On the existence and stability of a nonlinear wave system with variable exponents

Abstract

Problems with variable exponents have attracted a great deal of attention lately and various existence, nonexistence and stability results have been established. The importance of such problems has manifested due to the recent advancement of science and technology and to the wide application in areas such as electrorheological fluids (smart fluids) which have the property that the viscosity changes drastically when exposed to heat or electrical fields. To tackle and understand these models, new sophisticated mathematical functional spaces have been introduced, such as the Lebesgue and Sobolev spaces with variable exponents. In this work, we are concerned with a system of wave equations with variable-exponent nonlinearities. This system can be regarded as a model for interaction between two fields describing the motion of two “smart” materials. We, first, establish the existence of global solutions then show that solutions of enough regularities stabilize to the rest state $(0, 0)$ either exponentially or polynomially depending on the range of the variable exponents. We also present some numerical tests to illustrate our theoretical findings.

Keywords

Variable-exponent nonlinearity system wave equations existence decay

1. Introduction

Let Ω be a bounded and regular domain of $R^{n}$ . Our main interest lies in the following system of wave equations: $\begin{matrix} (P) & \{\begin{matrix} u_{t t} - Δ u + | u_{t} |^{m (\cdot) - 2} u_{t} + f_{1} (u, v) = 0, & in Ω, t > 0, \\ v_{t t} - Δ v + | v_{t} |^{r (\cdot) - 2} v_{t} + f_{2} (u, v) = 0, & in Ω, t > 0, \\ u (\cdot, t) = v (\cdot, t) = 0, & on \partial Ω, t ⩾ 0, \\ (u (0), v (0)) = (u_{0}, v_{0}), (u_{t} (0), v_{t} (0)) = (u_{1}, v_{1}), & in Ω, \end{matrix} \end{matrix}$ where $\begin{matrix} (1.1) & \begin{matrix} f_{1} (u, v) = [a | u + v |^{2 (p (\cdot) + 1)} (u + v) + b | u |^{p (\cdot)} u | v |^{p (\cdot) + 2}], \\ f_{2} (u, v) = [a | u + v |^{2 (p (\cdot) + 1)} (u + v) + b | u |^{p (\cdot) + 2} | v |^{p (\cdot)} v], \end{matrix} \end{matrix}$ with $a, b > 0$ are constants and m, p and r are given continuous functions on $\overline{Ω}$ satisfying the log-Hölder continuity condition: $\begin{matrix} (1.2) & | q (x) - q (y) | ⩽ - \frac{A}{log | x - y |}, for all x, y \in Ω, | x - y | < δ, \end{matrix}$ where $A > 0$ and $0 < δ < 1$ . For m and r, we assume $\begin{array}{l} (1.3) & 2 ⩽ m_{1} = \underset{x \in Ω}{ess \inf} m (x) ⩽ m (x) ⩽ m_{2} = \underset{x \in Ω}{ess \sup} m (x) ⩽ \frac{2 n}{n - 2}, n > 3 . \\ (1.4) & 2 ⩽ r_{1} = \underset{x \in Ω}{ess \inf} r (x) ⩽ r (x) ⩽ r_{2} = \underset{x \in Ω}{ess \sup} r (x) ⩽ \frac{2 n}{n - 2}, n > 3 \end{array}$ and for the exponent p, we require that $\begin{matrix} (1.5) & - 1 < p_{1} = \underset{x \in Ω}{ess \inf} p (x) ⩽ p (x) ⩽ p_{2} = \underset{x \in Ω}{ess \sup} p (x) ⩽ \{\begin{matrix} \frac{3 - n}{n - 2}, & if n ⩾ 3 \\ + \infty, & if n = 1, 2 . \end{matrix} \end{matrix}$

To motivate our work, we recall some results regarding wave equations in the case when m, p, r are constants or variables. In fact, the stabilization of linear and nonlinear wave equations by means of internal or boundary feedbacks has attracted a considerable attention and a great effort has been devoted to establish decay rates. See, in this regard, early works by Zuazua [35], Komornik [16], Nakao [29–32], Martinez [22], Han [14], Agre [1] and Benaissa and Messaoudi [6]. Regarding decay results for arbitrary growth of the damping term, we mention the pioneer work of Lasiecka and Tataru [18,19], and a relatively recent work by Mustafa and Messaoudi [28].

With the advancement of sciences and technology, many physical and engineering models required more sophisticated mathematical functional spaces to be studied and well understood. For example, in fluid dynamics, the electrorheological fluids (smart fluids) have the property that the viscosity changes (often drastically) when exposed to an electrical field. The Lebesgue and Sobolev spaces with variable exponents proved to be efficient tools to study such problems as well as other models like fluids with temperature-dependent viscosity, nonlinear viscoelasticity, filtration processes through a porous media and image processing [7]. More details on these problems can be found in [5,8].

For hyperbolic problems involving variable-exponent nonlinearities, most of the work is devoted to the local existence and blow up of solutions. See, for instance, [2–4,13,25–27]. For the stability of solutions of wave equations with variable-exponent nonlinearity, there are not many works. Let us mention the work of Messaoudi et al. [23], where the following equation $\begin{matrix} u_{t t} - div (| \nabla u |^{r (\cdot) - 2} \nabla u) + | u_{t} |^{m (\cdot) - 2} u_{t} = 0, \end{matrix}$ was studied in a bounded domain Ω with a smooth boundary $\partial Ω$ and for exponents $m (\cdot) ⩾ r (\cdot) ⩾ 2$ . The authors in their work showed that the solution energy decays exponentially if $m \equiv 2$ and polynomially at the rate of $t^{2 / (m_{2} - 2)}$ , if $m_{2} = {esssup}_{x \in Ω} m (x) > 2$ . Also, Ghegal et al. [12] considered, in a bounded domain, the equation $\begin{matrix} u_{t t} - Δ u + | u_{t} |^{m (\cdot) - 2} u_{t} = | u |^{p (\cdot) - 2} u, \end{matrix}$ together with initial and Dirichlet-boundary condition, and proved under appropriate conditions on $m (\cdot)$ , $p (\cdot)$ , and the initial data, global existence and a stability result similar to that of [23]. Very recently, Li et al. [20] discussed the following equation $\begin{matrix} u_{t t} - div (| \nabla u |^{p (x) - 2} \nabla u) + | u_{t} |^{m (x) - 2} u_{t} = | u |^{q (x) - 2} u, \end{matrix}$ and established some uniform estimates of decay rates of the solution. They also showed that $u \equiv 0$ is asymptotic stable in terms of natural energy associated with the solution of the above equation. In addition, they gave some numerical examples to illustrate their results. We refer the reader to the recent review paper [24] for more results concerning stability and blow up in wave problems.

In this paper, we consider System (P) and prove the existence of solutions, using the well-known Galerkin method and certain compactness property due to Lions and Aubin [21]. We then establish the exponential and the polynomial decay of the solution under suitable assumptions on the variable exponents m, r, and p. To the best of our knowledge, such a problem has not been discussed before in the context of nonlinearity with variable exponents. In addition to the introduction, this paper has four other sections. In Section 2, we present some preliminaries, The well-posedness is given in details in Section 3. Our stability result and its proof are given in Section 4. Finally, we present some numerical tests to illustrate our theoretical results in Section 5.

2. Preliminaries

In this section, we present some basic knowledge about the Lebesgue and Sobolev spaces with variable exponents (see [9–11,15]). Let $q : Ω \to [1, \infty]$ be a measurable function, where Ω is a domain of $R^{n}$ . We define the Lebesgue space with a variable exponent $q (\cdot)$ by $\begin{matrix} L^{q (\cdot)} (Ω) : = {v : Ω \to R; measurable in Ω : ϱ_{q (\cdot)} (λ v) < \infty, for some λ > 0}, \end{matrix}$ where $\begin{matrix} ϱ_{q (\cdot)} (v) = \int_{Ω} \frac{1}{q (x)} {| v (x) |}^{q (x)} d x \end{matrix}$ is a modular. Equipped with the following Luxembourg-type norm $\begin{matrix} ‖ v ‖_{q (\cdot)} : = inf {λ > 0 : \int_{Ω} {| \frac{v (x)}{λ} |}^{q (x)} d x ⩽ 1}, \end{matrix}$ $L^{q (\cdot)} (Ω)$ is a Banach space (see [8]).

We, also, define the variable-exponent Sobolev space $W^{1, q (\cdot)} (Ω)$ as follows: $\begin{matrix} W^{1, q (\cdot)} (Ω) = {v \in L^{q (\cdot)} (Ω) such that \nabla v exists and | \nabla v | \in L^{q (\cdot)} (Ω)} . \end{matrix}$ This is a Banach space with respect to the norm $‖ v ‖_{W^{1, q (\cdot)} (Ω)} = ‖ v ‖_{q (\cdot)} + ‖ \nabla v ‖_{q (\cdot)}$ . In addition, we set $W_{0}^{1, q (\cdot)} (Ω)$ to be the closure of $C_{0}^{\infty} (Ω)$ in $W^{1, q (\cdot)} (Ω)$ . Here we note that the space $W_{0}^{1, q (\cdot)} (Ω)$ is usually defined differently for the variable-exponent case. However, both definitions are equivalent under (1.2) (see [8]). The dual of $W_{0}^{1, q (\cdot)} (Ω)$ is defined, similarly to the classical Sobolev spaces, by $W^{- 1, q^{'} (\cdot)} (Ω)$ , where $\frac{1}{q (\cdot)} + \frac{1}{q^{'} (\cdot)} = 1$ .

Let $\begin{matrix} q_{1} : = \underset{x \in Ω}{ess \inf} q (x), q_{2} : = \underset{x \in Ω}{ess \sup} q (x) . \end{matrix}$

Lemma 2.1 ([8]).

If $q : Ω \to [1, \infty)$ is a measurable function with $q_{2} < \infty$ , then $C_{0}^{\infty} (Ω)$ is dense in $L^{q (\cdot)} (Ω)$ .

Lemma 2.2 ([8]).

If $1 < q_{1} ⩽ q (x) ⩽ q_{2} < \infty$ holds, then $\begin{matrix} min {‖ w ‖_{q (\cdot)}^{q_{1}}, ‖ w ‖_{q (\cdot)}^{q_{2}}} ⩽ ϱ_{q (\cdot)} (w) ⩽ max {‖ w ‖_{q (\cdot)}^{q_{1}}, ‖ w ‖_{q (\cdot)}^{q_{2}}}, \end{matrix}$ for any $w \in L^{q (\cdot)} (Ω)$ .

Lemma 2.3 (Hölder’s Inequality [8]).

