Well-posedness and decay in a system of hyperbolic and biharmonic-wave equations with variable exponents and weak dampings

Abstract

In this paper, we consider a coupled system of hyperbolic and biharmonic-wave equations with variable exponents in the damping and coupling terms. In each equation, the damping term is modulated by a time-dependent coefficient a(t) (or b(t)). First, we state and prove a well-posedness theorem of global weak solutions, by exploiting Galerkin’s method and some compactness arguments. Then, using the multiplier method, we establish the decay rates of the solution energy, under suitable assumptions on the time-dependent coefficients and the range of the variable exponents. We end our work with some illustrative examples.

1 Introduction

1.1 Model

In this work, we consider the following coupled system of wave and plate equations, for the unknowns u and v,

where $T>0$, $\Omega$ is a bounded domain of ${\mathbb {R}}^{n}\left( n\in {{\mathbb {N}}^{*}} \right)$ with a smooth boundary $\partial \Omega$, $\frac{\partial v}{\partial \eta }$ denotes the external normal derivative of v on the boundary of $\Omega ;$ $a,b: [0, \infty ) \longrightarrow (0,\infty )$ are two non-increasing $C^{1}$-functions and m, r and p are given continuous functions on $\overline{\Omega }$ satisfying some conditions. See (H.1)–(H3).

1.2 Motivation

In recent years and with the advancement of sciences, equations and systems of nonlinear wave equations with variable exponents occur in the mathematical modeling of various physical phenomena such as flows of electro-rheological fluids or fluids with temperature dependent viscosity, filtration processes through a porous media and image processing, nonlinear viscoelasticity, or robotics, etc. For more details on the subject, the reader can see [1, 8]. Our coupled system of variable-exponent nonlinearities (P) can be regarded as a model of the interaction between two fields describing the motion of two nonlinear “smart” materials, such as the motion of a suspension bridge and the cables. This class of problems requires more sophisticated mathematical tools to be investigated and well understood. The Lebesgue and sobolev spaces with variable exponents proved to be the appropriate spaces for studying such problems.

1.3 Literature review

For a class of one wave equation, Antontsev [3] studied the equation

$$\begin{aligned} u_{tt}-{div}\left( a|\nabla u| ^{p(x,t) -2}\nabla u\right) -\alpha \Delta u_{t}-bu|u| ^{\sigma (x,t)-2}=f, \ \text {in }\Omega \times \left( 0,T\right) , \end{aligned}$$

where $\alpha >0$ is a constant and $a,b,p,\sigma$ are given functions. Under specific conditions on the exponents, he proved the local and global existence of weak solutions and a blow-up result. In Guo and Gao [11] took $\sigma (x,t) =r>2$ and established a finite-time blow-up result for certain solutions with positive initial energy. After that, Guo [12] applied an interpolation inequality and some energy inequalities to obtain an estimate of the lower bound for the blow-up time when the source is super-linear. In Sun et al. [22] looked into the following equation

$$\begin{aligned} u_{tt}-{div}\left( a(x,t) \nabla u\right) +c(x,t)u_{t}|u_{t}|^{q(x,t)-1}=b(x,t) u|u|^{p(x,t) -2}, \ \text {in }\Omega \times (0,T), \end{aligned}$$

established a blow-up result and gave lower and upper bounds for the blow-up time, under some conditions on the initial data. In addition, they provided numerical illustrations for their results. In Messaoudi and Talahmeh [17] considered the equation

$$\begin{aligned} u_{tt}-{div}\left( |\nabla u|^{r(x)-2}\nabla u\right) +au_{t}| u_{t}|^{m(x)-2}=bu|u| ^{p(x) -2},\ \text {in }\Omega \times ( 0,T) , \end{aligned}$$

where $a,b>0\$are two constants and $m, r, p\$ are given functions. They proved a finite-time blow-up result. In the absence of source term $(b=0),$ Messaoudi et al. [18] obtained decay estimates of solutions and presented two numerical applications as illustration for their theoretical results. After that, they gave in [19] an overview of results concerning decay and blow up for nonlinear wave equations involving variable and constant exponents. Very recently, Xiaolei et al. [23] used some energy estimates and some komornik’s inequality to establish an asymptotic stability of solutions to quasilinear hyperbolic equations with variable source and damping terms.

Concerning coupled systems of hyperbolic equations with variable exponents, we mention the work of Bouhoufani and Hamchi [5], where they proved the global existence of weak solution and established decay estimates of the energy depending on the variable exponents. See also thesis [6]. Messaoudi and Talahmeh [20] considered a system of wave equations, with damping and source terms of variable-exponent nonlinearities, and proved a blow-up result for solutions with negative initial energy. Very recently, Messaoudi et al. [21], studied a coupled hyperbolic system with variable exponents. They, obtained an existence and uniqueness result of a weak solution, showed that certain solutions, with positive initial energy, blow up in finite time and gave some numerical applications.

For the case of systems with biharmonic operators and variable-exponents of nonlinearities, we cite the work of Bouhoufani et al. [4], in which the authors considered, in a bounded domain, two biharmonic-wave equations with nonlinear dampings and source terms. They established a local existence, uniqueness and blow-up result for solutions with negative-intial energy, and illustrated their theoretical findings by presenting some numericals tests. In Bouhoufani et al. [7] proved a theorem of finite-time blow up, for certain solution with positive initial data, and obtained the global existence as well as the decay rates, for the same problem, under suitable assumptions on the exponents and the initial data. After that, Messaoudi et al. [16] studied a coupled system of Laplacian and bi-Laplacian equations with dampings and source terms. They established existence, uniqueness, a long-time asymptotic behavior and blow up for solutions with positive-initial energy. In their case, with the presence of source terms, the solutions are local and they could blow up or exist globally (in time), depending on the range of the variable exponents and the initial data. However, in our study, the local solution doesn’t cease to exist in finite time, due to the nature of the coupling terms we have in our problem.

1.4 Main contribution

In this work, we intend to prove the local and global well-posedness for the problem (P) and establish explicit decay rates of the solution energy depending on the range of the variable exponents m, r and the time-dependent coefficients a and b. This paper consists of six sections. After the introduction, we recall the definitions of the variable-exponent Lebesgue and Sobolev spaces as well as some important lemmas related to these spaces. In section three, we state and establish the existence result of a weak solution of problem (P). Section four is devoted to the statement and the proof of our aim result of stabilty. In section five, we present some illustrative examples and end with a conclusion.

2 Preliminaries

2.1 Definitions and essential tools

In this section, we presente some important facts from [2, 10, 13] related to the Lebesgue and Sobolev spaces with variable exponents. Let $q: \Omega \longrightarrow \left[ 1, \infty \right)$ be a measurable function, where $\Omega$ is a domain of ${\mathbb {R}}^{n}.$ We define the Lebesgue space with a variable exponent by

$$\begin{aligned} L^{q(.)}(\Omega ) = \left\{ f: \Omega \longrightarrow {\mathbb {R}} \ \text {measurable in} \ \Omega : \ \varrho _{q(.)}(\lambda f) < +\infty , \ \text {for some} \ \lambda >0 \right\} , \end{aligned}$$

where

$$\begin{aligned} \varrho _{q(.)}(f)=\int _{\Omega }|f(x)|^{q(x)}dx. \end{aligned}$$

Endowed with the following Luxembourg-type norm

$$\begin{aligned} \Vert f\Vert _{q(.)}:= \text {inf} \ \left\{ \lambda >0: \int _{\Omega } \left| \frac{f(x)}{\lambda } \right| ^{q(x)}dx \le 1\right\} . \end{aligned}$$

$L^{q(.)}(\Omega )$ is a Banach space (see [2, 13]). We, also, define the variable exponent Sobolev space $W^{1,q(.)}(\Omega )$ as follows:

$$\begin{aligned} W^{1,q(.)}(\Omega )=\left\{ f \in L^{q(.)}(\Omega ) \ \text {such that} \ \nabla f \ \text {exists and}\; |\nabla f| \in L^{q(.)}(\Omega ) \right\} . \end{aligned}$$

This is a Banach space with respect to the norm

$$\begin{aligned} \Vert f\Vert _{W^{1,q(.)}(\Omega )}= \Vert f\Vert _{q(.)}+\Vert \nabla f\Vert _{q(.)}. \end{aligned}$$

Definition 2.1

We say that a function $q: \Omega \longrightarrow {\mathbb {R}}$ is log-Hölder continuous on $\Omega ,$ if there exists a constant $\theta >0,$ such that for all $0< \delta <1,$ we have

$$\begin{aligned} | q(x)-q(y) | \le -\frac{\theta }{\log |x-y|}, \text {for}\; a.e. \; x,y \in \Omega , \ with \ | x-y |< \delta . \end{aligned}$$

In addition, for q satisfying the log-Hölder continuity, we denote by $W^{1,q(.)}_{0}( \Omega )$ the closure of $C_{0}^{\infty }( \Omega )$ in $W^{1,q(.)}( \Omega )$ and by $W^{-1,q'(.)}( \Omega )$ the dual space of $W^{1,q(.)}_{0}( \Omega )$, in the same way as the usual Sobolev spaces, where $\frac{1}{q(.)}+ \frac{1}{q'(.)}=1$.

