Open Access

Young measure solutions for a fourth-order wave equation with variable growth

Boundary Value Problems20152015:123

https://doi.org/10.1186/s13661-015-0386-5

Received: 22 May 2015

Accepted: 3 July 2015

Published: 30 July 2015

Abstract

In this paper, we study the existence of Young measure solutions to a fourth-order wave equation with variable exponent nonlinearity on a bounded domain. The asymptotic behavior of the Young measure solutions is also investigated by applying a lemma developed by Nakao.

Keywords

nonlinear wave equationYoung measure solution: variable exponent spacenonlinear elastodynamicsGalerkin’s approximationasymptotic stability

MSC

35L3535B4035D99

1 Introduction

In this paper, we consider the initial boundary value problem of the following model:
$$\begin{aligned} & \frac{\partial^{2}u}{\partial t^{2}}+\Delta\bigl(|\Delta u|^{p(x)-2}\Delta u \bigr)+|u|^{q(x)-2}u+a \biggl(\int_{\Omega}|u|^{q(x)}\,dx \biggr)\frac{\partial u}{\partial t}=f(x,t), \quad (x,t)\in Q_{T}, \\ &u=\Delta u=0, \quad (x,t)\in\partial\Omega\times(0,T), \\ &u(x,0)=u_{0}(x), \qquad\frac{\partial u(x,0)}{\partial t}=u_{1}(x), \quad x\in \Omega, \end{aligned}$$
(1.1)
where \(\Omega\subset\mathbb{R}^{N}\) (\(N\geq3\)) is a bounded domain with smooth boundary Ω, \(0< T<\infty\) is a given constant, and \(Q_{T}=\Omega\times(0,T)\). The coefficient \(a:[0,\infty)\rightarrow (0,\infty)\) and the exponents \(p,q:\overline{\Omega}\rightarrow(1,\infty )\) are given continuous functions and \(f:Q_{T}\rightarrow\mathbb{R}\). PDEs with variable exponent growth conditions are usually called equations with nonstandard growth conditions. After Kováčik and Rákosník first discussed the variable exponent Lebesgue space \(L^{p(x)}\) and Sobolev space \(W^{k,p(x)}\) in [1], a lot of research has been done concerning these kinds of variable exponent spaces; see for example [2, 3] for the properties of such spaces and [413] for the applications of variable exponent spaces on partial differential equations. In [14] Růžička presented the mathematical theory for the application of variable exponent spaces in electro-rheological fluids. Problems with variable exponent growth conditions also appear in the mathematical modeling of stationary thermo-rheological viscous flows of non-Newtonian fluids [15] and nonlinear elastics [1618]. Another field of application of equations with variable exponent growth conditions is image restoration [19].
We claim that the Young measure solutions of problem (1.1) can be approximated by the following problem with a viscosity term \(\varepsilon \Delta^{2}\frac{\partial u}{\partial t}\) (\(\varepsilon>0\)):
$$\begin{aligned} & \frac{\partial^{2}u}{\partial t^{2}} +\Delta\bigl(|\Delta u|^{p(x)-2}\Delta u \bigr)+|u|^{q(x)-2}u \\ &\quad{}+a \biggl(\int_{\Omega}|u|^{q(x)}\,dx \biggr)\frac{\partial u}{\partial t}+\varepsilon\Delta^{2}\frac{\partial u}{\partial t}=f(x,t),\quad (x,t)\in Q_{T}, \\ &u=\Delta u=0,\quad (x,t)\in\partial\Omega\times(0,T), \\ &u(x,0)=u_{0}(x),\qquad \frac{\partial u(x,0)}{\partial t}=u_{1}(x),\quad x\in\Omega. \end{aligned}$$
(1.2)
When \(p(x)\equiv2\) and the space dimension \(N=1\), problems of the type (1.2) are a class of essential fourth-order wave equations appearing in elastoplastic-microstructure models. They govern the longitudinal motion of an elastoplastic bar and antiplane shearing deformation; see [20]. For \(p(x)\equiv2\) and the multidimensional case, Chen and Yang [21] discussed the global existence, asymptotic behavior and blow-up of solutions to the initial boundary problem of the equation with weak damping term \(\frac{\partial u}{\partial t}\); see also Messaoudi [22] for wave equations with nonlinear damping. For the analysis of nonlinear second-order hyperbolic equations with damping, we refer to the seminal work of Lions and Strauss [23]; see also Friedman and Nec̆as [24], and Emmrich and Thalhammer [25, 26]. In recent years, hyperbolic equations with variable exponent growth conditions were studied by Antontsev in [27], Haehnle and Prohl in [28], Pinasco in [29]; see also Autuori et al. in [30, 31] for the Kirchhoff equations with \(p(x)\)-growth. It is to be noted here that the viscosity term \(\Delta^{2}\frac{\partial u}{\partial t}\) plays a key role in the proof of the global existence. The global existence results of weak solutions for second-order wave equations (even if \(p(x)\equiv \mathrm{constant}\neq2\)) without the viscosity term \(\Delta^{2}\frac{\partial u}{\partial t}\) have been found only in one space dimension; see DiPerna [32] and Shearer [33]. To the best of our knowledge, the equations without the viscosity term are studied only in [3437]. In that work, the concept of Young measure solutions has been introduced and applied to dynamic problems and wave equations.