Let $p, q, s ⩾ 1$ be measurable functions defined on Ω such that $\begin{matrix} \frac{1}{s (y)} = \frac{1}{p (y)} + \frac{1}{q (y)}, for a.e. y \in Ω . \end{matrix}$ If $f \in L^{p (\cdot)} (Ω)$ and $g \in L^{q (\cdot)} (Ω)$ , then $f g \in L^{s (\cdot)} (Ω)$ and $\begin{matrix} ‖ f g ‖_{s (\cdot)} ⩽ 2 ‖ f ‖_{p (\cdot)} ‖ g ‖_{q (\cdot)} . \end{matrix}$

Lemma 2.4.
If $1 < p_{1} ⩽ p (x) ⩽ p_{2} < \infty$ holds, then $\begin{matrix} (2.1) & \int_{Ω} | v |^{p (\cdot)} d x ⩽ ‖ v ‖_{p_{1}}^{p_{1}} + ‖ v ‖_{p_{2}}^{p_{2}}, \end{matrix}$ for any $v \in L^{p (\cdot)} (Ω)$ .
Lemma 2.5 (Poincaré’s Inequality [8]).

Let Ω be a bounded domain of $R^{n}$ and $q (\cdot)$ satisfies ( 1.2 ), then $\begin{matrix} ‖ v ‖_{q (\cdot)} ⩽ C ‖ \nabla v ‖_{q (\cdot)}, for all v \in W_{0}^{1, q (\cdot)} (Ω), \end{matrix}$ where the positive constant C depends on $q_{1}$ , $q_{2}$ and Ω only. In particular, the space $W_{0}^{1, p (\cdot)} (Ω)$ has an equivalent norm given by $‖ v ‖_{W_{0}^{1, q (\cdot)} (Ω)} = ‖ \nabla v ‖_{q (\cdot)}$ .

Lemma 2.6 ([8]).

Let Ω be a bounded domain in $R^{n}$ with a smooth boundary $\partial Ω$ . Assume that $q \in C (\overline{Ω})$ such that $\begin{matrix} (2.2) & 2 ⩽ q_{1} ⩽ q (x) ⩽ q_{2} < \frac{2 n}{n - 2}, n ⩾ 3 . \end{matrix}$

Then the embedding $H_{0}^{1} (Ω) ↪ L^{p (\cdot)} (Ω)$ is continuous and compact.

By recalling the definition of $f_{1} (u, v)$ and $f_{2} (u, v)$ in (1.1), one can easily verify that $\begin{matrix} (2.3) & u f_{1} (u, v) + v f_{2} (u, v) = 2 (p (x) + 2) F (u, v), \forall (u, v) \in R^{2}, \end{matrix}$ where $\begin{matrix} F (u, v) = \frac{1}{2 (p (x) + 2)} [a | u + v |^{2 (p (x) + 2)} + 2 b | u v |^{p (x) + 2}] . \end{matrix}$

3. Existence

In this section, we state and prove an existence result of problem (P).

Lemma 3.1.
There exists a positive constant c such that $\begin{matrix} (3.1) & \begin{matrix} \int_{Ω} F (u, v) d x ⩽ c (‖ \nabla u ‖_{2}^{2 (p_{1} + 2)} + ‖ \nabla v ‖_{2}^{2 (p_{1} + 2)} + ‖ \nabla u ‖_{2}^{2 (p_{2} + 2)} + ‖ \nabla v ‖_{2}^{2 (p_{2} + 2)}) . \end{matrix} \end{matrix}$
Proof.
Using Lemma 2.1. in [34], we have $\begin{matrix} (3.2) & F (u, v) ⩽ \frac{c}{2 (p (x) + 2)} (| u |^{2 (p (x) + 2)} + | v |^{2 (p (x) + 2)}) . \end{matrix}$

