\overline{Q}}^{-1}\left(\frac{\kappa {\lambda }_i\left(t\right)}{{\xi }_i\left(t\right)}\right),i=1,2.$

It is inferred from (42)–(44) that

$$\begin{array}{ccc}\hfill {\mathcal{K}}_1^{\prime }\left(t\right)& \le & -mE\left(t\right){\overline{Q}}^{\prime }\left({\epsilon }_0\frac{E\left(t\right)}{E\left(0\right)}\right)+c{\epsilon }_0\frac{E\left(t\right)}{E\left(0\right)}{\overline{Q}}^{\prime }\left({\epsilon }_0\frac{E\left(t\right)}{E\left(0\right)}\right)+c\kappa \left(\frac{{\lambda }_1\left(t\right)}{{\xi }_1\left(t\right)}+\frac{{\lambda }_2\left(t\right)}{{\xi }_2\left(t\right)}\right)\hfill \end{array}$$

is obtained by multiplying the last inequality by $\xi \left(t\right)$ and using the fact that, as ${\epsilon }_0\frac{E\left(t\right)}{E\left(0\right)}<r,{\overline{Q}}^{\prime }\left({\epsilon }_0\frac{E\left(t\right)}{E\left(0\right)}\right)=Q^{\prime }\left({\epsilon }_0\frac{E\left(t\right)}{E\left(0\right)}\right)$ and ${\lambda }_i\left(t\right)\le -c{E}^{\prime }\left(t\right)$ (for $i=1,2)$, that

$$\begin{array}{ccc}\hfill \xi \left(t\right){\mathcal{K}}_1^{\prime }\left(t\right)& \le & -mE\left(t\right)\xi \left(t\right)Q^{\prime }\left({\epsilon }_0\frac{E\left(t\right)}{E\left(0\right)}\right)+c{\epsilon }_0\frac{E\left(t\right)}{E\left(0\right)}\xi \left(t\right)Q^{\prime }\left({\epsilon }_0\frac{E\left(t\right)}{E\left(0\right)}\right)-c{E}^{\prime }\left(t\right).\hfill \end{array}$$

Consequently, by letting ${\mathcal{K}}_2=\xi {\mathcal{K}}_1+cE$, we have: ${\alpha }_1{\mathcal{K}}_2\left(t\right)\le E\left(t\right)\le {\alpha }_2{\mathcal{K}}_2\left(t\right),$ for some ${\alpha }_1,{\alpha }_2>0$.

Hence, we conclude that, for some constant ${\beta }_1>0$ and for all $t\ge t_1$

$$\begin{array}{c}\hfill {\mathcal{K}}_2^{\prime }\left(t\right)\le -{\beta }_1\xi \left(t\right)\frac{E\left(t\right)}{E\left(0\right)}{Q}^{\prime }\left({\epsilon }_0\frac{E\left(t\right)}{E\left(0\right)}\right):=-{\beta }_1\xi \left(t\right)Q_2\left(\frac{E\left(t\right)}{E\left(0\right)}\right),\end{array}$$

(45)

where $Q_2\left(t\right)=t{Q}^{\prime }\left({\epsilon }_{0}t\right)$. Since $Q_2^{\prime }\left(t\right)=Q^{\prime }\left({\epsilon }_{0}t\right)+{\epsilon }_{0}t{Q}^{″}\left({\epsilon }_{0}t\right)$, then, using the strict convexity of Q on $(0,r]$, we reach that $Q_2^{\prime }\left(t\right),Q_2\left(t\right)>0$ on $(0,1]$. Thus, with $H\left(t\right)=\frac{{\alpha }_1{\mathcal{K}}_2\left(t\right)}{E\left(0\right)}$ and using the fact that ${\mathcal{K}}_2\sim E$ and (45), we have

$$H\left(t\right)\sim E\left(t\right),$$

(46)

and for some ${\beta }_2>0$,

$$H^{\prime }\left(t\right)\le -{\beta }_2\xi \left(t\right)Q_2\left(H\left(t\right)\right),\forall t\ge t_1.$$