Thus motivated, in the present paper, we prove the global existence of Young measure solutions of problem (1.1), we first construct Young measure solutions as the limit of the sequence of solutions of problem (1.2). Then we give a decay estimate to the Young measure solutions of problem (1.1).

Our work is organized as follows. In Section 2, we give some necessary definitions and properties of variable exponent Lebesgue spaces and Sobolev spaces. In Section 3, we obtain the existence of weak solutions of problem (1.2) by Galerkin’s approximation method. In Section 4, under some conditions, from the sequence of solutions of problem (1.2) and some a priori estimates, we get the existence of Young measure solutions by letting \(\varepsilon\rightarrow0\). In Section 5, we investigate the decay property of Young measure solutions and get a decay rate estimate by using Nakao’s lemma.

2 Preliminaries

In this section, we first recall some necessary properties of variable exponent Lebesgue spaces and Sobolev spaces; see [13] for the details.

Let \(\Omega\subset\mathbb{R}^{N}\) be a domain. A measurable function \(p:\Omega\rightarrow[1,\infty)\) is called a variable exponent and we define \(p^{-}=\operatorname{ess}\inf_{x\in\Omega}p(x)\) and \(p^{+}=\operatorname{ess}\sup_{x\in\Omega}p(x)\). If \(p^{+}\) is finite, then the exponent p is said to be bounded. The variable exponent Lebesgue space is
$$L^{p(x)}(\Omega)=\biggl\{ u:\Omega\rightarrow\mathbb{R}\mbox{ is a measurable function}; \rho_{p(x)}(u)=\int_{\Omega}\bigl|u(x)\bigr|^{p(x)}\,dx< \infty \biggr\} $$
with the Luxemburg norm
$$\|u\|_{L^{p(x)}(\Omega)}=\inf\bigl\{ \lambda>0 : \rho_{p(x)}\bigl(\lambda ^{-1}u\bigr)\leq1\bigr\} , $$
then \(L^{p(x)}(\Omega)\) is a Banach space, and when p is bounded, we have the following relations:
$$\begin{aligned} \min\bigl\{ \|u\|^{p^{-}}_{L^{p(x)}(\Omega)},\|u\|^{p^{+}}_{L^{p(x)}(\Omega)} \bigr\} \leq\rho_{p(x)}(u) \leq\max\bigl\{ \|u\|^{p^{-}}_{L^{p(x)}(\Omega)}, \|u\|^{p^{+}}_{L^{p(x)}(\Omega )}\bigr\} . \end{aligned}$$
That is, if p is bounded, then norm convergence is equivalent to convergence with respect to the modular \(\rho_{p(x)}\). For a bounded exponent the dual space \((L^{p(x)}(\Omega))^{\prime}\) can be identified with \(L^{p^{\prime}(x)}(\Omega)\), where the conjugate exponent \(p^{\prime}(x)\) is defined by \(p^{\prime}(x)=\frac {p(x)}{p(x)-1}\) for each \(x\in\Omega\). If \(1< p^{-}\leq p^{+}<\infty\), then \(L^{p(x)}(\Omega)\) is separable and reflexive.
In the variable exponent Lebesgue space, Hölder’s inequality is still valid; see [1], Theorem 2.1. For all \(u\in L^{p(x)}(\Omega)\), \(v\in L^{p^{\prime}(x)}(\Omega)\) with \(p(x)\in(1,\infty)\) the following inequality holds:
$$\int_{\Omega}|uv|\,dx\leq \biggl(\frac{1}{p^{-}}+ \frac{1}{(p^{\prime})^{-}} \biggr)\|u\| _{L^{p(x)}(\Omega)}\|v\|_{L^{p^{\prime}(x)}(\Omega)} \leq2\|u \|_{L^{p(x)}(\Omega)}\|v\|_{L^{p^{\prime}(x)}(\Omega)}. $$