Using the boundedness of $p (x)$ , Lemma 2.4 and the embedding theorem, we have $\begin{array}{rcl} \int_{Ω} F (u, v) d x & ⩽ & c (\int_{Ω} | u |^{2 (p_{2} + 2)} d x + \int_{Ω} | u |^{2 (p_{1} + 2)} d x + \int_{Ω} | v |^{2 (p_{2} + 2)} d x + \int_{Ω} | v |^{2 (p_{1} + 2)} d x) \\ (3.3) & ⩽ & c (‖ \nabla u ‖_{2}^{2 (p_{1} + 2)} + ‖ \nabla v ‖_{2}^{2 (p_{1} + 2)} + ‖ \nabla u ‖_{2}^{2 (p_{2} + 2)} + ‖ \nabla v ‖_{2}^{2 (p_{2} + 2)}) . \end{array}$ □
Definition 3.1.
Let $(u_{0}, u_{1}), (v_{0}, v_{1}) \in H_{0}^{1} (Ω) \times L^{2} (Ω)$ . Any pair of functions $\begin{array}{l} (u, v) \in L^{\infty} ([0, T], H_{0}^{1} (Ω)), (u_{t}, v_{t}) \in L^{\infty} ([0, T], L^{2} (Ω)), \\ u_{t} \in L^{m (\cdot)} (Ω \times (0, \infty)), v_{t} \in L^{r (\cdot)} (Ω \times (0, \infty)) \end{array}$ is called a weak solution of (P) if $\begin{matrix} (3.4) & \begin{matrix} \{\begin{matrix} \frac{d}{d t} \int_{Ω} u_{t} (x, t) ϕ (x) d x + \int_{Ω} \nabla u (x, t) . \nabla ϕ (x) d x \\ + \int_{Ω} {| u_{t} |}^{m (\cdot) - 2} u_{t} ϕ (x) d x + \int_{Ω} ϕ (x) f_{1} (u, v) d x = 0 \\ \frac{d}{d t} \int_{Ω} v_{t} (x, t) ψ (x) d x + \int_{Ω} \nabla v (x, t) . \nabla ψ (x) d x \\ + \int_{Ω} {| v_{t} |}^{r (\cdot) - 2} v_{t} ψ (x) d x + \int_{Ω} ψ (x) f_{2} (u, v) d x = 0 \\ u (0) = u_{0}, u_{t} (0) = u_{1}, v (0) = v_{0}, v_{t} (0) = v_{1}, \end{matrix} \end{matrix} \end{matrix}$ for a.e. $t \in [0, T]$ and all test functions $ϕ, ψ \in H_{0}^{1} (Ω)$ .
Theorem 3.2.
Let $(u_{0}, u_{1}), (v_{0}, v_{1}) \in H_{0}^{1} (Ω) \times L^{2} (Ω)$ be given and assume that conditions ( 1.2 )–( 1.5 ) hold. Then problem ( P ) has a unique weak solution such that $\begin{array}{l} (u, v) \in L^{\infty} ([0, T), H_{0}^{1} (Ω)), u_{t} \in L^{\infty} ([0, T), L^{2} (Ω)) \cap L^{m (\cdot)} (Ω \times (0, T)), \\ v_{t} \in L^{\infty} ([0, T), L^{2} (Ω)) \cap L^{r (\cdot)} (Ω \times (0, T)), \end{array}$ for any $T > 0$ .
Proof.
We use the standard Faedo-Galerkin method to prove our result. Let ${w_{j}}_{j = 1}^{\infty}$ be the eigenfunctions of the Laplacian operator subject to Dirichlet boundary conditions. Then ${w_{j}}_{j = 1}^{\infty}$ is orthogonal basis of $H_{0}^{1} (Ω)$ as well as of $L^{2} (Ω)$ . Let $V_{k} = s p a n {w_{1}, w_{2}, \dots, w_{k}}$ and the projections of initial data on the finite-dimensional subspace $V_{k}$ are given by $\begin{matrix} u_{0}^{k} = \sum_{j = 1}^{k} a_{j} w_{j}, v_{0}^{k} = \sum_{j = 1}^{k} b_{j} w_{j}, u_{1}^{k} = \sum_{j = 1}^{k} c_{j} w_{j}, v_{1}^{k} = \sum_{j = 1}^{k} d_{j} w_{j}, \end{matrix}$ where $\begin{matrix} (3.5) & \{\begin{matrix} u_{0}^{k} \to u_{0} and v_{0}^{k} \to v_{0} in H_{0}^{1} (Ω) \\ and \\ u_{1}^{k} \to u_{1} and v_{1}^{k} \to v_{1} in L^{2} (Ω) . \end{matrix} \end{matrix}$ We search for solutions of the form $\begin{matrix} u^{k} (x) = \sum_{j = 1}^{k} h^{j, k} (t) w_{j} (x) and v^{k} (x) = \sum_{j = 1}^{k} g^{j, k} (t) w_{j} (x) \end{matrix}$ for the approximate system in $V_{k}$ $\begin{matrix} (3.6) & \{\begin{matrix} {⟨ u_{t t}^{k}, w_{j} ⟩}_{L^{2} (Ω)} + {⟨ \nabla u^{k}, \nabla w_{j} ⟩}_{L^{2} (Ω)} + {⟨ {| u_{t}^{k} |}^{m (x) - 2} u_{t}^{k}, w_{j} ⟩}_{L^{2} (Ω)} \\ + {⟨ f_{1} (u^{k}, v^{k}), w_{j} ⟩}_{L^{2} (Ω)} = 0, j = 1, 2, \dots, k, \\ {⟨ v_{t t}^{k}, w_{j} ⟩}_{L^{2} (Ω)} + {⟨ \nabla v^{k}, \nabla w_{j} ⟩}_{L^{2} (Ω)} + {⟨ {| v_{t}^{k} |}^{m (x) - 2} v_{t}^{k}, w_{j} ⟩}_{L^{2} (Ω)} \\ + {⟨ f_{2} (u^{k}, v^{k}), w_{j} ⟩}_{L^{2} (Ω)} = 0, j = 1, 2, \dots, k, \\ u^{k} (0) = u_{0}^{k}, u_{t}^{k} (0) = u_{1}^{k}, v^{k} (0) = v_{0}^{k}, v_{t}^{k} (0) = v_{1}^{k}, \end{matrix} \end{matrix}$ This leads to a system of ODE’s for unknown functions $h^{j, k}$ and $g^{j, k}$ . Based on standard existence theory for ODE, the system (3.6) admits a solution $(u^{k}, v^{k})$ on a maximal time interval $[0, t_{k})$ , $0 < t_{k} < T$ , for each $k \in N$ . In fact $t_{k} = T$ and to show this, we multiply the first equation by ${(h^{j, k})}^{'}$ and the second equation by ${(g^{j, k})}^{'}$ in (3.6), sum over $j = 1, 2, \dots k$ and add the two equations to obtain $\begin{matrix} (3.7) & \begin{matrix} [b] & \frac{1}{2} \frac{d}{d t} ({‖ u_{t}^{k} ‖}_{2}^{2} + {‖ v_{t}^{k} ‖}_{2}^{2} + {‖ \nabla u^{k} ‖}_{2}^{2} + {‖ \nabla v^{k} ‖}_{2}^{2}) \\ = - \int_{Ω} {| u_{t}^{k} (s) |}^{m (\cdot) + 1} d x - \int_{Ω} {| v_{t}^{k} (s) |}^{r (\cdot) + 1} d x - \int_{Ω} (u_{t}^{k} f_{1} (u^{k}, v^{k}) + v_{t}^{k} f_{2} (u^{k}, v^{k})) d x . \end{matrix} \end{matrix}$ Recalling that $f_{1} = \frac{\partial F}{\partial u}$ , $f_{2} = \frac{\partial F}{\partial v}$ , we have $\begin{matrix} (3.8) & \begin{matrix} \frac{d}{d t} E^{k} (t) = - \int_{Ω} {| u_{t}^{k} (t) |}^{m (\cdot) + 1} d x - \int_{Ω} {| v_{t}^{k} (t) |}^{r (\cdot) + 1} d x ⩽ 0, \end{matrix} \end{matrix}$ where $\begin{matrix} (3.9) & E^{k} (t) : = \frac{1}{2} [{‖ u_{t}^{k} ‖}_{2}^{2} + {‖ v_{t}^{k} ‖}_{2}^{2}] + \frac{1}{2} [{‖ \nabla u^{k} ‖}_{2}^{2} + {‖ \nabla v^{k} ‖}_{2}^{2}] + \int_{Ω} F (u^{k}, v^{k}) d x . \end{matrix}$ Using (3.1), (3.5) and (3.8), we have, for some positive constant C independent of t and k, $\begin{matrix} (3.10) & E^{k} (t) ⩽ E^{k} (0) ⩽ C, 0 ⩽ t ⩽ T . \end{matrix}$ Integrate (3.7) over $(0, t)$ to obtain $\begin{matrix} (3.11) & \begin{matrix} \frac{1}{2} ({‖ u_{t}^{k} ‖}_{2}^{2} + {‖ \nabla u^{k} ‖}_{2}^{2} + {‖ v_{t}^{k} ‖}_{2}^{2} + {‖ v^{k} ‖}_{2}^{2}) \\ + \int_{0}^{t} \int_{Ω} {| u_{t}^{k} (s) |}^{m (\cdot) + 1} d x d s + \int_{0}^{t} \int_{Ω} {| v_{t}^{k} (s) |}^{r (\cdot) + 1} d x d s \\ = \frac{1}{2} ({‖ \nabla u_{0}^{k} ‖}_{2}^{2} + {‖ u_{1}^{k} ‖}_{2}^{2} + {‖ \nabla v_{0}^{k} ‖}_{2}^{2} + {‖ v_{1}^{k} ‖}_{2}^{2}) \\ - \int_{0}^{t} \int_{Ω} (u_{t}^{k} f_{1} (u^{k}, v^{k}) + v_{t}^{k} f_{2} (u^{k}, v^{k})) d x d s \end{matrix} \end{matrix}$ Now, we estimate the last two terms on the right-hand side of (3.11) as follows $\begin{matrix} (3.12) & \begin{matrix} | f_{1} (u^{k}, v^{k}) | & ⩽ C ({| u^{k} + v^{k} |}^{2 p (x) + 3} + {| u^{k} |}^{p (x) + 1} {| v^{k} |}^{p (x) + 2}) \\ ⩽ C ({| u^{k} |}^{2 p (x) + 3} + {| v^{k} |}^{2 p (x) + 3} + {| u^{k} |}^{p (x) + 1} {| v^{k} |}^{p (x) + 2}) . \end{matrix} \end{matrix}$ From (3.12) and Young’s inequality, with $\begin{matrix} q (x) = \frac{2 p (x) + 3}{p (x) + 1}, q^{'} (x) = \frac{2 p (x) + 3}{p (x) + 2}, \end{matrix}$ we get $\begin{matrix} {| u^{k} |}^{p (x) + 1} {| v^{k} |}^{p (x) + 2} ⩽ \frac{1}{2} {| u^{k} |}^{2 p (x) + 3} + C (x) {| v^{k} |}^{2 p (x) + 3}, \end{matrix}$ where $\begin{matrix} C (x) = \frac{p (x) + 2}{2 p (x) + 3} {(\frac{2 (p (x) + 1)}{(2 p (x) + 3)})}^{\frac{p (x) + 1}{p (x) + 2}} . \end{matrix}$ Then $C (x) ⩽ C$ since $p (x)$ is bounded. Hence, we have for some positive constant D, $\begin{matrix} | f_{1} (u^{k}, v^{k}) | ⩽ D [{| u^{k} |}^{2 p (x) + 3} + {| v^{k} |}^{2 p (x) + 3}] . \end{matrix}$ Consequently, by using Young’s inequality, Lemma 2.4 and the embedding theorem, (3.8) and (3.9), we arrive at $\begin{array}{l} \int_{Ω} u_{t}^{k} f_{1} (u^{k}, v^{k}) d x \\ ⩽ \frac{1}{2} {‖ u_{t}^{k} ‖}_{2}^{2} + \frac{D^{2}}{2} \int_{Ω} {| f_{1} (u^{k}, v^{k}) |}^{2} d x \\ ⩽ \frac{1}{2} {‖ u_{t}^{k} ‖}_{2}^{2} + \frac{c D^{2}}{2} \int_{Ω} {| u^{k} |}^{2 (2 p (x) + 3)} d x + \frac{c D^{2}}{2} \int_{Ω} {| v^{k} |}^{2 (2 p (x) + 3)} d x \\ ⩽ \frac{1}{2} {‖ u_{t}^{k} ‖}_{2}^{2} + \frac{c D^{2}}{2} \int_{Ω} ({| u^{k} |}^{2 (2 p_{2} + 3)} + {| v^{k} |}^{2 (2 p_{2} + 3)} + {| u^{k} |}^{2 (2 p_{1} + 3)} + {| v^{k} |}^{2 (2 p_{1} + 3)}) d x \\ ⩽ \frac{1}{2} {‖ u_{t}^{k} ‖}_{2}^{2} + \frac{c D^{2}}{2} (| | \nabla u^{k} | |_{2}^{2 (2 p_{2} + 3)} + | | \nabla v^{k} | |_{2}^{2 (2 p_{2} + 3)} + | | \nabla u^{k} | |_{2}^{2 (2 p_{1} + 3)} + | | \nabla v^{k} | |_{2}^{2 (2 p_{1} + 3)}) \\ ⩽ \frac{1}{2} {‖ u_{t}^{k} ‖}_{2}^{2} + \frac{c D^{2}}{2} ({(E (0))}^{4 (p_{2} + 1)} [| | \nabla u^{k} | |_{2}^{2} + | | \nabla v^{k} | |_{2}^{2}] + {(E (0))}^{4 (p_{1} + 1)} [| | \nabla u^{k} | |_{2}^{2} + | | \nabla v^{k} | |_{2}^{2}]) \\ ⩽ \frac{1}{2} {‖ u_{t}^{k} ‖}_{2}^{2} + \frac{c D^{2}}{2} [| | \nabla u^{k} | |_{2}^{2} + | | \nabla v^{k} | |_{2}^{2}] \\ (3.13) & ⩽ c ({‖ u_{t}^{k} ‖}_{2}^{2} + | | \nabla u^{k} | |_{2}^{2} + | | \nabla v^{k} | |_{2}^{2}), \end{array}$ where c is a generic positive constant. Similarly, we have $\begin{matrix} (3.14) & \begin{matrix} \int_{Ω} v_{t}^{k} f_{2} (u^{k}, v^{k}) d x ⩽ c ({‖ v_{t}^{k} ‖}_{2}^{2} + | | \nabla u^{k} | |_{2}^{2} + | | \nabla v^{k} | |_{2}^{2}) . \end{matrix} \end{matrix}$ Then, we infer from (3.11)-(3.14) that $\begin{matrix} (3.15) & \begin{matrix} \frac{1}{2} ({‖ u_{t}^{k} ‖}_{2}^{2} + {‖ \nabla u^{k} ‖}_{2}^{2} + {‖ v_{t}^{k} ‖}_{2}^{2} + {‖ v^{k} ‖}_{2}^{2}) \\ + \int_{0}^{t} \int_{Ω} {| u_{t}^{k} (s) |}^{m (\cdot) + 1} d x d s + \int_{0}^{t} \int_{Ω} {| v_{t}^{k} (s) |}^{r (\cdot) + 1} d x d s \\ ⩽ c + c \int_{0}^{t} ({‖ u_{t}^{k} ‖}_{2}^{2} + {‖ \nabla u^{k} ‖}_{2}^{2} + {‖ v_{t}^{k} ‖}_{2}^{2} + {‖ v^{k} ‖}_{2}^{2}) . \end{matrix} \end{matrix}$ Using Gronwall’s inequality, (3.15) gives $\begin{matrix} (3.16) & \begin{matrix} {‖ u_{t}^{k} ‖}_{2}^{2} + {‖ \nabla u^{k} ‖}_{2}^{2} + {‖ v_{t}^{k} ‖}_{2}^{2} + {‖ v^{k} ‖}_{2}^{2} \\ + \int_{0}^{t} \int_{Ω} {| u_{t}^{k} (s) |}^{m (\cdot) + 1} d x d s + \int_{0}^{t} \int_{Ω} {| v_{t}^{k} (s) |}^{r (\cdot) + 1} d x d s ⩽ C_{T}, \forall 0 ⩽ t ⩽ T, \end{matrix} \end{matrix}$ where $C_{T}$ is a constant independent of k and t. Therefore, $\begin{matrix} (3.17) & \{\begin{matrix} (u^{k}) and (v^{k}) & are bounded sequences in L^{\infty} (0, T; H_{0}^{1} (Ω)), \\ (u_{t}^{k}) & is a bounded sequence in L^{\infty} (0, T; L^{2} (Ω)) \cap L^{m (\cdot)} (Ω \times (0, T)), \\ (v_{t}^{k}) & is a bounded sequence in L^{\infty} (0, T; L^{2} (Ω)) \cap L^{r (\cdot)} (Ω \times (0, T)) . \end{matrix} \end{matrix}$ Therefore, there exist subsequences of $(u^{k})$ and $(v^{k})$ still denoted by $(u^{k})$ and $(v^{k})$ such that $\begin{matrix} (3.18) & \begin{matrix} \{\begin{matrix} u^{k} ⇀^{} u and v^{k} ⇀^{} v in L^{\infty} (0, T; H_{0}^{1} (Ω)), \\ u_{t}^{k} ⇀^{} u_{t} and v_{t}^{k} ⇀^{} v_{t} in L^{\infty} (0, T; L^{2} (Ω)) . \end{matrix} \end{matrix} \end{matrix}$ Making use of the Aubin-Lions Theorem, we find, up to subsequences, that $\begin{matrix} u^{k} \to u and v^{k} \to v strongly in L^{2} (0, T; L^{2} (Ω)) \end{matrix}$ and $\begin{matrix} u^{k} \to u and v^{k} \to v a.e. in Ω \times (0, T) . \end{matrix}$ Therefore, since $f_{i}$ is continuous, $\begin{matrix} (3.19) & f_{i} (u^{k}, v^{k}) \to f_{i} (u, v) a.e. in Ω \times (0, T), for i = 1, 2 . \end{matrix}$ Now using the arguments of (3.13), (3.14) and (3.17), we deduce that $\begin{matrix} (3.20) & \begin{matrix} \int_{0}^{T} {‖ f_{i} (u^{k}, v^{k}) ‖}_{2}^{2} d t ⩽ c \int_{0}^{T} ({‖ \nabla u^{k} ‖}_{2}^{2} + {‖ \nabla v^{k} ‖}_{2}^{2}) d t ⩽ C_{T} T, i = 1, 2 \end{matrix} \end{matrix}$ which implies that $f_{i} (u^{k}, v^{k})$ is bounded in $L^{2} (Ω \times (0, T))$ . Combining this and (3.19) and owing to Lions’ lemma ([21], pp. 12), we deduce that $\begin{matrix} (3.21) & f_{i} (u^{k}, v^{k}) ⇀ f_{i} (u, v) in L^{2} (Ω \times (0, T)), for i = 1, 2 . \end{matrix}$ Since $(u_{t}^{k})$ is bounded in $L^{m (\cdot)} (Ω \times (0, T))$ , then $({| u_{t}^{k} |}^{m (\cdot) - 2} u_{t}^{k})$ is bounded in $L^{\frac{m (\cdot)}{m (\cdot) - 1}} (Ω \times (0, T))$ . Hence, up to a subsequence, $\begin{matrix} (3.22) & {| u_{t}^{k} |}^{m (\cdot) - 2} u_{t}^{k} ⇀ ψ_{1} in L^{\frac{m (\cdot)}{m (\cdot) - 1}} (Ω \times (0, T)) . \end{matrix}$ Similarly, we have $\begin{matrix} (3.23) & {| v_{t}^{k} |}^{r (\cdot) - 2} v_{t}^{k} ⇀ ψ_{2} in L^{\frac{r (\cdot)}{r (\cdot) - 1}} (Ω \times (0, T)) . \end{matrix}$ Now, our task is to show that $ψ_{1} = {| u_{t} |}^{m (\cdot) - 2} u_{t}$ and $ψ_{2} = {| v_{t} |}^{r (\cdot) - 2} v_{t}$ . For this purpose, we integrate (3.6) over $(0, t)$ to obtain $\begin{matrix} (3.24) & \begin{matrix} \int_{Ω} u_{t}^{k} (t) w_{j} d x - \int_{Ω} u_{1}^{k} w_{j} d x + \int_{0}^{t} \int_{Ω} \nabla u^{k} (s) . \nabla w_{j} d x d s \\ + \int_{Ω} \int_{0}^{t} {| u_{s}^{k} (s) |}^{m (\cdot) - 2} u_{s}^{k} (s) w_{j} d s d x \\ + \int_{0}^{t} \int_{Ω} w_{j} f_{1} (u^{k}, v^{k}) d x d s = 0 \\ \int_{Ω} v_{t}^{k} (t) w_{j} d x - \int_{Ω} v_{1}^{k} w_{j} d x + \int_{0}^{t} \int_{Ω} \nabla v^{k} (s) . \nabla w_{j} d x d s \\ + \int_{Ω} \int_{0}^{t} {| v_{s}^{k} (s) |}^{r (\cdot) - 2} v_{s}^{k} (s) w_{j} d s d x \\ + \int_{0}^{t} \int_{Ω} w_{j} f_{2} (u^{k}, v^{k}) d x d s = 0, \forall j = 1, 2, \dots, k . \end{matrix} \end{matrix}$ Convergences (3.5), (3.18), (3.21) and (3.22) allow us to pass to the limit in both equations in (3.24), as $k \to + \infty$ , and get $\begin{matrix} (3.25) & \begin{matrix} \int_{Ω} u_{t} (t) w_{j} d x - \int_{Ω} u_{1} w_{j} d x + \int_{0}^{t} \int_{Ω} \nabla u (s) . \nabla w_{j} d x d s \\ + \int_{Ω} \int_{0}^{t} ψ_{1} (s) w_{j} d s d x + \int_{0}^{t} \int_{Ω} w_{j} f_{1} (u, v) d x d s = 0 \\ \int_{Ω} v_{t} (t) w_{j} d x - \int_{Ω} v_{1} w_{j} d x + \int_{0}^{t} \int_{Ω} \nabla v (s) . \nabla w_{j} d x d s \\ + \int_{Ω} \int_{0}^{t} ψ_{2} (s) w_{j} d s d x + \int_{0}^{t} \int_{Ω} w_{j} f_{2} (u, v) d x d s = 0, \forall j = 1, 2, \dots \end{matrix} \end{matrix}$ which implies that (3.25) is valid for any $w \in H_{0}^{1} (Ω)$ . Using the fact that the left-hand sides of both equations in (3.25) are an absolutely continuous functions, hence they are differentiable for a.e $t \in (0, T)$ , and we get $\begin{matrix} (3.26) & \begin{matrix} \frac{d}{d t} \int_{Ω} u_{t} (x, t) w (x) d x + \int_{Ω} \nabla u (x, t) . \nabla w (x) d x \\ + \int_{Ω} ψ_{1} (t) w (x) d x + \int_{Ω} w f_{1} (u, v) d x = 0, \\ \frac{d}{d t} \int_{Ω} v_{t} (x, t) w (x) d x + \int_{Ω} \nabla v (x, t) . \nabla w (x) d x \\ + \int_{Ω} ψ_{2} (t) w (x) d x + \int_{Ω} w f_{2} (u, v) d x = 0, \forall w \in H_{0}^{1} (Ω), for a.e. t \in [0, T] . \end{matrix} \end{matrix}$