Lemma 2.2

[2, 13] (Young’s Inequality) Let $r, q, s \ge 1$ be measurable functions defined on $\Omega$, such that

$$\begin{aligned} \frac{1}{s(y)}=\frac{1}{r(y)}+\frac{1}{q(y)}, \ \text {for} \ a.e \ y \in \Omega . \end{aligned}$$

Then, for all $a,b\ge 0,$ we have

$$\begin{aligned} \frac{(ab)^{s(.)}}{s(.)}\le \frac{(a)^{r(.)}}{r(.)}+\frac{(b)^{q(.)}}{q(.)}. \end{aligned}$$

Lemma 2.3

[2, 13] If $1<q^{-}\le q(y) \le q^{+}<+\infty$ hold then, for any $f \in L^{q(.)}(\Omega )$, we have

• (i)
  $$\begin{aligned} \text {min} \left\{ \Vert f \Vert _{q(.)}^{q^{-}},\Vert f \Vert _{q(.)}^{q^{+}}\right\} \le \varrho _{q(.)}(f) \le \text {max} \left\{ \Vert f\Vert _{q(.)}^{q^{-}},\Vert f \Vert _{q(.)}^{q^{+}}\right\} \end{aligned}$$
• (ii)
  $$\begin{aligned} \varrho _{q(.)}(f) \le ||f||_{q^{-}}^{q^{-}} + ||f||_{q^{+}}^{q^{+}}, \end{aligned}$$

  where

  $$\begin{aligned} q^{-}=ess \inf _{x\in \Omega }\ q(x) \ \text {and} \ q^{+}=ess \sup _{x\in \Omega }\ q(x). \end{aligned}$$

Lemma 2.4

(Embedding Property )[9] Let $q: \Omega \longrightarrow \left[ 1, \infty \right)$ be a measurable function and $k \ge 1$ be an integer. Suppose that r is a log-Hölder continuous function on $\Omega ,$ such that, for all $x\in \Omega ,$ we have

$$\begin{aligned} \left\{ \begin{array}{ll} k \le q^{-} \le q(x) \le q^{+}< \frac{nr(x)}{n-kr(x)},&{} \text {if} \ r^{+}< \frac{n}{k},\\ k \le q^{-} \le q^{+} < \infty ,&{} \text {if} \ r^{+} \ge \frac{n}{k}. \end{array}\right. \end{aligned}$$

Then, the embedding $W^{k,r(.)}_{0}( \Omega ) \hookrightarrow L^{q(.)}(\Omega )$ is continuous and compact.

To establish our stability result, the following Lemma is necessary.

Lemma 2.5

[15] Let $E: {\mathbb {R}}_{+}\longrightarrow {\mathbb {R}}_{+}$ be a non-increasing function and $\sigma : {\mathbb {R}}_{+}\longrightarrow {\mathbb {R}}_{+}$ be an increasing $C^{1}$-function, with $\sigma (0)=0$ and $\sigma (t) \longrightarrow +\infty$ as $t \longrightarrow \infty$. Assume that there exist $q \ge 0$ and $C>0,$ such that

$$\begin{aligned} \int _{S}^{\infty } \sigma '(t) E(t)^{q+1}dt\le CE(S), \ 0\le S < \infty . \end{aligned}$$

Then, there exist two positive constants c and w such that, for all $t \ge 0$,

$$\begin{aligned} E\left( t\right) \le \left\{ \begin{array}{ll} \frac{c }{\left[ 1+\sigma (t) \right] ^{1 / q}},&{} \ if \ q>0, \\ ce^{-\omega \sigma (t)}, &{} \ if \ q=0. \end{array}\right. \end{aligned}$$

2.2 Notations and assumptions

Throughout this paper, we denote by ${\mathcal {V}}$ the following space

$$\begin{aligned} {\mathcal {V}}=\lbrace u \in H^{2}(\Omega ): \ u= \frac{\partial u}{\partial \eta }=0 \ on \ \partial \Omega \rbrace =H_{0}^{2}(\Omega ). \end{aligned}$$

So, ${\mathcal {V}}$ is a separable Hilbert space endowed with the inner product and norm, respectively,

$$\begin{aligned} (u,v)_{ {\mathcal {V}} }=\int _{\Omega } (\Delta u) (\Delta v) dx \ and \ \left\| u \right\| _{ {\mathcal {V}}}=\left\| \Delta u \right\| _{L^{2}(\Omega )}. \end{aligned}$$

The assumptions on m, r and p, that will be used in the sequel, are as follows. For all $x\in \overline{\Omega }$, we suppose that

and

where

$$\begin{aligned} m^{-}= & {} \inf _{x\in \overline{\Omega }}\ m\left( x\right) , \ m^{+}=\ \sup _{x\in \overline{\Omega }} \ m\left( x\right) ,\\ r^{-}= & {} \inf _{x\in \overline{\Omega }}\ r\left( x\right) , \ r^{+}=\ \sup _{x\in \overline{\Omega }}\ r\left( x\right) ,\\ p^{-}= & {} \inf _{x\in \overline{\Omega }}\ p\left( x\right) \ \text {and} \ p^{+}=\ \sup _{x\in \overline{\Omega }} \ p\left( x\right) . \end{aligned}$$

Remark 2.6

Since m, r and p are $C^{1}(\overline{\Omega })$, then they satisfy the log-Hölder continuity condition.

3 Global well-posedness result

In this section, our goal is to prove a local and global existence theorem of weak solutions of (P). For this purpose, we introduce the definition of a weak solution of problem (P).

Definition 3.1

Consider $(u_{0},u_{1}) \in H^{1}_{0}(\Omega ) \times L^{2}(\Omega )$ and $(v_{0},v_{1}) \in {\mathcal {V}} \times L^{2}(\Omega )$. A pair of functions (u, v) is said to be a weak solution of (P) on $\left[ 0,T\right)$ if

$$\begin{aligned} \left| \begin{array}{ll} u \in L^{\infty } \left( [0,T); H^{1}_{0}(\Omega ) \right) , v \in L^{\infty } \left( [0,T); {\mathcal {V}} \right) \\ u_{t} \in L^{\infty }\left( [0,T); L^{2}(\Omega )\right) \cap L_{a}^{m(.)}\left( \Omega \times (0,T)\right) , \\ v_{t} \in L^{\infty }\left( [0,T); L^{2}(\Omega )\right) \cap L_{b}^{r(.)}\left( \Omega \times (0,T)\right) \end{array}\right. \end{aligned}$$