If \(0<|\Omega|<\infty\) and p, q are variable exponents such that \(p(x)\leq q(x)\) for each \(x\in\Omega\), then there exists a continuous embedding \(L^{p(x)}(\Omega)\hookrightarrow L^{q(x)}(\Omega)\).

Definition 2.1

(see [2])

We say that a bounded exponent \(p: \Omega\rightarrow\mathbb{R}\) is log-Hölder continuous if there is a constant \(C>0\) such that
$$\bigl|p(y)-p(z)\bigr|\log|y-z|\leq C $$
for all points \(y, z\in\Omega\).
The variable exponent Sobolev space \(W^{k,p(x)}(\Omega)\) is defined as
$$W^{k,p(x)}(\Omega)= \bigl\{ u\in L^{p(x)}(\Omega): D^{\alpha}u\in L^{p(x)}(\Omega), |\alpha|\leq k \bigr\} $$
and equipped with the norm
$$\|u\|_{W^{k,p(x)}(\Omega)}=\sum_{|\alpha|\leq k}\bigl\| D^{\alpha}u\bigr\| _{L^{p(x)}(\Omega)}, $$
then the space \(W^{k,p(x)}(\Omega)\) is a Banach space. The space \(W^{k,p(x)}_{0}(\Omega)\) is defined as the closure of \(C^{\infty }_{0}(\Omega)\) with the above norm. If \(1< p^{-}\leq p^{+}<\infty\), then the space \(W^{k,p(x)}(\Omega)\) is separable and reflexive; If \(p:\Omega \rightarrow(1,\infty)\) is a bounded log-Hölder continuous function, then \(C_{0}^{\infty}(\Omega)\) is dense in \(W_{0}^{k,p(x)}(\Omega)\).

Theorem 2.1

(see [2])

Let \(\Omega\subset\mathbb{R}^{N}\) be a bounded domain and assume that \(p: \mathbb{R}^{N}\rightarrow(1, \infty)\) is a bounded log-Hölder continuous exponent such that \(p^{-}>1\), then for any \(u\in W^{1, p(x)}_{0}(\Omega)\) we have
$$\|u\|_{L^{p(x)}(\Omega)}\leq c \|\nabla u\|_{L^{p(x)}(\Omega)}, $$
where the constant c only depends on the dimension N, \(|\Omega|\) and the log-Hölder constant of p.

Theorem 2.2

(see [2])

Let Ω be a bounded domain with smooth boundary. Assume that \(p:\Omega\rightarrow(1,\infty)\) is a bounded log-Hölder continuous function with \(p^{+}<\frac{N}{k}\) and \(q:\Omega\rightarrow(1,\infty)\) is a bounded measurable function with \(q(x)\leq p^{*}=\frac {Np(x)}{N-kp(x)}\). Then there is a continuous embedding
$$\begin{aligned} W^{k,p(x)}(\Omega)\hookrightarrow L^{q(x)}(\Omega), \end{aligned}$$
where the embedding constant depends on \(|\Omega|\), N, \(q^{+}\) and the log-Hölder constant of p.