Now, we define $\begin{matrix} (3.27) & \begin{matrix} X^{k} = \int_{0}^{T} \int_{Ω} ({| u_{t}^{k} |}^{m (\cdot) - 2} u_{t}^{k} - {| \tilde{u} |}^{m (\cdot) - 2} \tilde{u}) (u_{t}^{k} - \tilde{u}) d x d t ⩾ 0, \\ \forall \tilde{u} \in L^{m (\cdot)} ((0, T), H_{0}^{1} (Ω)) \end{matrix} \end{matrix}$ and $\begin{matrix} (3.28) & \begin{matrix} Y^{k} = \int_{0}^{T} \int_{Ω} ({| v_{t}^{k} |}^{r (\cdot) - 2} v_{t}^{k} - {| \tilde{v} |}^{r (\cdot) - 2} \tilde{v}) (v_{t}^{k} - \tilde{v}) d x d t ⩾ 0, \forall \tilde{v} \in L^{r (\cdot)} ((0, T), H_{0}^{1} (Ω)) . \end{matrix} \end{matrix}$ This is true by the following elementary inequality: $\begin{matrix} (3.29) & ({| a |}^{q (x) - 2} a - {| b |}^{q (x) - 2} b) (a - b) ⩾ 0, for a, b \in R . \end{matrix}$ So, by using the first equation in (3.6), after integrating over $(0, T)$ and letting $w = u_{t}^{k}$ , we get $\begin{matrix} X^{k} = & \frac{1}{2} ({‖ \nabla u_{0}^{k} ‖}_{2}^{2} + {‖ u_{1}^{k} ‖}_{2}^{2}) - \int_{0}^{T} \int_{Ω} u_{t}^{k} f_{1} (u_{k}, v_{k}) d x d t \\ - \frac{1}{2} ({‖ u_{t}^{k} ‖}_{2}^{2} + {‖ \nabla u^{k} ‖}_{2}^{2}) - \int_{0}^{T} \int_{Ω} {| u_{t}^{k} |}^{m (\cdot) - 2} u_{t}^{k} \tilde{u} d x d t \\ - \int_{0}^{T} \int_{Ω} {| \tilde{u} |}^{m - 2} \tilde{u} (u_{t}^{k} - \tilde{u}) d x d t . \end{matrix}$ Taking $k \to + \infty$ , we obtain $\begin{matrix} (3.30) & \begin{matrix} 0 ⩽ & lim sup X^{k} \\ = & \frac{1}{2} ({‖ \nabla u_{0} (t) ‖}_{2}^{2} + {‖ u_{1} ‖}_{2}^{2}) - \int_{0}^{T} \int_{Ω} u_{t} f_{1} (u, v) d x d t - \frac{1}{2} ({‖ u_{t} ‖}_{2}^{2} + {‖ \nabla u ‖}_{2}^{2}) \\ - \int_{0}^{T} \int_{Ω} ψ_{1} (t) \tilde{u} d x d t - \int_{0}^{T} \int_{Ω} {| \tilde{u} |}^{m (\cdot) - 2} \tilde{u} (u_{t} - \tilde{u}) d x d t . \end{matrix} \end{matrix}$ Replacing w by $u_{t}$ in the first equation of (3.26) and integrating over $(0, T)$ , we obtain $\begin{matrix} (3.31) & \begin{matrix} - \frac{1}{2} ({‖ \nabla u_{0} (t) ‖}_{2}^{2} + {‖ u_{1} ‖}_{2}^{2}) + \int_{0}^{T} \int_{Ω} u_{t} f_{1} (u, v) d x d t \\ + \frac{1}{2} ({‖ u_{t} ‖}_{2}^{2} + {‖ \nabla u ‖}_{2}^{2}) + \int_{0}^{T} \int_{Ω} ψ_{1} u_{t} d x d t = 0 . \end{matrix} \end{matrix}$ Combining (3.30) and (3.31), we arrive at $\begin{matrix} 0 & ⩽ lim sup X^{k} \\ = \int_{0}^{T} \int_{Ω} ψ_{1} u_{t} d x d t - \int_{0}^{T} \int_{Ω} ψ_{1} \tilde{u} d x d t - \int_{0}^{T} \int_{Ω} {| \tilde{u} |}^{m (\cdot) - 2} \tilde{u} (u_{t} - \tilde{u}) d x d t \\ ⩽ \int_{0}^{T} \int_{Ω} (ψ_{1} - {| \tilde{u} |}^{m (\cdot) - 2} \tilde{u}) (u_{t} - \tilde{u}) d x d t . \end{matrix}$ Hence, $\begin{matrix} \int_{0}^{T} \int_{Ω} (ψ - {| \tilde{u} |}^{m - 2} \tilde{u}) (u_{t} - \tilde{u}) d x d t ⩾ 0, \forall \tilde{u} \in L^{m (\cdot)} (Ω \times (0, T)) \end{matrix}$ by density of $H_{0}^{1} (Ω)$ in $L^{m (\cdot)} (Ω)$ .

Let $\tilde{u} = λ z + u_{t}$ , $z \in L^{m (\cdot)} (Ω \times (0, T))$ . So, we get, $\forall λ \neq 0$ , $\begin{matrix} - λ \int_{0}^{T} \int_{Ω} (ψ - {| λ z + u_{t} |}^{m (\cdot) - 2} (λ z + u_{t})) z d x d t ⩽ 0, z \in L^{m (\cdot)} (Ω \times (0, T)) . \end{matrix}$ Let $λ > 0$ . So we have $\begin{matrix} \int_{0}^{T} \int_{Ω} (ψ - {| λ z + u_{t} |}^{m (\cdot) - 2} (λ z + u_{t})) z d x d t ⩽ 0, z \in L^{m (\cdot)} (Ω \times (0, T)) . \end{matrix}$ As $λ \to 0$ , we get $\begin{matrix} (3.32) & \int_{0}^{T} \int_{Ω} (ψ - {| u_{t} |}^{m (\cdot) - 2} u_{t}) z d x d t ⩽ 0, z \in L^{m (\cdot)} (Ω \times (0, T)) . \end{matrix}$ Similarly, for $λ < 0$ , we get $\begin{matrix} (3.33) & \int_{0}^{T} \int_{Ω} (ψ - {| u_{t} |}^{m (\cdot) - 2} u_{t}) z d x d t ⩾ 0, z \in L^{m (\cdot)} (Ω \times (0, T)) . \end{matrix}$ Thus, (3.32) and (3.33) imply that $ψ_{1} = {| u_{t} |}^{m (\cdot) - 2} u_{t}$ .