(3.1)

and (u, v) satisfies,

$$\begin{aligned}&\int _{\Omega } u_{t}\phi \ dx - \int _{\Omega }^{}u_{1} \phi \ dx+ \int _{0}^{t} \int _{\Omega } a(\tau ) |u_{t}|^{m(x)-2} u_{t} \phi \ dx d \tau \\&\quad +\int _{0}^{t}\int _{\Omega } \nabla u.\nabla \phi \ dx d \tau +\int _{0}^{t}\int _{\Omega } |u|^{p(x)-2} u |v|^{p(x)} \phi \ dx d \tau =0 \end{aligned}$$

and

$$\begin{aligned}&\int _{\Omega } v_{t}\psi dx - \int _{\Omega }^{}v_{1} \psi \ dx+ \int _{0}^{t} \int _{\Omega } b (\tau ) |v_{t}|^{r(x)-2}v_{t} \psi \ dx d \tau \\&\quad +\int _{0}^{t}\int _{\Omega } (\Delta v) (\Delta \psi ) \ dx d \tau +\int _{0}^{t}\int _{\Omega } |v|^{p(x)-2} v |u|^{p(x)} \psi \ dx d\tau =0, \end{aligned}$$

for all $(\phi , \psi )\in H^{1}_{0}(\Omega )\times {\mathcal {V}}$ and all $t\in \left( 0,T\right)$, with $\left( u(.,0),v(.,0)\right) =(u_{0},v_{0}), \left( u_{t}(.,0),v_{t}(.,0)\right) =(u_{1},v_{1})$. Here,

$$\begin{aligned} L_{a}^{m(.)}\left( \Omega \times (0,T)\right) =\left\{ w: \Omega \times (0,T) \longrightarrow {\mathbb {R}}; \int _{0}^{T} \int _{\Omega } a (\tau ) |w(x)|^{m(x)} dx d \tau <+\infty \right\} . \end{aligned}$$

and

$$\begin{aligned} L_{b}^{r(.)}\left( \Omega \times (0,T)\right) =\left\{ w: \Omega \times (0,T) \longrightarrow {\mathbb {R}}; \int _{0}^{T} \int _{\Omega } b (\tau ) |w(x)|^{r(x)} dx d \tau <+\infty \right\} . \end{aligned}$$

We have the following well-posedness result.

Theorem 3.2

Assume that (H.1)–(H.3) are satisfied. Then, for any initial data $(u_{0},u_{1}) \in H^{1}_{0}(\Omega ) \times L^{2}(\Omega )$ and $(v_{0},v_{1}) \in {\mathcal {V}} \times L^{2}(\Omega )$, there exists a weak solution (u, v) of (P) (in the sense of Definition 3.1) defined in $\left[ 0,T\right)$, for all $T>0$.

Proof

We procced in several steps:

Step 1. Consider $T>0$ fixed but arbitrary. Let $\left\{ \omega _{j} \right\} _{j=1}^{\infty }$ be an orthonormal basis of ${\mathcal {V}}$ and $V_{k}=span \left\{ \omega _{1}, \omega _{2},\ldots , \omega _{k} \right\}$ be the subspace generated by the first k vectors $\omega _1,\omega _2,\ldots , \omega _k$. Consider

$$\begin{aligned} u^{k}(t)=\Sigma _{j=1}^{k} a_{j}(t)\omega _{j} \ \text {and} \ v^{k}(t)=\Sigma _{j=1}^{k} b_{j}(t)\omega _{j}, \ t \in (0,T), \end{aligned}$$

such that $(u^{k},v^{k})$ satisfy

$$\begin{aligned} \left\{ \begin{array}{ll} \int _{\Omega }u_{tt}^{k}(t)\omega _{j}dx+\int _{\Omega } \nabla u^{k}(t). \nabla \omega _{j}dx+\int _{\Omega }a(t)\left| u^{k}_{t}(t)\right| ^{m\left( x\right) -2}u^{k}_{t}(t)\omega _{j}dx \\ =-\int _{\Omega }|u^{k}(t)| ^{p(x) -2}u^{k}(t)|v^{k}(t)| ^{p(x)}\omega _{j}dx, \\ \int _{\Omega }v_{tt}^{k}(t)\omega _{j}dx+\int _{\Omega } ( \Delta v^{k}(t))( \Delta \omega _{j}) dx+\int _{\Omega }b(t)\left| v^{k}_{t}(t)\right| ^{r\left( x\right) -2}v^{k}_{t}(t)\omega _{j}dx \\ =-\int _{\Omega }|v^{k}(t)| ^{p(x) -2}v^{k}(t)|u^{k}(t)| ^{p(x)}\omega _{j}dx, \end{array}\right. \end{aligned}$$

(3.2)

for $j=1, 2,\ldots ,k$, with the initial data

$$\begin{aligned}&u^{k}(0)=u^{k}_{0}=\Sigma _{i=1}^{k} \left\langle u_{0},\omega _{i}\right\rangle \omega _{i}, \ u^{k}_{t}(0)=u^{k}_{1}=\Sigma _{i=1}^{k} \left\langle u_{1},\omega _{i}\right\rangle \omega _{i} \nonumber \\&v^{k}(0)=v^{k}_{0}=\Sigma _{i=1}^{k} \left\langle v_{0},\omega _{i}\right\rangle \omega _{i}, \ v^{k}_{t}(0)=v^{k}_{1}=\Sigma _{i=1}^{k} \left\langle v_{1},\omega _{i}\right\rangle \omega _{i}, \end{aligned}$$

(3.3)

such that

$$\begin{aligned} \left. \begin{array}{ll} (u^{k}_{0},v^{k}_{0}) \longrightarrow (u_{0},v_{0}) \ \ \text {in} \ H^{1}_{0} (\Omega )\times {\mathcal {V}}, \\ (u_1^{k},v_{1}^{k}) \longrightarrow (u_{1},v_{1}) \ \ \text {in} \ L^{2}(\Omega )\times L^{2}(\Omega ). \end{array}\right. \end{aligned}$$

(3.4)

For any $k\ge 1$, Eq. (3.2) generate a system of k nonlinear ordinary differential equations, which admits a unique local solution $(u^{k},v^{k})$ defined on $[0,t_{k}),$ $0 < t_{k} \le T$, by the standard theory of ODE. In the following step, our purpose is to extend this solution to [0, T),  for any $k\ge 1$.

Step 2. Multiplying both sides of (3.2)$_{1}$ and (3.2)$_{2}$ by $a'_{j}(t)$ and $b'_{j}(t)$, respectively, using Green’s formula and the boundary conditions, and then summing each result over j,  from 1 to k, we obtain, for all $0< t \le t_{k}$,

$$\begin{aligned}&\frac{1}{2}\frac{d}{dt}\left[ ||u_{t}^{k}||_{2}^{2}+ || \nabla u^{k}||_{2}^{2} \right] + \int _{\Omega } a(t)\left| u^{k}_{t}\right| ^{m\left( x\right) }dx \nonumber \\&\quad =- \int _{\Omega }u^{k}|u^{k}| ^{p(x) -2}|v^{k}| ^{p(x)} \vert u^{k}_{t} \vert dx \end{aligned}$$

(3.5)

and

$$\begin{aligned}&\frac{1}{2} \frac{d}{dt}\left[ ||v_{t}^{k}||_{2}^{2}+ || \Delta v^{k}||_{2}^{2} \right] + \int _{\Omega }b(t)\left| v^{k}_{t}\right| ^{r\left( x\right) }dx \nonumber \\&\quad =- \int _{\Omega }v ^{k}|v^{k}| ^{p(x) -2}|u^{k}| ^{p(x)} \vert v^{k}_{t}\vert dx. \end{aligned}$$

(3.6)

Adding (3.5) and (3.6), we get

$$\begin{aligned}&\frac{1}{2}\frac{d}{dt} \left[ \Vert u^{k}_{t}\Vert ^{2}_{2}+ \Vert v^{k}_{t}\Vert ^{2}_{2}+\Vert \nabla u^{k} \Vert ^{2}_{2}+\Vert \Delta v^{k} \Vert ^{2}_{2} \right] +\frac{d}{dt} \int _{\Omega } \frac{\vert u^{k} v^{k}\vert ^{p(x)}}{p(x)} dx \nonumber \\&\quad + \int _{\Omega }\left( a(t)\left| u^{k}_{t} \right| ^{m\left( x\right) }+ b(t)\left| v^{k}_{t} \right| ^{r\left( x\right) }\right) dx \le 0. \end{aligned}$$

(3.7)