Theorem 2.3

(see [38])

Let \(\Omega\subset\mathbb{R}^{N}\) be a bounded domain with smooth boundary. Suppose that p is a bounded log-Hölder continuous functions in Ω, with \(p^{-}>1\). Then there exists a constant \(C>0\) depending only on N, Ω and the log-Hölder constant of p such that for each \(u\in W_{0}^{2,p(x)}(\Omega)\), the following inequality holds:
$$\begin{aligned} \|u\|_{W_{0}^{2,p(x)}(\Omega)}\leq C\|\Delta u\|_{L^{p(x)}(\Omega)}. \end{aligned}$$

Proposition 2.1

(see [29, 39])

Let Ω be a bounded domain in \(\mathbb{R}^{N}\) and let \(\{\omega_{i}\} _{i=1}^{\infty}\) be an orthogonal basis in \(L^{2}(\Omega)\). Then for any \(\varepsilon>0\), there exists a positive number \(N_{\varepsilon}\) such that
$$\begin{aligned} \|u\|_{L^{2}(\Omega)}\leq \Biggl(\sum^{N_{\varepsilon}}_{i=1} \biggl(\int_{\Omega}u\omega_{i}\,dx \biggr)^{2} \Biggr)^{\frac{1}{2}}+\varepsilon\|u\| _{W_{0}^{1,p}(\Omega)} \end{aligned}$$
for all \(u\in W_{0}^{1,p}(\Omega)\) where \(2\leq p<\infty\).

The following theorem gives a relation between almost everywhere convergence and weak convergence.

Theorem 2.4

(see [7])

Let \(p: \Omega\rightarrow\mathbb{R}\) be a bounded log-Hölder continuous function with \(p^{-}>1\). If \(\{u_{n}\}^{\infty}_{n=1}\) is bounded in \(L^{p(x)}(Q_{T})\) and \(u_{n}\rightarrow u\) a.e. in \(Q_{T}\) as \(n\rightarrow\infty\), then there exists a subsequence of \(\{ u_{n}\}\) still denoted by \(\{u_{n}\}\) such that \(u_{n}\rightharpoonup u\) weakly in \(L^{p(x)}(Q_{T})\) as \(n\rightarrow\infty\).

Denote by \(C_{0}(\mathbb{R}^{N})\) (\(N\geq1\)) the closure of continuous functions in \(\mathbb{R}^{N}\) with compact support. The dual of \(C_{0}(\mathbb{R}^{N})\) can be identified with the space \(\mathcal {M}(\mathbb{R}^{N})\) of signed Radon measures with finite mass via the pairing
$$\begin{aligned} \langle\mu,f\rangle=\int_{\mathbb{R}^{N}}f\,d\mu. \end{aligned}$$
A map \(\mu:E\rightarrow\mathcal{M}(\mathbb{R}^{N})\) (\(E\subset\mathbb {R}^{N}\)) is called weak measurable if the functions \(x\rightarrow \langle\mu(x),f\rangle\) are measurable for all \(f\in C_{0}(\mathbb {R}^{N})\). We write \(\mu_{x}\) instead of \(\mu(x)\).

Theorem 2.5

(see [40], Theorem 3.1)

Let \(E\subset\mathbb{R}^{N}\) be a measurable set of finite measure and let \(z_{j}:E\rightarrow\mathbb{R}^{N}\) be a sequence of measurable functions. Then there exist a subsequence \(z_{j_{k}}\) and a weak measurable map \(\nu:E\rightarrow\mathcal{M}(\mathbb{R}^{N})\) such that
  1. (i)

    \(\nu_{x}\geq0\), \(\|\nu_{x}\|_{\mathcal{M}(\mathbb{R}^{N})}= \int_{\mathbb{R}^{d}}\,d\nu_{x}\leq1\), for a.e. \(x\in E\).