Using (3.28) and repeating the same steps as before, we can prove that $ψ_{2} = {| v_{t} |}^{r (\cdot) - 2} v_{t}$ . Hence (3.26) becomes $\begin{matrix} \frac{d}{d t} \int_{Ω} u_{t} (x, t) w (x) d x + \int_{Ω} \nabla u (x, t) . \nabla w (x) d x + \int_{Ω} w f_{1} (u, v) d x \\ + \int_{Ω} {| u_{t} |}^{m (\cdot) - 2} u_{t} w (x) d x = 0, \forall w \in H_{0}^{1} (Ω) \\ \frac{d}{d t} \int_{Ω} v_{t} (x, t) w (x) d x + \int_{Ω} \nabla v (x, t) . \nabla w (x) d x + \int_{Ω} w f_{2} (u, v) d x \\ + \int_{Ω} {| v_{t} |}^{r (\cdot) - 2} v_{t} w (x) d x = 0, \forall w \in H_{0}^{1} (Ω) . \end{matrix}$

To handle the initial conditions, we note that $\begin{matrix} (3.34) & \begin{matrix} u^{k} ⇀ u and v^{k} ⇀ v weakly in L^{2} (0, T; H_{0}^{1} (Ω)) \\ u_{t}^{k} ⇀ u_{t} and v_{t}^{k} ⇀ v_{t} weakly in L^{2} (0, T; L^{2} (Ω)) \end{matrix} \end{matrix}$ Thus, using Lion’s lemma [21] and (3.5), we easily obtain $\begin{matrix} u (x, 0) = u_{0} (x) and v (x, 0) = v_{0} (x) . \end{matrix}$ As in [17], we multiply (3.6) by $ϕ \in C_{0}^{\infty} (0, T)$ and integrate over $(0, T)$ to obtain, for any $w \in V_{k}$ , $\begin{matrix} (3.35) & \begin{matrix} - \int_{0}^{T} \int_{Ω} u_{t}^{k} w ϕ^{'} d x d t \\ = - \int_{0}^{T} \int_{Ω} \nabla u^{k} . \nabla w ϕ d x d t - \int_{0}^{T} \int_{Ω} {| u_{t}^{k} |}^{m (\cdot) - 2} u_{t}^{k} w ϕ d x d t \\ - \int_{0}^{T} \int_{Ω} w ϕ f_{1} (u^{k}, v^{k}) d x d t \\ - \int_{0}^{T} \int_{Ω} v_{t}^{k} w ϕ^{'} d x d t \\ = - \int_{0}^{T} \int_{Ω} \nabla v^{k} . \nabla w ϕ d x d t - \int_{0}^{T} \int_{Ω} {| v_{t}^{k} |}^{r (\cdot) - 2} v_{t}^{k} w ϕ d x d t \\ - \int_{0}^{T} \int_{Ω} w ϕ f_{2} (u^{k}, v^{k}) d x d t \end{matrix} \end{matrix}$ As $k \to + \infty$ , we have for any $w \in H_{0}^{1} (Ω)$ and any $ϕ \in C_{0}^{\infty} ((0, T))$ , $\begin{matrix} (3.36) & \begin{matrix} - \int_{0}^{T} \int_{Ω} u_{t} w ϕ^{'} d x d t \\ = - \int_{0}^{T} \int_{Ω} \nabla u . \nabla w ϕ d x d t - \int_{0}^{T} \int_{Ω} w ϕ f_{1} (u, v) d x d t \\ - \int_{0}^{T} \int_{Ω} {| u_{t} |}^{m (\cdot) - 2} u_{t} w ϕ d x d t \\ - \int_{0}^{T} \int_{Ω} u_{t} w ϕ^{'} d x d t \\ = - \int_{0}^{T} \int_{Ω} \nabla u . \nabla w ϕ d x d t - \int_{0}^{T} \int_{Ω} w ϕ f_{1} (u, v) d x d t \\ - \int_{0}^{T} \int_{Ω} {| u_{t} |}^{r (\cdot) - 2} u_{t} w ϕ d x d t . \end{matrix} \end{matrix}$ This means (see [17]), $\begin{matrix} u_{t t}, v_{t t} \in L^{2} ([0, T), H^{- 1} (Ω)) . \end{matrix}$ Recalling that $u_{t}, v_{t} \in L^{2} ((0, T), L^{2} (Ω))$ , we obtain $\begin{matrix} u_{t}, v_{t} \in C ([0, T), H^{- 1} (Ω)) . \end{matrix}$ So, $u_{t}^{k} (x, 0)$ makes sense and $\begin{matrix} u_{t}^{k} (\cdot, 0) \to u_{t} (\cdot, 0) and v_{t}^{k} (\cdot, 0) \to v_{t} (\cdot, 0) in H^{- 1} (Ω) \end{matrix}$ But $\begin{matrix} u_{t}^{k} (\cdot, 0) = u_{1}^{k} \to u_{1} and v_{t}^{k} (\cdot, 0) = v_{1}^{k} \to v_{1} in L^{2} (Ω) \end{matrix}$ Hence, $\begin{matrix} u_{t} (x, 0) = u_{1} (x) and v_{t} (x, 0) = v_{1} (x) . \end{matrix}$

For uniqueness, let us assume that problem (P) has two solutions $(u_{1}, v_{1})$ and $(u_{2}, v_{2})$ . Then, $(w_{1}, w_{2}) = (u_{1} - u_{2}, v_{1} - v_{2})$ satisfies $\begin{matrix} (3.37) & \{\begin{matrix} {(w_{1})}_{t t} - Δ w_{1} + ({| {(u_{1})}_{t} |}^{m (\cdot) - 2} {(u_{1})}_{t} - {| {(u_{2})}_{t} |}^{m (\cdot) - 2} {(u_{2})}_{t}) \\ + f_{1} (u_{1}, v_{1}) - f_{1} (u_{2}, v_{2}) = 0, in Ω \times (0, T), \\ {(w_{2})}_{t t} - Δ (w_{2}) + ({| {(v_{1})}_{t} |}^{r (\cdot) - 2} {(v_{1})}_{t} - {| {(v_{2})}_{t} |}^{r (\cdot) - 2} {(v_{2})}_{t}) \\ + f_{2} (u_{1}, v_{1}) - f_{2} (u_{2}, v_{2}) = 0, in Ω \times (0, T), \\ w_{1} = w_{2} = 0, on \partial Ω \times (0, T), \\ w_{1} (x, 0) = w_{2} (x, 0) = 0, {(w_{1})}_{t} (x, 0) = {(w_{2})}_{t} (x, 0) = 0, in Ω \times (0, T) . \end{matrix} \end{matrix}$ Now, multiply the first equation in (3.37) by ${(w_{1})}_{t}$ and the second one by ${(w_{2})}_{t}$ , add the resulting equations and integrate over $Ω \times (0, t)$ to obtain $\begin{array}{l} {‖ {(w_{1})}_{t} ‖}_{2}^{2} + {‖ \nabla (w_{1}) ‖}_{2}^{2} + {‖ {(w_{2})}_{t} ‖}_{2}^{2} + {‖ \nabla (w_{2}) ‖}_{2}^{2} \\ + 2 \int_{0}^{t} \int_{Ω} ({| {(u_{1})}_{t} |}^{m (\cdot) - 2} {(u_{1})}_{t} - {| {(u_{2})}_{t} |}^{m (\cdot) - 2} {(u_{2})}_{t}) ({(u_{1})}_{t} - {(u_{2})}_{t}) d x d s \\ + 2 \int_{0}^{t} \int_{Ω} ({| {(v_{1})}_{t} |}^{r (\cdot) - 2} {(v_{1})}_{t} - {| {(v_{2})}_{t} |}^{r (\cdot) - 2} {(v_{2})}_{t}) ({(v_{1})}_{t} - {(v_{2})}_{t}) d x d s \\ = \int_{0}^{t} \int_{Ω} {(w_{1})}_{t} (f_{1} (u_{2}, v_{2}) - f_{1} (u_{1}, v_{1})) d x d s \\ (3.38) & + \int_{0}^{t} \int_{Ω} {(w_{2})}_{t} (f_{2} (u_{2}, v_{2}) - f_{2} (u_{1}, v_{1})) d x d s . \end{array}$ Following similar arguments used to obtain (3.13) and (3.14), we find $\begin{matrix} (3.39) & \begin{matrix} \int_{Ω} {(w_{1})}_{t} (f_{1} (u_{2}, v_{2}) - f_{1} (u_{1}, v_{1})) d x d s + \int_{Ω} {(w_{2})}_{t} (f_{2} (u_{2}, v_{2}) - f_{2} (u_{1}, v_{1})) d x d s \\ ⩽ c ({‖ {(w_{1})}_{t} ‖}_{2}^{2} + {‖ \nabla (w_{1}) ‖}_{2}^{2} + {‖ {(w_{2})}_{t} ‖}_{2}^{2} + {‖ \nabla (w_{2}) ‖}_{2}^{2}) . \end{matrix} \end{matrix}$ Hence, by using inequality (3.29), (3.38) and (3.39) and recalling Gronwall’s lemma, then we deduce that $\begin{matrix} {‖ {(w_{1})}_{t} ‖}_{2}^{2} + {‖ \nabla (w_{1}) ‖}_{2}^{2} + {‖ {(w_{2})}_{t} ‖}_{2}^{2} + {‖ \nabla (w_{2}) ‖}_{2}^{2} = 0 \end{matrix}$ which means that $(u_{1}, v_{1}) = (u_{2}, v_{2})$ . This completes the proof. □

4. Decay results

In this section we state and prove our main decay result. For this purpose, we assume that the solution of (P) has the required regularity to justify the calculations. We, then, define the energy functional $E (t)$ associated to our system by $\begin{matrix} (4.1) & E (t) : = \frac{1}{2} [‖ u_{t} ‖_{2}^{2} + ‖ v_{t} ‖_{2}^{2}] + \frac{1}{2} [‖ \nabla u ‖_{2}^{2} + ‖ \nabla v ‖_{2}^{2}] + \int_{Ω} F (u, v) d x . \end{matrix}$ By multiplying the first equation in (P) by $u_{t}$ and the second equation by $v_{t}$ , integrating over Ω, using integration by parts and summing up, we get $\begin{matrix} (4.2) & E^{'} (t) = - \int_{Ω} | u_{t} |^{m (x) + 1} d x - \int_{Ω} | v_{t} |^{r (x) + 1} d x ⩽ 0 . \end{matrix}$ Before giving the decay result, we need the following lemma.

Lemma 4.1 (Komornik [16]).