We then integrate (3.7) over (0, t), with $t \le t_k$, to arrive at

$$\begin{aligned}&\frac{1}{2}\left[ \Vert u^{k}_{t}\Vert ^{2}_{2}+ \Vert v^{k}_{t}\Vert ^{2}_{2}+\Vert \nabla u^{k} \Vert ^{2}_{2}+\Vert \Delta v^{k} \Vert ^{2}_{2} \right] + \int _{\Omega } \frac{\vert u^{k} v^{k}\vert ^{p(x)}}{p(x)} dx \nonumber \\&\quad + \int _{0 }^{t}\int _{\Omega }\left( a(\tau )\left| u^{k}_{t}(.,\tau ) \right| ^{m\left( x\right) }+ b(\tau )\left| v^{k}_{t}(., \tau )\right| ^{r\left( x\right) }\right) dxd\tau \nonumber \\&\quad \le \frac{1}{2} \left[ \Vert u^{k}_{1} \Vert ^{2}_{2}+ \Vert v^{k}_{1}\Vert ^{2}_{2}+\Vert \nabla u^{k}_{0} \Vert ^{2}_{2}+\Vert \Delta v^{k}_{0} \Vert ^{2}_{2}\right] + \int _{\Omega } \frac{\vert u_{0}^{k} v_{0}^{k}\vert ^{p(x)}}{p(x)} dx =\tilde{C}, \end{aligned}$$

(3.8)

thanks to the convergence (3.4). Note here that the last term in the right-hand side of (3.8) is finite by (H.3). It can be established exactly by the same calculations of (3.16) below. Also, under the hypothesise (H.3), we conclude that

$$\begin{aligned}&\Vert u^{k}_{t}\Vert ^{2}_{2}+ \Vert v^{k}_{t}\Vert ^{2}_{2}+\Vert \nabla u^{k} \Vert ^{2}_{2}+\Vert \Delta v^{k} \Vert ^{2}_{2}\\&\quad + \int _{0 }^{t}\int _{\Omega }\left( a(\tau ) \left| u^{k}_{t}(.,\tau ) \right| ^{m\left( x\right) }+ b(\tau )\left| v^{k}_{t}(. ,\tau ) \right| ^{r\left( x\right) }\right) dxd\tau \le C_{T} , \forall \ 0 \le t \le t_{k}, \end{aligned}$$

where $C_{T}$ is a constant independante of t and k. Therefore, we can extend the $(u_k)_k$ and $(v_k)_k$ on [0, T). Moreover, we have

$$\begin{aligned} \left\{ \begin{array}{ll} (u^{k})_{k} { isboundedin} L^{\infty } \left( (0, T), H_{0}^{1}( \Omega ) \right) , \\ (v^{k})_{k} { isboundedin} L^{\infty } \left( (0, T), {\mathcal {V}} \right) , \\ (u_{t}^{k})_{k}{} { isboundedin}L^{\infty }\left( (0, T), L^{2}(\Omega )\right) \cap L_{a}^{m(.)}\left( \Omega \times (0,T)\right) , \\ (v_{t}^{k}) _{k}{} { isboundedin}L^{\infty }\left( (0, T), L^{2}(\Omega )\right) \cap L_{b}^{r(.)} \left( \Omega \times (0,T)\right) . \end{array}\right. \end{aligned}$$

(3.9)

Step 3. From (3.9), there exist two subsequences of $(u^{k})_{k}$ and $(v^{k})_{k}$, still denoted by $(u^{k})_{k}$ and $(v^{k})_{k}$ (for simplicity), respectively, and two functions $u,v: \Omega \times [0,T) \longrightarrow {\mathbb {R}}$, such that

$$\begin{aligned} \left\{ \begin{array}{ll} u^{k} \rightharpoonup ^{*} u \ \text {in} \ L^{\infty } \left( (0, T), H_{0}^{1}( \Omega ) \right) , \\ v^{k} \rightharpoonup ^{*} v \ \text {in} \ L^{\infty } \left( (0, T), {\mathcal {V}} \right) , \\ (u_{t}^{k}, v_{t}^{k}) \rightharpoonup ^{*} (u_{t},v_{t}) \ \text {in} \ L^{\infty }\left( (0, T),L^{2}(\Omega )\right) \times L^{\infty }\left( (0, T),L^{2}(\Omega )\right) , \\ (u_{t}^{k}, v_{t}^{k}) \rightharpoonup (u_{t},v_{t}) \ \text {in} \ L_{a}^{m(.)}\left( \Omega \times (0,T)\right) \times L_{b}^{r(.)}\left( \Omega \times (0,T)\right) . \end{array}\right. \end{aligned}$$

(3.10)

Next, we show that

$$\begin{aligned} |u^{k}| ^{p(.) -2}u^{k}|v^{k}| ^{p(.)} \rightharpoonup |u| ^{p(.) -2}u|v| ^{p(.)} \ \text {in} \ L^{2} \left( \Omega \times (0,T)\right) \end{aligned}$$

(3.11)

and

$$\begin{aligned} |v^{k}| ^{p(.) -2}v^{k}|u^{k}| ^{p(.)} \rightharpoonup |v| ^{p(.) -2}v|u| ^{p(.)} \ \text {in} \ L^{2} \left( \Omega \times (0,T)\right) . \end{aligned}$$

(3.12)

By the convergences (3.10)$_{1}$ and (3.10)$_{2}$, the fact that $H^{i}_{0}(\Omega )\hookrightarrow ^{compact} L^{2}(\Omega )$ $( i=\overline{1,2})$ and invoking Lions’ Theorem [14], there exist two subsequences of $(u^{k})_{k}$ and $(v^{k})_{k}$, denoted by $(u^{k})_{k}$ and $(v^{k})_{k}$, respectively, such that

$$\begin{aligned} u^{k}\longrightarrow u \ \text {and} \ v^{k}\longrightarrow v \ \ \text {strongly in} \ L^{2}\left( (0, T), L^{2}(\Omega )\right) \end{aligned}$$

and

$$\begin{aligned} u^{k}\longrightarrow u \ \text {and} \ v^{k}\longrightarrow v \ \ \text {a.e. in} \ \Omega \times (0, T), \end{aligned}$$

(3.13)

for all $T>0$. The continuity of the function:

$$\begin{aligned} (u,v) \mapsto \left( |u| ^{p(.) -2}u|v| ^{p(.)}, |v| ^{p(.) -2}v|u| ^{p(.)}\right) \end{aligned}$$

and the convergences (3.13) lead to

$$\begin{aligned} |u^{k}| ^{p(.) -2}u^{k}|v^{k}| ^{p(.)} \rightarrow |u| ^{p(.) -2}u|v| ^{p(.)} \ \ \text {a.e. in } \ \Omega \times (0,T) \end{aligned}$$

(3.14)

and

$$\begin{aligned} |v^{k}| ^{p(.) -2}v^{k}|u^{k}| ^{p(.)} \rightarrow |v| ^{p(.) -2}v|u| ^{p(.)} \ \ \text {a.e. in } \ \Omega \times (0,T). \end{aligned}$$

On the other hand, applying Young’s inequality, with

$$\begin{aligned} q(x) = \frac{2p(x)-1}{p(x)-1} \ \text {and} \ q'(x) = \frac{2p(x)-1}{p(x)}, \end{aligned}$$

we obtain, for a.e. $x\in \Omega ,$

$$\begin{aligned} \left| u^{k}\right| ^{p(x)-1}\left| v^{k}\right| ^{p(x)} \le \frac{1}{2}\left| u^{k}\right| ^{2p(x)-1} + C(x)\left| v^{k}\right| ^{2p(x)-1} , \end{aligned}$$

where

$$\begin{aligned} C(x) = \frac{p(x)}{2p(x)-1}\left( \frac{2p(x)}{(2p(x)-1)}\right) ^{\frac{p(x)-1}{p(x)}}. \end{aligned}$$

Since p is bounded on $\Omega ,$ C(x) is bounded too. Hence, it comes, for some $C_{1}>0$ and for a.e. $x\in \Omega ,$

$$\begin{aligned} \left| u^{k}\right| ^{p(x)-1}\left| v^{k}\right| ^{p(x)} \le C_{1} \left[ \left| u^{k}\right| ^{2p(x)-1} + \left| v^{k}\right| ^{2p(x)-1} \right] . \end{aligned}$$

(3.15)