     
  2. (ii)
    For all \(f\in C_{0}(\mathbb{R}^{N})\)
    $$\begin{aligned} f(z_{j_{k}})\rightharpoonup\langle\nu_{x},f\rangle \quad\textit{weakly}*\textit{ in } L^{\infty}(E). \end{aligned}$$
     
  3. (iii)
    Let \(K\subset\mathbb{R}^{N}\) be compact. Then
    $$\begin{aligned} \operatorname{supp}\nu_{x}\subset K \quad\textit{if } \operatorname{dist}(z_{j_{k}},K) \rightarrow 0 \textit{ in measure}. \end{aligned}$$
     
  4. (iv)
    Furthermore one has
    $$\begin{aligned} \bigl(\mathrm{i}^{\prime}\bigr)\quad \|\nu_{x} \|_{\mathcal{M}(\mathbb{R}^{N})}=1, \quad\textit{for a.e. } x\in E \end{aligned}$$
    if and only if the sequence does not escape to infinity, i.e. if
    $$\begin{aligned} \lim_{L\rightarrow\infty}\sup_{k}\bigl|\{x\in E:|z_{j_{k}}|\geq L\}\bigr|=0. \end{aligned}$$
    (2.1)
     
  5. (v)

    If (i′) holds, if \(A\subset E\) is measurable and \(f\in C(\mathbb{R}^{N})\) and if \(f(z_{j_{k}})\) is relatively weakly compact in \(L^{1}(A)\), then \(f(z_{j_{k}})\rightharpoonup\langle\nu_{x},f\rangle\) weakly in \(L^{1}(A)\).

     
  6. (vi)

    If (i′) holds, then in (iii) one can replace ‘if’ by ‘if and only if’.

     

Remark 2.1

If for some \(s>0\) and all \(j\in\mathbb{N}\)
$$\begin{aligned} \int_{E} |z_{j}|^{s}\leq C \end{aligned}$$
then (2.1) holds.

3 Existence of weak solutions of problem (1.2)

In this section, let \(\varepsilon\in(0,1)\) fixed, we prove the existence of weak solutions for problem (1.2). Our main hypotheses are the following:
  1. (H1)
    \(p,q:\overline{\Omega}\rightarrow(1,\infty)\) are two log-Hölder continuous functions satisfying
    $$\begin{aligned} \max \biggl\{ 1,\frac{2N}{N+4} \biggr\} < p^{-}=\inf_{\overline{\Omega }}p(x) \leq p^{+}=\sup_{\overline{\Omega}}p(x)< \frac{N}{2} \end{aligned}$$
    and
    $$\begin{aligned} 1< q^{-}=\inf_{\overline{\Omega}}q(x)\leq q(x)< \frac{Np(x)}{N-2p(x)}, \quad \mbox{for } x\in\overline{\Omega}. \end{aligned}$$
    Here \(\overline{\Omega}\) denotes the closure of Ω.
     
  2. (H2)
    \(a\in C([0,\infty))\) and there exists a constant \(a_{0}>0\) such that
    $$\begin{aligned} a(s)\geq a_{0}>0, \quad\mbox{for } s\in[0,\infty). \end{aligned}$$
     
  3. (H3)

    \(u_{0}\in W^{2,p(x)}(\Omega)\cap W_{0}^{2,2}(\Omega)\), \(u_{1}\in L^{2}(\Omega)\), \(f\in L^{2}(Q_{T})\).

     

Definition 3.1

A function \(u_{\varepsilon}:Q_{T}\rightarrow\mathbb{R}\) is called a weak solution of (1.2), if
$$\begin{aligned} \textstyle\begin{cases} u_{\varepsilon}\in L^{\infty}(0,T; W_{0}^{2,p(x)}(\Omega) )\cap L^{\infty}(0,T;L^{q(x)}(\Omega))\cap C (0,T;W_{0}^{2,2}( \Omega) ),\\ \frac{\partial u_{\varepsilon}}{\partial t}\in L^{\infty}(0,T;L^{2}(\Omega) ) \cap L^{2} (0,T;W_{0}^{2,2}(\Omega)), \end{cases}\displaystyle \end{aligned}$$
and
$$\begin{aligned} &{-}\int_{\Omega}\frac{\partial u_{\varepsilon}(x,\tau)}{\partial t} \varphi (x,\tau)\,dx-\int _{Q_{T}}\frac{\partial u_{\varepsilon}}{\partial t}\frac {\partial\varphi}{\partial t} \,dx\,dt+\int _{Q_{T}} |\Delta u_{\varepsilon}|^{p(x)-2}\Delta u_{\varepsilon}\Delta\varphi\\ &\qquad{}+\varepsilon\Delta\frac {\partial u_{\varepsilon}}{\partial t}\Delta\varphi\,dx\,dt +\int_{Q_{T}} a \biggl(\int_{\Omega}|u_{\varepsilon}|^{q(x)}\,dx \biggr)\frac {\partial u_{\varepsilon}}{\partial t}\varphi \,dx\,dt\\ &\quad=\int_{Q_{T}}f\varphi \,dx\,dt+\int_{\Omega}u_{1}\varphi(x,0)\,dx, \end{aligned}$$
for all \(\varphi\in C^{1}(0,T;C_{0}^{\infty}(\Omega))\) and \(\tau\in(0,T]\).