Let $E : R^{+} \to R^{+}$ be a nonincreasing function. Assume that there exist constants $c, α, σ > 0$ such that $\begin{matrix} (4.3) & \int_{s}^{\infty} E^{1 + σ} (t) d t ⩽ \frac{1}{ω} E^{σ} (0) E (s) = \tilde{c} E (s), \forall s > 0 . \end{matrix}$ Then, $\forall t ⩾ 0$ , $\begin{matrix} (4.4) & E (t) ⩽ \{\begin{matrix} \frac{c E (0)}{{(1 + t)}^{1 / σ}}, & if σ > 0 \\ c E (0) e^{- ω t}, & if σ = 0 . \end{matrix} \end{matrix}$

Our main result is

Theorem 4.2.
Suppose that the conditions of Theorem 3.2 hold. then there exist two constants $c, α > 0$ , independent of t and may depend on $E (0)$ , such that the energy $E (t)$ satisfies, $\forall t ⩾ 0$ , $\begin{matrix} (4.5) & E (t) ⩽ \{\begin{matrix} \frac{c}{{(1 + t)}^{2 / (σ - 2)}}, & if (m_{2}, r_{2}) \neq (2, 2), \\ c e^{- α t}, & if m (\cdot) = r (\cdot) = 2, \end{matrix} \end{matrix}$ where $σ = max {m_{2}, r_{2}}$ .
Proof.
By multiplying the first equation in (P) by $u E^{q} (t)$ , and the second equation by $v E^{q} (t)$ , for $q > 0$ to be specified later, integrating over $Ω \times (s, T)$ , $T > s$ , using integration by parts and summing up, we get $\begin{matrix} \int_{s}^{T} E^{q} (t) \int_{Ω} (u u_{t t} - u Δ u + | u_{t} |^{m (x) - 2} u_{t} u + u f_{1}) d x d t \\ + \int_{s}^{T} E^{q} (t) \int_{Ω} (v v_{t t} - v Δ v + | v_{t} |^{r (x) - 2} v_{t} v + v f_{2}) d x d t = 0 . \end{matrix}$ which implies that $\begin{matrix} (4.6) & \begin{matrix} \int_{s}^{T} E^{q} (t) \frac{d}{d t} \int_{Ω} (u u_{t} + v v_{t}) d x d t + \int_{s}^{T} E^{q} (t) \int_{Ω} [u_{t}^{2} + v_{t}^{2} + | \nabla u |^{2} + | \nabla v |^{2}] d x d t \\ - 2 \int_{s}^{T} E^{q} (t) \int_{Ω} (u_{t}^{2} + v_{t}^{2}) d x d t + \int_{s}^{T} E^{q} (t) \int_{Ω} [| u_{t} |^{m (x) - 2} u_{t} u + | v_{t} |^{r (x) - 2} v_{t} v] d x d t \\ + 2 \int_{s}^{T} E^{q} (t) \int_{Ω} (p (x) + 2) F (u, v) d x d t = 0 . \end{matrix} \end{matrix}$ By using the definition of $E (t)$ , (1.4), and the relation $\begin{matrix} \frac{d}{d t} (E^{q} (t) \int_{Ω} (u u_{t} + v v_{t}) d x) = q E^{q - 1} (t) E^{'} (t) \int_{Ω} (u u_{t} + v v_{t}) d x + E^{q} (t) \frac{d}{d t} \int_{Ω} (u u_{t} + v v_{t}) d x, \end{matrix}$ equation (4.6) becomes $\begin{matrix} (4.7) & \begin{matrix} 2 \int_{s}^{T} E^{q + 1} (t) d t \\ ⩽ - \int_{s}^{T} \frac{d}{d t} [E^{q} (t) \int_{Ω} (u u_{t} + v v_{t}) d x] d t + q \int_{s}^{T} E^{q - 1} (t) E^{'} (t) \int_{Ω} (u u_{t} + v v_{t}) d x d t \\ + 2 \int_{s}^{T} E^{q} (t) \int_{Ω} (u_{t}^{2} + v_{t}^{2}) d x d t \\ - \int_{s}^{T} E^{q} (t) \int_{Ω} [| u_{t} |^{m (x) - 2} u_{t} u + | v_{t} |^{r (x) - 2} v_{t} v] d x d t . \end{matrix} \end{matrix}$ Estimates. $\begin{array}{l} ∙ & | - \int_{s}^{T} \frac{d}{d t} [E^{q} (t) \int_{Ω} (u u_{t} + v v_{t}) d x] d t | \\ = | E^{q} (s) \int_{Ω} (u u_{t} (x, s) + v v_{t} (x, s)) d x \\ - E^{q} (T) \int_{Ω} (u u_{t} (x, T) + v v_{t} (x, T)) d x | \\ ⩽ E^{q} (s) [\frac{1}{2} \int_{Ω} u^{2} (x, s) d x + \frac{1}{2} \int_{Ω} u_{t}^{2} (x, s) d x \\ + \frac{1}{2} \int_{Ω} v^{2} (x, s) d x + \frac{1}{2} \int_{Ω} v_{t}^{2} (x, s) d x] \\ + E^{q} (T) [\frac{1}{2} \int_{Ω} u^{2} (x, T) d x + \frac{1}{2} \int_{Ω} u_{t}^{2} (x, T) d x \\ + \frac{1}{2} \int_{Ω} v^{2} (x, T) d x + \frac{1}{2} \int_{Ω} v_{t}^{2} (x, T) d x] \\ ⩽ E^{q} (s) [\frac{1}{2} c_{} \int_{Ω} {| \nabla u (x, s) |}^{2} d x + \frac{1}{2} \int_{Ω} u_{t}^{2} (x, s) d x \\ + \frac{1}{2} c_{} \int_{Ω} {| \nabla v (x, s) |}^{2} d x + \frac{1}{2} \int_{Ω} v_{t}^{2} (x, s) d x] \\ + E^{q} (T) [\frac{1}{2} c_{} \int_{Ω} {| \nabla u (x, T) |}^{2} d x + \frac{1}{2} \int_{Ω} u_{t}^{2} (x, T) d x \\ + \frac{1}{2} c_{} \int_{Ω} {| \nabla v (x, T) |}^{2} d x + \frac{1}{2} \int_{Ω} v_{t}^{2} (x, T) d x] \\ ⩽ C (E^{q + 1} (s) + E^{q + 1} (T)) \\ (4.8) & ⩽ C E^{q} (0) E (s) = \tilde{C} E (s) . \end{array}$ where $c_{*}$ is the embedding constant and $\tilde{C}$ is a generic positive constant. $\begin{array}{l} (4.9) & \begin{matrix} ∙ | q \int_{s}^{T} E^{q - 1} (t) E^{'} (t) \int_{Ω} (u u_{t} + v v_{t}) d x d t | & ⩽ - c \int_{s}^{T} E^{q - 1} (t) E^{'} (t) E (t) d t \\ = - c \int_{s}^{T} E^{q} (t) E^{'} (t) d t \\ ⩽ c E^{q + 1} (s) ⩽ \tilde{C} E (s), \end{matrix} \\ (4.10) & \begin{matrix} ∙ | 2 \int_{s}^{T} E^{q} (t) \int_{Ω} (u_{t}^{2} + v_{t}^{2}) d x d t | ⩽ & | 2 \int_{s}^{T} E^{q} (t) \int_{Ω} u_{t}^{2} d x d t | \\ + | 2 \int_{s}^{T} E^{q} (t) \int_{Ω} v_{t}^{2} d x d t | . \end{matrix} \end{array}$

We estimate the first term of the right-hand side of (4.10) and the second term is handled in a similar way. For this purpose, we set as in [16], $\begin{matrix} Ω_{+} = {x \in Ω / | u_{t} (x, t) | ⩾ 1}, Ω_{-} = {x \in Ω / | u_{t} (x, t) | < 1} \end{matrix}$ and exploit Hölder’s and Young’s inequalities and (1.3)–(1.5) as follows $\begin{array}{rcl} ∙ | 2 \int_{s}^{T} E^{q} (t) \int_{Ω} u_{t}^{2} d x d t | & = & | 2 \int_{s}^{T} E^{q} (t) [\int_{Ω_{-}} u_{t}^{2} + \int_{Ω_{+}} u_{t}^{2}] d x d t | \\ ⩽ & c \int_{s}^{T} E^{q} (t) [{(\int_{Ω_{-}} | u_{t} |^{m_{2}})}^{2 / m_{2}} + {(\int_{Ω_{+}} | u_{t} |^{m_{1}})}^{2 / m_{1}}] d x d t \\ ⩽ & c \int_{s}^{T} E^{q} (t) [{(\int_{Ω} | u_{t} |^{m (x)})}^{2 / m_{2}} + {(\int_{Ω} | u_{t} |^{m (x)})}^{2 / m_{1}}] d x d t \\ ⩽ & c \int_{s}^{T} E^{q} (t) {(- E^{'} (t))}^{2 / m_{2}} d t + c \int_{s}^{T} E^{q} (t) {(- E^{'} (t))}^{2 / m_{1}} d t . \end{array}$ Here, we distinguish three cases.

Case 1. $m_{1} = m_{2} = 2$ ( $m (\cdot) \equiv 2$ ). $\begin{array}{rcl} (4.11) & | 2 \int_{s}^{T} E^{q} (t) \int_{Ω} u_{t}^{2} d x d t | & ⩽ & c \int_{s}^{T} E^{q} (t) (- E^{'} (t)) d t ⩽ \tilde{C} E (s) . \end{array}$

Case 2. $m_{1} > 2$ . By using Young’s inequality, we have $\forall ε > 0$ , $\begin{array}{rcl} | 2 \int_{s}^{T} E^{q} (t) \int_{Ω} u_{t}^{2} d x d t | & ⩽ & c ε \int_{s}^{T} {(E (t))}^{\frac{q m_{2}}{m_{2} - 2}} d t + c ε \int_{s}^{T} {(E (t))}^{\frac{q m_{1}}{m_{1} - 2}} d t + c_{ε} \int_{s}^{T} (- E^{'} (t)) d t \\ (4.12) & ⩽ & c ε [1 + {(E (0))}^{\frac{2 q (m_{2} - m_{1})}{(m_{2} - 2) (m_{1} - 2)}}] \int_{s}^{T} {(E (t))}^{\frac{q m_{2}}{m_{2} - 2}} d t + c_{ε} E (s) . \end{array}$

Case 3. $m_{1} = 2 < m_{2}$ , then, similarly we have, $\forall ε > 0$ , $\begin{array}{rcl} | 2 \int_{s}^{T} E^{q} (t) \int_{Ω} u_{t}^{2} d x d t | & ⩽ & c ε \int_{s}^{T} {(E (t))}^{\frac{q m_{2}}{m_{2} - 2}} d t + c \int_{s}^{T} E^{q} (t) (- E^{'} (t)) d t \\ + c_{ε} \int_{s}^{T} (- E^{'} (t)) d t \\ (4.13) & ⩽ & c ε \int_{s}^{T} {(E (t))}^{\frac{q m_{2}}{m_{2} - 2}} d t + c_{ε} E (s) . \end{array}$ In the same way, we estimate the second term of the right-hand side of (4.10) to get three similar cases.