Integrating the latter over $(t_1,t)$ yields

$$\begin{array}{c}\hfill {\int }_{t_1}^t\frac{-H^{\prime }\left(s\right)}{Q_2\left(H\left(s\right)\right)}ds\ge {\beta }_2{\int }_{t_1}^t\xi \left(s\right)ds,\end{array}$$

which leads to

$$\begin{array}{c}\hfill \frac{1}{{\epsilon }_0}{\int }_{{\epsilon }_{0}H\left(t\right)}^{{\epsilon }_{0}H\left(t_1\right)}\frac{1}{s{Q}_2^{\prime }\left(s\right)}ds\ge {\beta }_2{\int }_{t_1}^t\xi \left(s\right)ds.\end{array}$$

Lastly, since the function $Q_1$ given by $Q_1\left(t\right)={\int }_t^r\frac{1}{s{Q}^{\prime }\left(s\right)}ds$ is strictly decreasing on $(0,r]$ and ${\displaystyle \underset{t\to 0}{lim}{Q}_1\left(t\right)=+\infty }$, we deduce that

$$H\left(t\right)\le \frac{1}{{\epsilon }_0}{Q}_1^{-1}\left({\beta }_1{\int }_{t_1}^t\xi \left(s\right)ds\right).$$

Combining the latter with (46), one can claim that (13) holds. □

In the following remark, we may extend our previous results in the case where we take nonlinear coupling terms instead of the linear ones used in system (1) and also for a quasi-linear version, where the material densities vary according to the velocity.

Remark 3. 

1. 

We consider system (1) with $f_1(u,v)$ (respectively, $f_2(u,v)$) instead of $\alpha v$ (respectively, $\alpha u$), that is

$$\left\{\begin{array}{ccc}{\displaystyle u_{tt}-\gamma \Delta u_{tt}+{\Delta }^{2}u-{\int }_0^t{g}_1(t-s){\Delta }^{2}u\left(s\right)ds+f_1(u,v)=0}\hfill & \mathit{in}\hfill & \Omega \times (0,\infty )\hfill \\ {\displaystyle v_{tt}-\Delta v+{\int }_0^t{g}_2(t-s)\Delta v\left(s\right)ds+f_2(u,v)=0}\hfill & \mathit{in}\hfill & \Omega \times (0,\infty )\hfill \\ u={\partial }_{\nu }u=0\hfill & \mathit{on}\hfill & {\Gamma }_0\times (0,\infty )\hfill \\ {\mathbf{B}}_{1}u-{\mathbf{B}}_1\left\{{\displaystyle {\int }_0^t{g}_1(t-s)u\left(s\right)ds}\right\}=0\hfill & \mathit{on}\hfill & {\Gamma }_1\times (0,\infty )\hfill \\ {\mathbf{B}}_{2}u-\gamma {\partial }_{\nu }{u}_{tt}-{\mathbf{B}}_2\left\{{\displaystyle {\int }_0^t{g}_1(t-s)u\left(s\right)ds}\right\}=0\hfill & \mathit{on}\hfill & {\Gamma }_1\times (0,\infty )\hfill \\ v=0\hfill & \mathit{on}\hfill & \Gamma \times (0,\infty )\hfill \\ u\left(0\right)=u^0,u_t\left(0\right)=u^1,v\left(0\right)=v^0,v_t\left(0\right)=v^1\hfill & \mathit{in}\hfill & \Omega ,\hfill \end{array}\right.$$

where $f_i,i=1,2$, satisfy.

$f_i:{\mathbb{R}}^2\to \mathbb{R}$ (for $i=1,2$) are $C^1$ functions and there exists a positive function F such that

$$f_1(x_1,x_2)=\frac{\partial F}{\partial x_1},f_2(x_1,x_2)=\frac{\partial F}{\partial x_2},x_1{f}_1(x_1,x_2)+x_2{f}_2(x_1,x_2)-F(x_1,x_2)\ge 0,$$

and

$$\begin{array}{c}\hfill \left|\frac{\partial f_i}{\partial x_1}(x_1,x_2)\right|+\left|\frac{\partial f_i}{\partial x_2}(x_1,x_2)\right|\le d(1+|x_1{|}^{{\beta }_{i1}-1}+|x_2{|}^{{\beta }_{i2}-1}),\forall (x_1,x_2)\in {\mathbb{R}}^2,\end{array}$$

for some constant $d>0$ and ${\beta }_{ij}\ge 1$ for $i,j=1,2.$

By using the same method derived here, we may prove that the above system is well-posed and a general decay rate can be established, as in (13).