We choose a sequence \(\{\omega_{j}\}_{j=1}^{\infty}\subset C_{0}^{\infty}(\Omega )\) such that \(C_{0}^{\infty}(\Omega)\subset\overline{\bigcup_{n=1}^{\infty}V_{n}}^{C^{2}(\bar{\Omega})} \) and \(\{\omega_{j}\}_{j=1}^{\infty}\) is a complete orthogonal basis in \(L^{2}(\Omega)\), where \(V_{n}=\operatorname{span}\{ \omega_{1}, \omega_{2}, \ldots, \omega_{n}\}\); see [7, 39].

Since \(\bigcup_{n=1}^{\infty}V_{n}\) is dense in \(C^{2}(\bar{\Omega})\), we have the following lemma.

Lemma 3.1

(see [41])

For the function \(u_{0}\in W^{2,p(x)}(\Omega )\cap W_{0}^{2,2}(\Omega)\), there exists a sequence \(\psi_{n}\) with \(\psi _{n}\in V_{n}\) such that \(\psi_{n}\rightarrow u_{0}\) in \(W^{2,p(x)}(\Omega)\cap W_{0}^{2,2}(\Omega)\) as \(n\rightarrow\infty\).

Proof

For \(u_{0}\in W^{2,p(x)}(\Omega)\cap W_{0}^{2,2}(\Omega)\), there exists a sequence \(\{v_{n}\}\) in \(C_{0}^{\infty}(\Omega)\) such that \(v_{n}\rightarrow u_{0}\) in \(W^{2,p(x)}(\Omega)\cap W_{0}^{2,2}(\Omega)\). Since \(\{v_{n}\}\subset C_{0}^{\infty}(\Omega)\subset\overline{\bigcup_{m=1}^{\infty}V_{m}}^{C^{2}(\bar{\Omega})}\), we can find a sequence \(\{v_{n}^{k}\}\subset\bigcup_{m=1}^{\infty}V_{m}\) such that, for each \(n\in\mathbb{N}\), we have \(v_{n}^{k}\rightarrow u_{n}\) in \(C^{2}(\overline{\Omega})\) as \(k\rightarrow\infty\). For \(\frac{1}{2^{n}}\), there exists \(k_{n}\geq1\) such that \(\|v_{n}^{k_{n}}-u_{n}\|_{C^{2}(\overline{\Omega})}\leq\frac{1}{2^{n}}\). Thus
$$\begin{aligned} \bigl\| v_{n}^{k_{n}}-u_{0}\bigr\| _{W^{2,p(x)}(\Omega)\cap W_{0}^{2,2}(\Omega)}\leq C\bigl\| v_{n}^{k_{n}}-v_{n}\bigr\| _{C^{2}(\overline{\Omega})}+ \|v_{n}-u_{0}\|_{W^{2,p(x)}(\Omega )\cap W_{0}^{2,2}(\Omega)}. \end{aligned}$$
That is, \(v_{n}^{k_{n}}\rightarrow u_{0}\) in \(W^{2,p(x)}(\Omega)\cap W_{0}^{2,2}(\Omega)\) as \(n\rightarrow\infty\). Denote \(u_{n}=v_{n}^{k_{n}}\). Since \(u_{n}\in\bigcup_{m=1}^{\infty}V_{m}\), there exists \(V_{m_{n}}\) such that \(u_{n}\in V_{m_{n}}\); without loss of generality, we assume that \(V_{m_{1}}\subset V_{m_{2}}\) as \(m_{1}\leq m_{2}\). We assume that \(m_{1}>1\) and define \(\psi_{n}\) as follows: \(\psi_{n}(x)=0\), \(n=1,\ldots,m_{1}-1\); \(\psi_{n}=u_{1}\), \(n=m_{1},\ldots,m_{2}-1\); \(\psi_{n}=u_{2}\), \(n=m_{2},\ldots,m_{3}-1;\ldots \) , then we obtain the sequence \(\{\psi_{n}\}\) and \(\psi_{n}\rightarrow u_{0}\) in \(W^{2,p(x)}(\Omega)\cap W_{0}^{2,2}(\Omega)\) as \(n\rightarrow \infty\). □