Case 1. $r_{1} = r_{2} = 2$ ( $r (\cdot) \equiv 2$ ). $\begin{array}{rcl} (4.14) & | 2 \int_{s}^{T} E^{q} (t) \int_{Ω} v_{t}^{2} d x d t | ⩽ \tilde{C} E (s) . \end{array}$

Case 2. $r_{1} > 2$ . $\begin{array}{rcl} | 2 \int_{s}^{T} E^{q} (t) \int_{Ω} v_{t}^{2} d x d t | & ⩽ & c ε \int_{s}^{T} {(E (t))}^{\frac{q r_{2}}{r_{2} - 2}} d t + c ε \int_{s}^{T} {(E (t))}^{\frac{q r_{1}}{r_{1} - 2}} d t \\ + c_{ε} \int_{s}^{T} (- E^{'} (t)) d t \\ ⩽ & c ε [1 + {(E (0))}^{\frac{2 q (r_{2} - r_{1})}{(r_{2} - 2) (r_{1} - 2)}}] \int_{s}^{T} {(E (t))}^{\frac{q r_{2}}{r_{2} - 2}} d t \\ (4.15) & + c_{ε} E (s) . \end{array}$

Case 3. $r_{1} = 2 < r_{2}$ . $\begin{array}{rcl} (4.16) & | 2 \int_{s}^{T} E^{q} (t) \int_{Ω} v_{t}^{2} d x d t | & ⩽ & c ε \int_{s}^{T} {(E (t))}^{\frac{q r_{2}}{r_{2} - 2}} d t + c_{ε} E (s) . \end{array}$ For the last term of (4.7), we have $\begin{array}{l} ∙ & | \int_{s}^{T} E^{q} (t) \int_{Ω} (| u_{t} |^{m (x) - 2} u_{t} u + | v_{t} |^{r (x) - 2} v_{t} v) d x d t | \\ (4.17) & ⩽ \int_{s}^{T} E^{q} (t) \int_{Ω} | u_{t} |^{m (x) - 1} | u | d x d t + \int_{s}^{T} E^{q} (t) \int_{Ω} | v_{t} |^{r (x) - 1} | v | d x d t . \end{array}$ We handle the first term of the right-hand side of (4.17), the other term is treated similarly. We use Young’s inequality with $s (x) = \frac{m (x)}{m (x) - 1}$ , $s^{'} (x) = m (x)$ , so for a.e. $x \in Ω$ , we have $\begin{matrix} | u_{t} |^{m (x) - 1} | u | ⩽ ε | u |^{m (x)} + c_{ε} (x) | u_{t} |^{m (x)}, \end{matrix}$ where $\begin{matrix} c_{ε} (x) = ε^{1 - m (x)} {(m (x))}^{- m (x)} {(m (x) - 1)}^{m (x) - 1} . \end{matrix}$ Therefore, $\begin{array}{l} | \int_{s}^{T} E^{q} (t) \int_{Ω} | u_{t} |^{m (x) - 2} u_{t} u d x d t | \\ ⩽ ε \int_{s}^{T} E^{q} (t) \int_{Ω} | u |^{m (x)} d x d t + \int_{s}^{T} E^{q} (t) \int_{Ω} c_{ε} (x) | u_{t} |^{m (x)} d x d t \\ ⩽ ε \int_{s}^{T} E^{q} (t) [\int_{Ω} | u |^{m_{1}} d x + \int_{Ω} | u |^{m_{2}} d x] d t + \int_{s}^{T} E^{q} (t) \int_{Ω} c_{ε} (x) | u_{t} |^{m (x)} d x d t . \end{array}$ By using the embedding, we obtain $\begin{array}{rcl} | \int_{s}^{T} E^{q} (t) \int_{Ω} | u_{t} |^{m (x) - 2} u_{t} u d x d t | & ⩽ & ε \int_{s}^{T} E^{q} (t) [c_{1} ‖ \nabla u ‖_{2}^{m_{1}} + c_{2} ‖ \nabla u ‖_{2}^{m_{2}}] d t \\ + \int_{s}^{T} E^{q} (t) \int_{Ω} c_{ε} (x) | u_{t} |^{m (x)} d x d t, \end{array}$ where $c_{1}$ and $c_{2}$ are positive constants independent of ε. We then recall (4.1) to get $\begin{array}{l} | \int_{s}^{T} E^{q} (t) \int_{Ω} | u_{t} |^{m (x) - 2} u_{t} u d x d t | \\ ⩽ ε \int_{s}^{T} E^{q} (t) [c_{1} {(2 E (t))}^{\frac{m_{1}}{2}} + c_{2} {(2 E (t))}^{\frac{m_{2}}{2}}] d t \\ + \int_{s}^{T} E^{q} (t) \int_{Ω} c_{ε} (x) | u_{t} |^{m (x)} d x d t \\ ⩽ ε c_{1}^{'} \int_{s}^{T} E^{q + 1} (t) {(E (t))}^{\frac{m_{1}}{2} - 1} d t + ε c_{2}^{'} \int_{s}^{T} E^{q + 1} (t) {(E (t))}^{\frac{m_{2}}{2} - 1} d t \\ + \int_{s}^{T} E^{q} (t) \int_{Ω} c_{ε} (x) | u_{t} |^{m (x)} d x d t \\ ⩽ c^{'} ε ({(E (0))}^{\frac{m_{1}}{2} - 1} + {(E (0))}^{\frac{m_{2}}{2} - 1}) \int_{s}^{T} E^{q + 1} (t) d t \\ (4.18) & + \int_{s}^{T} E^{q} (t) \int_{Ω} c_{ε} (x) | u_{t} |^{m (x)} d x d t, \end{array}$ where $c_{1}^{'}$ , $c_{2}^{'}$ and $c^{'}$ are positive constants independent of ε.

In the same way, the second term of the right-hand side of (4.17) is estimated as $\begin{array}{rcl} | \int_{s}^{T} E^{q} (t) \int_{Ω} | v_{t} |^{r (x) - 2} v_{t} v d x d t | & ⩽ & k^{'} ε ({(E (0))}^{\frac{r_{1}}{2} - 1} + {(E (0))}^{\frac{r_{2}}{2} - 1}) \int_{s}^{T} E^{q + 1} (t) d t \\ + \int_{s}^{T} E^{q} (t) \int_{Ω} k_{ε} (x) | v_{t} |^{r (x)} d x d t, \end{array}$ where $k^{'}$ is a positive constant independent of ε and $\begin{matrix} k_{ε} (x) = ε^{1 - r (x)} {(r (x))}^{- r (x)} {(r (x) - 1)}^{r (x) - 1} . \end{matrix}$ Hence (4.17) becomes $\begin{array}{l} | \int_{s}^{T} E^{q} (t) \int_{Ω} (| u_{t} |^{m (x) - 2} u_{t} u + | v_{t} |^{r (x) - 2} v_{t} v) d x d t | \\ ⩽ ε [c^{'} ({(E (0))}^{\frac{m_{1}}{2} - 1} + {(E (0))}^{\frac{m_{2}}{2} - 1}) + k^{'} ({(E (0))}^{\frac{r_{1}}{2} - 1} + {(E (0))}^{\frac{r_{2}}{2} - 1})] \int_{s}^{T} E^{q + 1} (t) d t \\ (4.19) & + \int_{s}^{T} E^{q} (t) \int_{Ω} c_{ε} (x) | u_{t} |^{m (x)} d x d t + \int_{s}^{T} E^{q} (t) \int_{Ω} k_{ε} (x) | v_{t} |^{r (x)} d x d t . \end{array}$ At this point, we consider two cases.

Case 1. $r_{2} = m_{2} = 2$ ( $r (\cdot) \equiv m (\cdot) \equiv 2$ ).

In this case, a combination of (4.7)- (4.11), (4.14), (4.18), and (4.19) leads to $\begin{array}{rcl} 2 \int_{s}^{T} E^{q + 1} (t) d t & ⩽ & \tilde{C} E (s) + (c^{'} + k^{'}) ε \int_{s}^{T} E^{q + 1} (t) d t \\ + \int_{s}^{T} E^{q} (t) \int_{Ω} c_{ε} (x) | u_{t} |^{m (x)} d x d t + \int_{s}^{T} E^{q} (t) \int_{Ω} k_{ε} (x) | v_{t} |^{r (x)} d x d t . \end{array}$ We choose ε so small so that $(c^{'} + k^{'}) ε < 1$ . Once ε is fixed, then $c_{ε} (x) ⩽ M$ and $k_{ε} (x) ⩽ M$ since $m (x)$ and $r (x)$ are bounded. Thus, we arrive at $\begin{array}{rcl} \int_{s}^{T} E^{q + 1} (t) d t & ⩽ & \tilde{c} E (s) + M \int_{s}^{T} E^{q} (t) \int_{Ω} | u_{t} |^{m (x)} d x d t + M \int_{s}^{T} E^{q} (t) \int_{Ω} | v_{t} |^{r (x)} d x d t \\ ⩽ & \tilde{c} E (s) - 2 M \int_{s}^{T} E^{q} (t) E^{'} (t) d t \\ ⩽ & \tilde{c} E (s) + \frac{2 M}{q + 1} E^{q + 1} (s) . \end{array}$ It suffices to take $q = 0$ , to arrive at $\begin{array}{rcl} \int_{s}^{T} E (t) d t & ⩽ & \tilde{c_{1}} E (s) . \end{array}$ By taking $T \to \infty$ , we get $\begin{matrix} \int_{s}^{\infty} E (t) d t ⩽ \tilde{c_{1}} E (s), \forall s > 0 . \end{matrix}$ Hence, Komornik’s lemma implies the exponential decay result.

Case 2. $m_{2} \neq r_{2}$ .

Without loss of generality, we assume that $m_{2} > r_{2} ⩾ 2$ . At this point, we distinguish two subcases.

Subcase 2.1. $r_{2} = 2$ .