From the assumption (H.3), invoking Lemma 2.3 and the embeddings result (Corollary 2.4), estimate (3.15) yields, for all $t\le t_{k}$ $(t_{k}\le T),$

$$\begin{aligned}&\int _{\Omega } |u^{k}| ^{2(p(x) -1)}|v^{k}| ^{2p(x)} dx \nonumber \\&\quad \le C_{1}^{2}\int _{\Omega }\left[ \left| u^{k}\right| ^{2p(x)-1} + \left| v^{k}\right| ^{2p(x)-1} \right] ^{2}dx \nonumber \\&\quad \le C \int _{\Omega }\left( \left| u^{k}\right| ^{2(2p^{+}-1)} + \left| u^{k}\right| ^{2(2p^{-}-1)} + \left| v^{k}\right| ^{2(2p^{+}-1)} + \left| v^{k}\right| ^{2(2p^{-}-1)}\right) dx \nonumber \\&\le C \left( \Vert \nabla u^{k}\Vert _{2}^{2(2p^{+}-1)} +\Vert \nabla u^{k}\Vert _{2}^{2(2p^{-}-1)} + \Vert \Delta v^{k} \Vert _{2}^{2(2p^{+}-1)} + \Vert \Delta v^{k}\Vert _{2}^{2(2p^{-}-1)} \right) \nonumber \\&\quad \le C, \end{aligned}$$

(3.16)

where $C>0$ is a generic positive constant. It follows, for some $\tilde{C}_{T}>0,$

$$\begin{aligned} \int _{0}^{T} \Vert |u^{k}| ^{p(.)-2}u^{k} |v^{k}| ^{p(.)} \Vert _{2}^{2}d\tau \le \tilde{C}_{T}, \end{aligned}$$

which means that $|u^{k}| ^{p(.) -2}u^{k}|v^{k}| ^{p(.)}$ is bounded in $L^{2}\left( \Omega \times (0,T)\right)$. This result with (3.14) allow us to establish (3.11), by virtue of Lions’ Lemma. In similar way, we obtain (3.12). For the damping terms, we claim that

$$\begin{aligned} a(.)| u_{t}^{k} | ^{m(.)-2} u_{t}^{k} \rightharpoonup \ a(.)| u_{t}| ^{m(.)-2} u_{t} \ \text{ in } \ L^{\frac{m(.)}{m(.)-1}}(\Omega \times (0,T)) \end{aligned}$$

and

$$\begin{aligned} b(.)|v_{t}^{k} | ^{r(.)-2}v_{t}^{k} \rightharpoonup \ b(.)| v_{t} | ^{r(.)-2}v_{t} \ \text{ in } \ L^{\frac{r(.)}{r(.)-1}}(\Omega \times (0,T)). \end{aligned}$$

Indeed, using Hölder’s inquality and the fact that $(u_{t}^{k},v_{t}^{k})_{k}$ is bounded in $L_{a}^{m(.)}(\Omega \times (0,T)) \times L_{b}^{r(.)}(\Omega \times (0,T))$, we infer that

$$\begin{aligned} (a(.)| u_{t}^{k} |^{m(.)-2}u^{k}_{t} )_{k} \ \text{ is } \text{ bounded } \text{ in } \ L^{\frac{m(.)}{m(.)-1}}(\Omega \times (0,T) ) \end{aligned}$$

and

$$\begin{aligned} (b(.)| v_{t}^{k} |^{m(.)-2}v^{k}_{t} )_{k} \ \text{ is } \text{ bounded } \text{ in } \ L^{\frac{r(.)}{r(.)-1}}(\Omega \times (0,T) ). \end{aligned}$$

Therefore, there exist two subsequences of $(| u_{t}^{k} | ^{m(.)-2}u^{k}_{t} )_{k}$ and $(|v_{t}^{k} |^{r(.)-2}v^{k}_{t} )_{k}$, denoted by $(|u_{t}^{k} |^{m(.)-2} u^{k}_{t})_{k}$ and $(|v_{t}^{k} |^{r(.)-2} v^{k}_{t})_{k}$, respectively, such that

$$\begin{aligned} a(.)|u_{t}^{k} |^{m(.)-2} u^{k}_{t} \rightharpoonup \Phi \ \text{ in } \ L^{\frac{m(.)}{m(.)-1}} (\Omega \times (0,T)) \end{aligned}$$

and

$$\begin{aligned} b(.)| v_{t}^{k} |^{r(.)-2} v^{k}_{t} \rightharpoonup \Psi \ \text{ in } \ L^{\frac{r(.)}{r(.)-1}} (\Omega \times (0,T)) \end{aligned}$$

By repeating the same steps of [21] for the sequences $(S_{k})_{k}, (\tilde{S}_k)_{k}$ defined, for all $k \ge 1,$ as

$$\begin{aligned} S_k= \int _0^T a(t) \int _{\Omega }\left( h(u^k_t)-h(z)\right) (u^k_t-z)dx dt, \end{aligned}$$

for $z \in L_{a}^{m(\cdot )}\left( (0,T),{\mathcal {V}} \right)$ and $h(z)=|z| ^{m(\cdot )-2}z$, and

$$\begin{aligned} \tilde{S}_k= \int _0^T b(t) \int _{\Omega }\left( h(v^k_t)-h(z)\right) (v^k_t-z)dxdt, \end{aligned}$$

for $z \in L_{b}^{r(\cdot )}\left( (0,T), H_{0}^{1}(\Omega ) \right)$ and $h(z)=|z| ^{r(\cdot )-2}z$, we easily show that

$$\begin{aligned} \Phi = a(.)| u_t | ^{m(\cdot )-2} u_t \quad \text { and } \quad \Psi =b(.)|v_t^k | ^{r(\cdot )-2}v_t^k \end{aligned}$$

and establish that (u, v) satisfies the two differential equations of (P), on $\Omega \times (0,T)$ (in the weak sense), for all $T>0$.

Step 4. As in [21], we easily establish that (u, v) satisfies the initial conditions. Finally, we conclude that (u, v) is a global weak solution of (P). $\square$

Remark 3.3

Note that the uniqueness of the solution remains open. However, if $a(.)=b(.),$ we can obtain uniqueness by repeating the same steps of [21].

4 The decay rates

In order to state and prove our stability result, we define the energy functional associated to problem $\left( P\right)$, by

$$\begin{aligned} E\left( t\right) =:\frac{1}{2}\left[ \left\| u_{t}\right\| _{2}^{2}+\left\| v_{t}\right\| _{2}^{2} + \left\| \nabla u \right\| _{2}^{2}+\left\| \Delta v \right\| _{2}^{2} \right] + \int _{\Omega }\frac{\left| uv\right| ^{p(x)}}{p(x)}dx, \end{aligned}$$

for all $t\in \left[ 0,T\right)$.

Multiplying the first equation of $\left( P\right)$ by $u_{t}$, the second one by $v_{t}$, integrating each result over $\Omega$, using Green’s formula and the boundary conditions, and then summing up, we obtain

$$\begin{aligned} E'(t)=-a(t)\int _{\Omega }|u_{t}| ^{m(x)}dx- b(t)\int _{\Omega }|v_{t}|^{r(x)}dx\le 0, \end{aligned}$$

(4.1)

for a.e $t\in \left[ 0,T\right)$.