2. 

By following the same approaches as in Section 3 and Section 4, we shall prove that the following quasi-linear coupled system

$$\left\{\begin{array}{ccc}{\displaystyle |u_t{|}^{\rho }{u}_{tt}-\gamma \Delta u_{tt}+{\Delta }^{2}u-{\int }_0^t{g}_1(t-s){\Delta }^{2}u\left(s\right)ds+f_1(u,v)=0}\hfill & \mathit{in}\hfill & \Omega \times (0,\infty )\hfill \\ {\displaystyle |v_t{|}^{\rho }{v}_{tt}-\Delta v_{tt}-\Delta v+{\int }_0^t{g}_2(t-s)\Delta v\left(s\right)ds+f_2(u,v)=0}\hfill & \mathit{in}\hfill & \Omega \times (0,\infty )\hfill \\ u={\partial }_{\nu }u=0\hfill & \mathit{on}\hfill & {\Gamma }_0\times (0,\infty )\hfill \\ {\mathbf{B}}_{1}u-{\mathbf{B}}_1\left\{{\displaystyle {\int }_0^t{g}_1(t-s)u\left(s\right)ds}\right\}=0\hfill & \mathit{on}\hfill & {\Gamma }_1\times (0,\infty )\hfill \\ {\mathbf{B}}_{2}u-\gamma {\partial }_{\nu }{u}_{tt}-{\mathbf{B}}_2\left\{{\displaystyle {\int }_0^t{g}_1(t-s)u\left(s\right)ds}\right\}=0\hfill & \mathit{on}\hfill & {\Gamma }_1\times (0,\infty )\hfill \\ v=0\hfill & \mathit{on}\hfill & \Gamma \times (0,\infty )\hfill \\ u\left(0\right)=u^0,u_t\left(0\right)=u^1,v\left(0\right)=v^0,v_t\left(0\right)=v^1\hfill & \mathit{in}\hfill & \Omega ,\hfill \end{array}\right.$$

with $\rho >0$, possess at least a weak solution $u\in C([0,T],V)\cap C^1([0,T],W),v\in C([0,T],H_0^1(\Omega ))\cap C^1([0,T],H_0^1(\Omega ))$; and moreover, we shall establish a general decay rate of energy as in (13).

4. Conclusions

This paper focuses on the existence and the asymptotic stability of solutions for a system of two coupled Kirchhoff plate and wave equations in a bounded domain of ${\mathbb{R}}^2$, subject only to viscoelasticity dissipative terms and with the presence of rotational forces (in the Kirchhoff plate equation). It should be noted that this model takes the memory effects into account, which may exist in some materials, particularly in low temperature. The first equation, in system (1), describes the motion of a plate, which is clamped along one portion of its boundary and has free vibrations on the other portion of the boundary, whereas the second one models the motion of a membrane. This work is motivated by previous results concerning coupled wave equations [13,18,19,20,21,22] and coupled wave–plate equations [24].

By using the Faedo–Galerkin method, we proved the existence of a unique global weak solution. Furthermore, by constructing an appropriate Lyapunov function, we showed the general decay of the energy associated with the system (1). As a future work, we aim to change the type of damping by considering, for example, the Balakrishnan–Taylor damping (of the form $(\nabla y,\nabla y_t)\Delta y$), strong damping (of the form ${\Delta }^2{y}_t)$ or past history terms in the Kirchhoff plate equation (of the form ${\displaystyle {\int }_0^{\infty }{g}_1\left(s\right){\Delta }^2(x,t-s)ds}$) or/and in the wave equation (of the form ${\displaystyle {\int }_0^{\infty }{g}_2\left(s\right)\Delta (x,t-s)ds}$).