In this subcase, combining (4.7)–(4.10), (4.13), (4.14), (4.18), and (4.19) to get $\begin{array}{rcl} 2 \int_{s}^{T} E^{q + 1} (t) d t & ⩽ & c_{ε} E (s) + c ε (1 + {(E (0))}^{\frac{m_{1}}{2} - 1} + {(E (0))}^{\frac{m_{2}}{2} - 1}) \int_{s}^{T} E^{q + 1} (t) d t \\ + ε \int_{s}^{T} {(E (t))}^{\frac{q m_{2}}{m_{2} - 2}} d t + \int_{s}^{T} E^{q} (t) \int_{Ω} c_{ε} (x) | u_{t} |^{m (x)} d x d t \\ (4.20) & + \int_{s}^{T} E^{q} (t) \int_{Ω} k_{ε} (x) | v_{t} |^{r (x)} d x d t . \end{array}$ We, then, pick q such that $\frac{q m_{2}}{m_{2} - 2} = q + 1$ , that is, $q = \frac{m_{2}}{2} - 1$ . Hence, we arrive at $\begin{array}{rcl} 2 \int_{s}^{T} E^{q + 1} (t) d t & ⩽ & c_{ε} E (s) + c ε (1 + {(E (0))}^{\frac{m_{1}}{2} - 1} + {(E (0))}^{\frac{m_{2}}{2} - 1}) \int_{s}^{T} E^{q + 1} (t) d t \\ + \int_{s}^{T} E^{q} (t) \int_{Ω} c_{ε} (x) | u_{t} |^{m (x)} d x d t + \int_{s}^{T} E^{q} (t) \int_{Ω} k_{ε} (x) | v_{t} |^{r (x)} d x d t . \end{array}$ We choose $ε > 0$ so small that $\begin{matrix} c ε (1 + {(E (0))}^{\frac{m_{1}}{2} - 1} + {(E (0))}^{\frac{m_{2}}{2} - 1}) < 1 . \end{matrix}$ Once ε is fixed, then $c_{ε} (x) ⩽ M$ and $k_{ε} (x) ⩽ M$ since $m (x)$ and $r (x)$ are bounded. Thus, we obtain $\begin{array}{rcl} \int_{s}^{T} E^{q + 1} (t) d t & ⩽ & \tilde{c} E (s) + M \int_{s}^{T} E^{q} (t) \int_{Ω} | u_{t} |^{m (x)} d x d t + M \int_{s}^{T} E^{q} (t) \int_{Ω} | v_{t} |^{r (x)} d x d t \\ ⩽ & \tilde{c} E (s) - 2 M \int_{s}^{T} E^{q} (t) E^{'} (t) d t \\ ⩽ & \tilde{c} E (s) + \frac{2 M}{q + 1} E^{q + 1} (s) ⩽ c (E (s) + E^{q + 1} (s)) \\ ⩽ & c (1 + E^{q} (0)) E (s) = \tilde{c_{1}} E (s), \forall T > s > 0 . \end{array}$ By taking $T \to \infty$ , we get $\begin{matrix} \int_{s}^{\infty} E^{\frac{m_{2}}{2}} (t) d t ⩽ \tilde{c_{1}} E (s), \forall s > 0 . \end{matrix}$ Subcase 2.2. $r_{2} > 2$ .

In this subcase, combining (4.7)–(4.10), (4.13), (4.16), (4.18), and (4.19) to get $\begin{array}{rcl} 2 \int_{s}^{T} E^{q + 1} (t) d t & ⩽ & c_{ε} E (s) + ε \int_{s}^{T} {(E (t))}^{\frac{q m_{2}}{m_{2} - 2}} d t + ε \int_{s}^{T} {(E (t))}^{\frac{q r_{2}}{r_{2} - 2}} d t \\ ⩽ & ε [c^{'} ({(E (0))}^{\frac{m_{1}}{2} - 1} + {(E (0))}^{\frac{m_{2}}{2} - 1}) \\ + k^{'} ({(E (0))}^{\frac{r_{1}}{2} - 1} + {(E (0))}^{\frac{r_{2}}{2} - 1})] \int_{s}^{T} E^{q + 1} (t) d t \\ + \int_{s}^{T} E^{q} (t) \int_{Ω} c_{ε} (x) | u_{t} |^{m (x)} d x d t + \int_{s}^{T} E^{q} (t) \int_{Ω} k_{ε} (x) | v_{t} |^{r (x)} d x d t . \end{array}$ Again, we take q such that $\frac{q m_{2}}{m_{2} - 2} = q + 1$ , that is, $q = \frac{m_{2}}{2} - 1$ . This implies $\frac{q r_{2}}{r_{2} - 2} > q + 1$ . Hence, we arrive at $\begin{array}{l} 2 \int_{s}^{T} E^{q + 1} (t) d t \\ ⩽ c ε (2 + {(E (0))}^{\frac{m_{1}}{2} - 1} + {(E (0))}^{\frac{m_{2}}{2} - 1} + {(E (0))}^{\frac{r_{1}}{2} - 1} + {(E (0))}^{\frac{r_{2}}{2} - 1}) \int_{s}^{T} E^{q + 1} (t) d t \\ + c_{ε} E (s) + \int_{s}^{T} E^{q} (t) \int_{Ω} c_{ε} (x) | u_{t} |^{m (x)} d x d t + \int_{s}^{T} E^{q} (t) \int_{Ω} k_{ε} (x) | v_{t} |^{r (x)} d x d t . \end{array}$ We then pick $ε > 0$ so small that $\begin{matrix} c ε (2 + {(E (0))}^{\frac{m_{1}}{2} - 1} + {(E (0))}^{\frac{m_{2}}{2} - 1} + {(E (0))}^{\frac{r_{1}}{2} - 1} + {(E (0))}^{\frac{r_{2}}{2} - 1}) < 1 . \end{matrix}$ Once ε is fixed, then $c_{ε} (x) ⩽ M$ and $k_{ε} (x) ⩽ M$ since $m (x)$ and $r (x)$ are bounded. Thus, we arrive at $\begin{array}{rcl} \int_{s}^{T} E^{q + 1} (t) d t & ⩽ & \tilde{c} E (s) + M \int_{s}^{T} E^{q} (t) \int_{Ω} | u_{t} |^{m (x)} d x d t \\ + M \int_{s}^{T} E^{q} (t) \int_{Ω} | v_{t} |^{r (x)} d x d t \\ ⩽ & \tilde{c} E (s) - 2 M \int_{s}^{T} E^{q} (t) E^{'} (t) d t \\ ⩽ & \tilde{c} E (s) + \frac{2 M}{q + 1} E^{q + 1} (s) \\ ⩽ & c (E (s) + E^{q + 1} (s)) \\ ⩽ & c (1 + E^{q} (0)) E (s) \\ = & \tilde{c_{1}} E (s), \forall T > s > 0 . \end{array}$ By taking $T \to \infty$ , we get $\begin{matrix} \int_{s}^{\infty} E^{\frac{m_{2}}{2}} (t) d t ⩽ \tilde{c_{1}} E (s), \forall s > 0 . \end{matrix}$ Hence, Komornik’s lemma implies the desired result. This completes the proof. □
Remark 4.1.
In our work, we considered exponents depending on the space variable only. An interesting and open question is to consider exponents depending on both x and t, and establish similar results under reasonable conditions.

5. Numerical tests

In this section, some numerical experiments have been performed to illustrate the theoretical results in Theorem 4.2. We solve the system (P) under specific initial data and Dirichlet boundary conditions. The problem (P) was discritized using a $P 1$ -finite-element method in space and Newmark method in time.

The spatial interval $Ω = [0, 1]$ is subdivided into 500 subintervals, where the spatial step $Δ x = 0.002$ , and the temporal interval $[0, T] = [0, 20]$ with a time step $Δ t = 10^{- 3}$ is deduced from the stability condition of Newmark method [33]. We run our code for 10000 time steps using the following initial conditions: $\begin{array}{rcl} (5.1) & \begin{array}{l} u (x, 0) = sin (π x) and v (x, 0) = - sin (π x) in [0, 1] \\ u_{t} (x, 0) = 1 and v_{t} (x, 0) = 1 in [0, 1] \end{array} \end{array}$ We consider three tests for the choice of functions $m (x)$ and $r (x)$ , where $\begin{matrix} p (x) = 2 - \frac{1}{1 + x} and a = b = 1 . \end{matrix}$ Here are the preformed tests.

Test 1: Based on the results of Theorem 4.2, we obtained an exponential decay of the energy: $\begin{matrix} E_{1} (t) ⩽ E_{f}^{1} (t) = c e^{- α t}, \end{matrix}$ for two positive constants c depending on $E_{1} (0)$ and α. For this test, we used the functions: $\begin{matrix} m (x) = 2 and r (x) = 2 . \end{matrix}$

Test 2: In test 2, we examine the second case of Theorem 4.2, where we have a polynomial decay of the energy: $\begin{matrix} E_{2} (t) ⩽ E_{f}^{2} (t) = \frac{c}{{(1 + t)}^{2 / (σ - 2)}}, \end{matrix}$ for two positive constants c depending on $E_{2} (0)$ and $σ = max {m (\cdot), r (\cdot)}$ . For this test, we used the functions: $\begin{matrix} m (x) = 2 and r (x) = 2 + \frac{1}{1 + x} . \end{matrix}$

Test 3: In test 3, we examined again the second case of Theorem 4.2 with different choices of m and r, where we have a polynomial decay of the energy: $\begin{matrix} E_{3} (t) ⩽ E_{f}^{3} (t) = \frac{c}{{(1 + t)}^{2 / (σ - 2)}}, \end{matrix}$ for two positive constants c depending on $E_{3} (0)$ and $σ = max {m (\cdot), r (\cdot)}$ . For this test, we used the functions: $\begin{matrix} m (x) = 2 + \frac{1}{1 + x} and r (x) = 2 + \frac{1}{1 + x} . \end{matrix}$

Fig. 1.

Test 1: Plot of the approximate solutions (left) $u (x, t)$ and (right) $v (x, t)$ .

Fig. 2.

Test 1: Damping cross section waves and energy decay.

Fig. 3.

Test 2: Plot of the approximate solutions (left) $u (x, t)$ and (right) $v (x, t)$ .

Fig. 4.

Test 2: Damping cross section waves and energy decay.

Fig. 5.

Test 3: Plot of the approximate solutions (left) $u (x, t)$ and (right) $v (x, t)$ .

Fig. 6.

Test 3: Damping cross section waves and energy decay.

The computational simulations show the decay types for the proposed tests. For Test 1, Fig. 1 represents the approximate solution $(u, v)$ of the system (P) in time. While, Fig. 2 (left) shows three cross-section cuts for the approximate solution $(u, v)$ at $x = 0.25$ , 0.5 and $x = 0.75$ ; and Fig. 2 (right) represents the energy functional $E_{1} (t)$ and the corresponding upper bound $E_{f}^{1} (t)$ . The same process was followed for Test 2 (Fig. 3 and Fig. 4) and Test 3 (Fig. 5 and Fig. 6). From these numerical tests, we observed that the obtained decay results are compatible with the theoretical results (Theorem 4.2).

Footnotes

Acknowledgements

The authors thank University of Sharjah, Birzeit University and KFUPM for their support. This work is partially funded by KFUPM under project No. SB191048.

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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On the existence and stability of a nonlinear wave system with variable exponents

Abstract

Keywords

1. Introduction

2. Preliminaries

Lemma 2.1 ([8]).

Lemma 2.2 ([8]).

Lemma 2.3 (Hölder’s Inequality [8]).

Lemma 2.4. If 1 < p 1 ⩽ p ( x ) ⩽ p 2 < ∞ holds, then (2.1) ∫ Ω | v | p ( · ) d x ⩽ ‖ v ‖ p 1 p 1 + ‖ v ‖ p 2 p 2 , for any v ∈ L p ( · ) ( Ω ) . Lemma 2.5 (Poincaré’s Inequality [8]).

Lemma 2.6 ([8]).

3. Existence

Lemma 4.1 (Komornik [16]).

Footnotes

Acknowledgements

Data availability

References

Lemma 2.4.
If $1 < p_{1} ⩽ p (x) ⩽ p_{2} < \infty$ holds, then $\begin{matrix} (2.1) & \int_{Ω} | v |^{p (\cdot)} d x ⩽ ‖ v ‖_{p_{1}}^{p_{1}} + ‖ v ‖_{p_{2}}^{p_{2}}, \end{matrix}$ for any $v \in L^{p (\cdot)} (Ω)$ .
Lemma 2.5 (Poincaré’s Inequality [8]).