Theorem 4.1

Assume that (H.1)–(H.3) hold, and that $\int _{0}^{\infty }a(s)ds= \int _{0}^{\infty } b(s) ds=+\infty$. Then, there exist two constants $c, \omega >0$ such that the solution of (P) satisfies, for all $t\ge 0$,

$$\begin{aligned} E\left( t\right) \le \left\{ \begin{array}{ll} ce^{-\omega \int _{0}^{t} \Gamma (s)ds}, &{} \ if \ \alpha ^{+} =2, \\ \frac{c}{ \left( 1+\int _{0}^{t} \Gamma (s)ds \right) ^{\frac{2}{\alpha ^{+}-2}}},&{} \ if \ \alpha ^{+}>2, \end{array}\right. \end{aligned}$$

where

$$\begin{aligned} \alpha ^{+}=max \ \lbrace m^{+},r^{+}\rbrace \ \text {and} \ \Gamma =min \lbrace a, b \rbrace . \end{aligned}$$

Proof

Let $S \in (0,T)$. Multiplying $\left( P\right) _{1}$ and $\left( P\right) _{2}$ by $\Gamma E^{q}u$ and $\Gamma E^{q}v$, respectively, for $q\ge 0$ (to be specified later), integrating each result over $\Omega \times \left( S,T\right)$ and using Green’s formula, we obtain

$$\begin{aligned} \int _{S }^{T}&\Gamma (t)E^{q}(t)\int _{\Omega }\left[ (uu_{t})_{t}-u_{t}^{2}+| \nabla u|^{2} +a(t)| u_{t}| ^{m(x) -2}u_{t}u\right] dxdt \nonumber \\&=-\int _{S }^{T}\Gamma (t)E^{q}(t) \int _{\Omega }|uv|^{p(x)}dxdt \end{aligned}$$

(4.2)

and

$$\begin{aligned} \int _{S }^{T}&\Gamma (t)E^{q}(t)\int _{\Omega }\left[ (vv_{t})_{t}-v_{t}^{2}+| \Delta v|^{2} +b(t)|v_{t}| ^{r(x) -2}v_{t}v\right] dxdt \nonumber \\&=-\int _{S }^{T}\Gamma (t)E^{q}(t) \int _{\Omega }|uv| ^{p(x)}dxdt. \end{aligned}$$

(4.3)

Adding and subtracting the following two terms

$$\begin{aligned} \left( -\int _{S }^{T} \Gamma (t) E^{q}(t) \int _{\Omega }u_{t}^{2}dxdt\right) \ \text {and} \ \left( -\int _{S }^{T} \Gamma (t)E^{q} (t)\int _{\Omega } v_{t}^{2} dxdt \right) \end{aligned}$$

to (4.2) and (4.3), respectively, and then adding the two resulting equations, we infer

$$\begin{aligned}&\int _{S }^{T} \Gamma E^{q}\int _{\Omega }(u_{t}^{2}+v_{t}^{2} +| \nabla u|^{2} +| \Delta v|^{2})dxdt \nonumber \\&\quad = -\int _{S }^{T} \Gamma E^{q}\int _{\Omega }(uu_{t}+vv_{t})_{t}dxdt + 2 \int _{S }^{T} \Gamma E^{q}\int _{\Omega }(u_{t}^{2}+v_{t}^{2})dxdt \nonumber \\&\quad -\int _{S }^{T}\Gamma E^{q}\int _{\Omega }\left( a | u_{t} |^{m(x) -2}u_{t}u +b | v_{t}| ^{r(x)-2}v_{t}v \right) dxdt \nonumber \\&\quad -2 \int _{S }^{T} \Gamma E^{q} \int _{\Omega } |uv|^{p(x)}dxdt. \end{aligned}$$

(4.4)

Recalling the expression of the energy, (4.4) leads to

$$\begin{aligned} 2\int _{S }^{T} \Gamma E^{q+1}dt=&-\int _{S }^{T} \Gamma E^{q}\int _{\Omega }(uu_{t}+vv_{t})_{t}dxdt + 2 \int _{S }^{T} \Gamma E^{q}\int _{\Omega }(u_{t}^{2}+v_{t}^{2})dxdt \nonumber \\&- \int _{S }^{T}\Gamma E^{q}\int _{\Omega }a \left( |u_{t}|^{m(x) -2}u_{t}u+b |v_{t}| ^{r(x) -2}v_{t}v \right) dxdt \nonumber \\&+\int _{S }^{T} \Gamma E^{q} \int _{\Omega } \left( \frac{2}{p(x)}-2\right) |uv|^{p(x)}dxdt. \end{aligned}$$

Since $p(x)>1,$ for all $x\in \Omega$, then

$$\begin{aligned} 2 \int _{S }^{T} \Gamma E^{q+1}dt \le&-\int _{S }^{T} \Gamma E^{q}\int _{\Omega }(uu_{t}+vv_{t})_{t}dxdt + 2 \int _{S }^{T} \Gamma E^{q}\int _{\Omega }(u_{t}^{2}+v_{t}^{2})dxdt \nonumber \\&- \int _{S }^{T}\Gamma E^{q}\int _{\Omega }\left( a| u_{t}|^{m(x) -2}u_{t}u+b | v_{t}| ^{r(x) -2}v_{t}v\right) dxdt. \end{aligned}$$

(4.5)

On the other hand, we have for $a.e. \ t \in \left[ S,T\right]$

$$\begin{aligned} \frac{d}{dt}\left( \Gamma E^{q}\int _{\Omega }\left( uu_{t}+vv_{t}\right) dx \right) =&\left( \Gamma E^{q} \right) ' \int _{\Omega }\left( uu_{t}+vv_{t} \right) dx + \Gamma E^{q}\int _{\Omega }\left( uu_{t}+vv_{t}\right) _{t}dx; \end{aligned}$$

which gives,

$$\begin{aligned} \Gamma E^{q}\int _{\Omega }\left( uu_{t}+vv_{t}\right) _{t}dx=&\frac{d}{dt}\left( \Gamma E^{q}\int _{\Omega }\left( uu_{t}+vv_{t}\right) dx\right) - \left( \Gamma E^{q} \right) ' \int _{\Omega }\left( uu_{t}+vv_{t} \right) dx. \end{aligned}$$

(4.6)

Substituting (4.6) into (4.5), we arrive at

$$\begin{aligned} 2\int _{S }^{T} \Gamma E^{q+1}dt&\le I_{1}+I_{2}+I_{3}+I_{4}, \end{aligned}$$

(4.7)

where

$$\begin{aligned} I_1=&-\left[ \Gamma E^{q}\int _{\Omega }(uu_{t}+vv_{t})dx \right] _{S}^{T}, \\ I_2=&\int _{S }^{T} (\Gamma ' E^{q}+q\Gamma E^{q-1})\int _{\Omega }(uu_{t}+vv_{t})dxdt, \\ I_3=&2 \int _{S }^{T} \Gamma E^{q}\int _{\Omega }(u_{t}^{2}+v_{t}^{2})dxdt, \\ I_4=&-\int _{S }^{T} \Gamma E^{q}\int _{\Omega }\left( a |u_{t}|^{m(x) -2}u_{t}u+b |v_{t}|^{r(x) -2}v_{t}v \right) dxdt. \end{aligned}$$

In what follows, we estimate $I_{i},$ for $i=1,\ldots ,4$.

First, using Young’s and Poincaré’s inequalities and the definition of E,  we obtain

$$\begin{aligned} \left| \int _{\Omega } \left( uu_{t}+vv_{t}\right) dx \right| \le \frac{c_{e}}{2}\left[ \Vert \nabla u \Vert ^{2}_{2}+\Vert \Delta v\Vert ^{2}_{2}+\Vert u_{t}|^{2}_{2}+\Vert v_{t}\Vert ^{2}_{2}\right] \le CE(t), \end{aligned}$$

(4.8)

where $c_{e}$ is the Poincaré constant. Therefore, recalling (4.1), we infer

$$\begin{aligned} I_{1} =&\Gamma (S)E^{q}(S) \int _{\Omega } \left( u\left( x,S\right) u_{t}\left( x,S\right) +v\left( x,S\right) v_{t}\left( x,S\right) \right) dx \nonumber \\&-\Gamma (T)E^{q}(T) \int _{\Omega } \left( u (x,T)u_{t}(x,T)+v(x,T)v_{t}\left( x,T\right) \right) dx \nonumber \\&\le C \left[ \Gamma (S)E^{q+1}(S)+\Gamma (T)E^{q+1}(T)\right] \le C \Gamma (S)E^{q+1}(S) \le CE(S), \end{aligned}$$

(4.9)

where C is a generic positive constant. Next, using $E'(t)\le 0,$ we get

$$\begin{aligned} I_{2}&\le C \int _{S }^{T} \left( \Gamma ' E^{q}+q\Gamma E^{q-1}E' \right) E(t)dt \nonumber \\&\le C \vert \int _{S }^{T} \Gamma ' E^{q+1} dt \vert + C \vert \int _{S }^{T} q \Gamma E^{q}E' dt \vert \nonumber \\&\le C E^{q+1}(S) \vert \int _{S }^{T} \Gamma 'dt \vert +C q \Gamma (S) \vert \int _{S }^{T} E^{q}E' dt \vert \nonumber \\&\le C E^{q+1}(S)\left[ \Gamma (S)-\Gamma (T) \right] + CE(S) \le CE(S). \end{aligned}$$

(4.10)

For the third term, we set

$$\begin{aligned} I_{3}=J_{1}+J_{2}, \end{aligned}$$

with

$$\begin{aligned} J_{1}= 2\int _{S }^{T} \Gamma E^{q} \int _{\Omega } \vert u_{t} \vert ^{2}dxdt \ \ \text {and} \ \ J_{1}=2 \int _{S }^{T} \Gamma E^{q}\int _{\Omega } \vert v_{t} \vert ^{2}dxdt. \end{aligned}$$

To estimate $J_{1}$, we consider the following partition of $\Omega$

$$\begin{aligned} \ \Omega _{+}=\left\{ x\in \Omega \ /\ \vert u_{t}(x,t) \vert \ge 1\right\} , \ \Omega _{-}=\left\{ x\in \Omega \ /\ | u_{t}(x,t) | <1\right\} . \end{aligned}$$

Therefore, by Hölder’s inequality and the definition of $\alpha ^{+}$, we obtain

$$\begin{aligned} J_{1}&=2\int _{S }^{T} \Gamma E^{q} \left[ \int _{\Omega _{-} } \vert u_{t} \vert ^{2}dx+ \int _{\Omega _{+} } \vert u_{t} \vert ^{2}dx\right] dt \nonumber \\&\le C \int _{S }^{T}\Gamma E^{q} \left( \int _{\Omega _{-} } \vert u_{t}\vert ^{ \alpha ^{+}}dx\right) ^{\frac{2}{\alpha ^{+}}} dt + C \int _{S }^{T} \Gamma E^{q} \int _{\Omega _{+} } \vert u_{t} \vert ^{m(x)}dxdt\\&\le C \int _{S }^{T}\Gamma E^{q} \left( \int _{\Omega _{-}} \vert u_{t} \vert ^{m(x)}dx\right) ^{\frac{2}{\alpha ^{+}}}dt+ C \int _{S }^{T} E^{q}\left( \Gamma \int _{\Omega _{+}} \vert u_{t} \vert ^{m(x)}dx\right) dt. \end{aligned}$$

This yields

$$\begin{aligned} J_{1}&\le C \int _{S }^{T} \Gamma ^{\frac{\alpha ^{+}-2}{ \alpha ^{+}}} E^{q} \left( \Gamma \int _{\Omega } \vert u_{t} \vert ^{m (x)}dx\right) ^{\frac{2}{\alpha ^{+}}} + C \int _{S }^{T} E^{q}\left( \Gamma \int _{\Omega } \vert u_{t}\vert ^{m(x)}dx\right) dt \\&\le C \int _{S }^{T}\Gamma ^{\frac{\alpha ^{+}-2}{ \alpha ^{+}}} E^{q}\left( a \int _{\Omega } \vert u_{t} \vert ^{m (x) }dx\right) ^{\frac{2}{\alpha ^{+}}}+ C \int _{S }^{T} E^{q}\left( a \int _{\Omega } \vert u_{t} \vert ^{m(x)}dx\right) dt \\&\le C \int _{S }^{T}\Gamma ^{\frac{\alpha ^{+}-2}{ \alpha ^{+}}} E^{q}\left( -E'\right) ^{\frac{2}{\alpha ^{+}}} dt + C \int _{S }^{T}E^{q} ( -E')dt \\&\le C \int _{S }^{T}\Gamma ^{\frac{\alpha ^{+}-2}{ \alpha ^{+}}} E^{q}\left( -E'\right) ^{\frac{2}{\alpha ^{+}}} dt+CE(S), \end{aligned}$$

using (4.1) and the definition of $\Gamma .$ Similarly, we find

$$\begin{aligned} J_{2}\le C \int _{S }^{T}\Gamma ^{\frac{\alpha ^{+}-2}{ \alpha ^{+}}} E^{q}\left( -E'\right) ^{\frac{2}{\alpha ^{+}}} dt+CE(S). \end{aligned}$$

Adding $J_{1}$ and $J_{2}$, it results

$$\begin{aligned} I_{3} \le C \int _{S }^{T}\Gamma ^{\frac{\alpha ^{+}-2}{ \alpha ^{+}}} E^{q}\left( -E'\right) ^{\frac{2}{\alpha ^{+}}} dt+CE(S). \end{aligned}$$

Two cases are possible:

Case 1: if $\alpha ^{+}=2$ then,

$$\begin{aligned} I_{3}&\le C\int _{S }^{T} E^{q}\left( -E^{'}\right) dt+ CE(S) \\&\le C\left[ E^{q+1}(S)-E^{q+1}(T) \right] +CE(S)\le CE(S). \end{aligned}$$

Case 2: if $\alpha ^{+}>2$, we exploit Young’s inequality, with $\delta = q+1 \ \text {and } \delta '= \left( q+1\right) /q$, to get, for all $\varepsilon >0,$

$$\begin{aligned} I_{3} \le \varepsilon C \int _{S }^{T} \Gamma ^{\frac{( \alpha ^{+}-2)(q+1)}{q \alpha ^{+}} } E^{q+1}dt+C_{\varepsilon } \int _{S }^{T} \left( -E'\right) ^{\frac{2(q+1)}{\alpha ^{+}}}dt+CE(S). \end{aligned}$$

If we take $\varepsilon =\frac{1}{2C}$ and $q=\frac{\alpha ^{+}}{2}-1$, then

$$\begin{aligned} I_{3}&\le \frac{1}{2} \int _{S }^{T} \Gamma E^{q+1}dt+ C_{\varepsilon } \int _{S }^{T} \left( -E'\right) dt+ CE(S) \nonumber \\&\le \frac{1}{2} \int _{S }^{T} \Gamma E^{q+1}dt+ CE(S). \end{aligned}$$

Therefore, for $\alpha ^{+} \ge 2,$

$$\begin{aligned} I_{3} \le \frac{1}{2} \int _{S }^{T} \Gamma E^{q+1}dt+ CE(S). \end{aligned}$$

(4.11)

Finally, we handle $I_{4}$ as follows. Since a and b are bounded functions on ${\mathbb {R}}_{+},$ then

$$\begin{aligned} I_{4} \le J_{3}+J_{4}, \end{aligned}$$

where

$$\begin{aligned} J_{3}= & {} C \int _{S }^{T} \Gamma E^{q}\int _{\Omega } \vert u \vert \vert u_{t} \vert ^{m(x)-1}dxdt \ \ \text {and} \ \ J_{4}=C \int _{S }^{T} \Gamma E^{q} \int _{\Omega } \vert v \vert \vert v_{t} \vert ^{r(x) -1}dxdt. \end{aligned}$$

Now, as in [18], applying Young’s inequality with

$$\begin{aligned} \delta (x)=\frac{m(x) }{m\left( x\right) -1} \ \text {and} \ \delta '(x) =m(x), \end{aligned}$$

we obtain, for all $\varepsilon >0,$

$$\begin{aligned} J_{3}&\le \int _{S }^{T} \Gamma E^{q} \left[ \varepsilon \int _{\Omega } \vert u \vert ^{m(x)}dx+ \int _{\Omega } C_{\varepsilon }(x) \vert u_{t} \vert ^{m(x)}dx\right] dt, \end{aligned}$$

where

$$\begin{aligned} C_{\varepsilon }(x)= \frac{\left[ m(x)-1\right] ^{m(x)-1}}{\left[ m(x)\right] ^{m(x)} \varepsilon ^{m(x)-1} }. \end{aligned}$$

Similarly,

$$\begin{aligned} J _{4}\le \int _{S }^{T} \Gamma E^{q}\left[ \varepsilon \int _{\Omega } \vert v \vert ^{r(x)}dx+ \int _{\Omega }C'_{\varepsilon }(x) \vert v_{t} \vert ^{r(x)}dx\right] dt, \end{aligned}$$

where

$$\begin{aligned} C'_{\varepsilon }(x)= \frac{\left[ r(x)-1\right] ^{r(x)-1}}{\left[ r(x) \right] ^{r(x)} \varepsilon ^{r(x)-1} }. \end{aligned}$$

By addition, we find

$$\begin{aligned} I_{4}&\le \int _{S }^{T} \Gamma E^{q}\int _{\Omega } \left( \varepsilon |u| ^{m(x)}+\varepsilon | v| ^{r(x)} + C_{\varepsilon }(x) | u_{t}| ^{m(x)}+C'_{\varepsilon }(x) |v_{t}| ^{r(x)} \right) dxdt. \end{aligned}$$

(4.12)

Under the hypothesis (H.1) and (H.2), and recalling that $m^{-},r^{-}\ge 2,$ we have the following estimate

$$\begin{aligned} J_{5}&= \varepsilon \int _{S }^{T}\Gamma E^{q}\int _{\Omega } ( |u | ^{m(x)}+| v| ^{r(x)}) dxdt\\&\le \varepsilon C\int _{S }^{T}\Gamma E^{q}\int _{\Omega } \left( | u | ^{m_{-}}+ | u | ^{m_{+}}+| v | ^{r_{-}}+ | v | ^{r_{+}}\right) dxdt \nonumber \\&\le \varepsilon C\int _{S }^{T}\Gamma E^{q} \left( \Vert \nabla u \Vert ^{m_{-}}_{2}+ \Vert \nabla u \Vert ^{m_{+}}_{2}+ \Vert \Delta v \Vert ^{r_{-}}_{2}+ \Vert \Delta v \Vert ^{r_{+}}_{2} \right) dt \nonumber \\&\le \varepsilon C\int _{S }^{T}\Gamma E^{q+1}\left( E ^{\frac{m_{-}}{2}-1}+ E ^{\frac{m_{+}}{2}-1} + E ^{\frac{r_{-}}{2}-1}+ E ^{\frac{r_{+}}{2}-1}\right) dt \nonumber \\&\le \varepsilon C \left( E(0) ^{\frac{m_{-}}{2}-1}+E(0) ^{\frac{m_{+}}{2}-1}+E(0) ^{\frac{r_{-}}{2}-1}+E(0) ^{\frac{r_{+}}{2}-1}\right) \int _{S }^{T}\Gamma E^{q+1}dt. \end{aligned}$$

Taking

$$\begin{aligned} \varepsilon = \frac{1}{2C}\left( E(0) ^{\frac{m_{-}}{2}-1}+E(0) ^{\frac{m_{+}}{2}-1}+E(0) ^{\frac{r_{-}}{2}-1}+E(0) ^{\frac{r_{+}}{2}-1}\right) ^{-1}, \end{aligned}$$

it results

$$\begin{aligned} J_{5}&\le \frac{1}{2} \int _{S }^{T}\Gamma E^{q+1}dt. \end{aligned}$$

Moreover, $C_{\varepsilon }(.)$ and $C'_{\varepsilon }(.)$, in (4.12), will be bounded since m(.) and r(.) are bounded. Consequently, inequality (4.12) leads to

$$\begin{aligned} I_{4}&\le \frac{1}{2}\int _{S }^{T}\Gamma E^{q+1}dt+C\int _{S }^{T}\Gamma E^{q}\left( |u_{t}|^{m(x)}+ |v_{t}|^{r(x)}\right) dxdt \nonumber \\&\le \frac{1}{2}\int _{S }^{T}\Gamma E^{q+1}dt+C\int _{S }^{T}E^{q} \left( a |u_{t}|^{m(x)}+ b |v_{t}|^{r(x)}\right) dxdt \nonumber \\&\le \frac{1}{2} \int _{S }^{T}\Gamma E^{q+1}dt+ C \int _{S }^{T} E^{q} (-E'(t))dt \nonumber \\&\le \frac{1}{2} \int _{S }^{T}\Gamma E^{q+1}dt +CE(S). \end{aligned}$$

(4.13)

Finally, by inserting (4.9), (4.10), (4.11) and (4.13) into (4.7), we find

$$\begin{aligned} \int _{S }^{T} \Gamma E^{q+1}(t)dt \le CE(S). \end{aligned}$$

Taking $T\longrightarrow \infty ,$ it yields

$$\begin{aligned} \int _{S }^{\infty } \Gamma E^{q+1}(t)dt\le CE(S). \end{aligned}$$

Invoking Lemma 2.5 with $\sigma (t)= \int _{0 }^{t}\Gamma (s)ds$, we obtain the desired result. $\square$

Remark 4.2

As a special case, when a and b are constants, we have the following corollary.

Corollary 4.3

Assume that assumptions (H.1)–(H.3) hold. Then, there exist two constants $c, \omega >0$ such that the solution of (P) satisfies, for all $t\ge 0$,

$$\begin{aligned} E\left( t\right) \le \left\{ \begin{array}{ll} c \left( 1+t \right) ^{\frac{2}{2- \alpha ^{+}}},&{} \ if \ \alpha ^{+}>2, \\ ce^{-\omega t }, &{} \ if \ \alpha ^{+} =2. \end{array}\right. \end{aligned}$$

5 Examples

Based on Theorem 4.1, we present the following examples to illustrate different types of the energy decay, depending on the value of $\alpha ^{+}$ and on the damping coefficients a(t) and b(t).

Example 5.1

For $a(t)=\frac{1}{c_1+t}, b(t)=\frac{1}{c_2+t}$, with $0< c_1 \le c_2,$ the estimate in Theorem 4.1 leads to

$$\begin{aligned} E\left( t\right) \le E_1\left( t\right) = \left\{ \begin{array}{ll} \frac{c }{\left( c_1+t \right) ^{\omega }} \ , &{} \ if \ \alpha ^{+} =2,\\ \frac{c}{ \left[ 1+\ln \left( 1+\frac{t}{c_1}\right) \right] ^{\frac{2}{ \alpha ^{+}-2} }} \ , &{} \ if \ \alpha ^{+}>2. \end{array}\right. \end{aligned}$$

Example 5.2

If $a(t)=\frac{1}{\sqrt{5}(1+t)}, b(t)=\frac{1}{2\sqrt{1+t}},$ then the solution energy of (P) decreases as

$$\begin{aligned} E_2\left( t\right) = \left\{ \begin{array}{ll} c_3 e^{-\omega \sqrt{1+t} } \ , &{} \ if \ \alpha ^{+} =2, \\ c / \left( 1+ t \right) ^{ \frac{1}{\alpha ^{+}-2}} \ ,&{} \ if \ \alpha ^{+}>2, \end{array}\right. \end{aligned}$$

for $c_3=c e^ \omega >0.$

Example 5.3

For $a(t)=\frac{1}{(\sqrt{3}+t)\ln (\sqrt{3}+t)}, b(t)=\frac{1}{(\sqrt{3}+t)^{2} ( \ln (\sqrt{3}+t))^{2}}$, the upper-bound function of the energy is

$$\begin{aligned} E_3\left( t\right) =\left\{ \begin{array}{ll} \frac{c_4 }{ \left( \ln (\sqrt{3}+ t) \right) ^{\omega } } \ , &{} \ if \ \alpha ^{+} =2, \\ c / \left[ 1+ \ln \left( \frac{\ln ( \sqrt{3}+ t)}{ \ln \sqrt{3} } \right) \right] ^{\frac{2}{\alpha ^{+}-2}} \ ,&{} \ if \ \alpha ^{+}>2, \end{array}\right. \end{aligned}$$

where $c_4= c \left( \frac{ \ln 3}{2} \right) ^{\omega }>0.$

6 Conclusion

In this work, we considered a coupled system of two weakly damped wave and plate equations, with Laplacian and bi-Laplacian operators and with variable exponent in the damping and coupling terms. We proved a theorem of global well-posedness and established different decay rates of the energy, depending on the variable exponents m(x) and r(x), and on the function coefficients a(t) and b(t). To illustrate our theoretical findings, we also gave some examples. This work generalizes many other works in the literature, in particular, those concerning the special case, where a and b are constants.