The existence of weak solutions of problem (1.1) is proved by Galerkin’s approximation. We shall find the sequence of approximate solutions in the form
$$u_{n}(x,t)=\sum_{j=1}^{n}\bigl( \eta_{n}(t)\bigr)_{j}\omega_{j}(x). $$
The unknown functions \((\eta_{n}(t))_{j}\) are determined by ordinary differential equations in the following.
We first define a vector-valued function \(P_{n}(t,\mu,\nu): [0,T]\times \mathbb{R}^{n}\times\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}\) as follows:
$$\begin{aligned} & \bigl(P_{n}(t,\mu,\nu) \bigr)_{i} \\ &\quad=\int_{\Omega}\Biggl|\sum_{j=1}^{n} \mu_{j}\Delta\omega_{j}\Biggr|^{p(x)-2}\Biggl(\sum _{j=1}^{n}\mu _{j}\Delta \omega_{j}\Biggr)\Delta\omega_{i} +\Biggl|\sum _{j=1}^{n}\mu_{j}\Delta \omega_{j}\Biggr|^{q(x)-2}\Biggl(\sum_{j=1}^{n} \mu_{j}\Delta \omega_{j}\Biggr)\Delta\omega_{i}\,dx \\ &\qquad{}+\int_{\Omega}\varepsilon\sum_{j=1}^{n} \nu_{j}\Delta\omega_{j}\Delta\omega _{i}+a \Biggl(\int_{\Omega}\Biggl|\sum_{j=1}^{n} \nu_{j}\omega_{j}\Biggr|^{q(x)}\,dx \Biggr)\omega _{i}\,dx,\quad i=1,\ldots,n, \end{aligned}$$
where \(\mu=(\mu_{1},\ldots,\mu_{n})\) and \(\nu=(\nu_{1},\ldots,\nu_{n})\). Now we consider the following Cauchy problem of second-order ordinary differential equations:
$$\begin{aligned} \textstyle\begin{cases} \eta''(t)+P_{n} (t,\eta(t),\eta^{\prime}(t))=F_{n}(t),\\ \eta(0)=U_{0n},\qquad \eta^{\prime}(0)=U_{1n}, \end{cases}\displaystyle \end{aligned}$$
(3.1)
where \((U_{0n})_{i}=\int_{\Omega}\psi_{n}\omega_{i}\,dx\), \((U_{1n})_{i}=\int_{\Omega}\phi_{n}\omega_{i}\,dx\), \(F_{n}=\int_{\Omega}f_{n}\omega_{i}\,dx\), \(\psi_{n}\in V_{n}\), \(\phi_{n}\in V_{n}\), \(f_{n}\in C_{0}^{\infty}(Q_{T})\), and \(\psi_{n}\rightarrow u_{0}\) strongly in \(W^{2,p(x)}(\Omega)\cap W_{0}^{1,2}(\Omega)\) (\(\psi _{n}\) from Lemma 3.1), \(\phi_{n}\rightarrow u_{1}\) strongly in \(L^{2}(\Omega)\), \(f_{n}\rightarrow f\) strongly in \(L^{2}(Q_{T})\) (since \(C_{0}^{\infty}(Q_{T})\) is dense in \(L^{2}(Q_{T})\)).
Let \(\eta^{\prime}(t)=X(t)\), \(Y(t)= (\eta(t), X(t) )\), and \(H_{n}(t,Y)= (X,F_{n} -P_{n}(t,\eta,X) )\). Then the problem (3.1) is transformed into the following problem:
$$\begin{aligned} \textstyle\begin{cases} Y^{\prime}(t)=H_{n} \bigl(t,Y(t)\bigr),\\ Y(0)=(U_{0n},U_{1n}). \end{cases}\displaystyle \end{aligned}$$
(3.2)
The assumption (H2) implies
$$\begin{aligned} P_{n}(t,\eta,X)X={}&P_{n}\bigl(t,\eta,\eta^{\prime}\bigr)\eta^{\prime}\\ ={}&\int_{\Omega}|\Delta u_{n}|^{p(x)-2}\Delta u_{n}\Delta\frac{\partial u_{n}}{\partial t}+|u_{n}|^{q(x)-2}u_{n} \frac{\partial u_{n}}{\partial t}\,dx +\varepsilon\int_{\Omega}\Delta \frac{\partial u_{n}}{\partial t}\Delta \frac{\partial u_{n}}{\partial t}\,dx \\ &{}+\int_{\Omega}a \biggl(\int_{\Omega}|u_{n}|^{q(x)}\,dx \biggr)\biggl|\frac{\partial u_{n}}{\partial t}\biggr|^{2}\,dx \\ \geq{}& \frac{d}{dt}\int_{\Omega}\frac{|\Delta u_{n}|^{p(x)}}{p(x)}+ \frac {|u_{n}|^{q(x)}}{q(x)}\,dx+\varepsilon\int_{\Omega}\biggl|\Delta \frac{\partial u_{n}}{\partial t}\biggr|^{2}\,dx+a_{0}\int_{\Omega}\biggl| \frac{\partial u_{n}}{\partial t}\biggr|^{2}\,dx. \end{aligned}$$
From (3.2) and Young’s inequality, we obtain
$$\begin{aligned} &Y^{\prime}Y+\frac{d}{dt}\int_{\Omega}\frac{|\Delta u_{n}|^{p(x)}}{p(x)}+\frac{|u_{n}|^{q(x)}}{q(x)}\,dx+\varepsilon\int_{\Omega}\biggl|\Delta\frac{\partial u_{n}}{\partial t}\biggr|^{2}\,dx+\frac{a_{0}}{2}\int _{\Omega}\biggl|\frac{\partial u_{n}}{\partial t}\biggr|^{2}\,dx \\ &\quad\leq C \biggl(\frac{1}{2}|Y|^{2}+\int _{\Omega}\frac{|\Delta u_{n}|^{p(x)}}{p(x)}\,dx+\int_{\Omega}f_{n}^{2}\,dx \biggr). \end{aligned}$$
(3.3)
Thus,
$$\begin{aligned} \frac{d}{dt} \biggl(\frac{1}{2}|Y|^{2}+\int _{\Omega}\frac{|\Delta u_{n}|^{p(x)}}{p(x)}+\frac{|u_{n}|^{q(x)}}{q(x)}\,dx \biggr)\leq C \biggl(\frac {1}{2}|Y|^{2}+\int_{\Omega}\frac{|\Delta u_{n}|^{p(x)}}{p(x)}\,dx+\int_{\Omega}f_{n}^{2}\,dx \biggr). \end{aligned}$$
Gronwall’s inequality and \(f_{n}\rightarrow f\) strongly in \(L^{2}(Q_{T})\) imply
$$\begin{aligned} |Y|^{2}+\int_{\Omega}\frac{|\Delta u_{n}|^{p(x)}}{p(x)}\,dx\leq C, \end{aligned}$$
(3.4)
where C is a constant independent of n and ε. Thus, \(|Y-Y(0)|\leq2 \sqrt{C}\). We denote
$$L_{n}=\max_{(t,Y)\in[0, T]\times B (Y(0),2 \sqrt{C} )}\bigl|H_{n}(t,Y)\bigr|,\qquad \tau_{n}=\min \biggl\{ T, \frac{2 \sqrt{C}}{L_{n}} \biggr\} , $$
where \(B (Y(0),2 \sqrt{C} )\) is the ball of radius \(2 \sqrt{C}\) with center at the point \(Y(0)\) in \(\mathbb{R}^{2n}\). From the definition of \(H(t,Y)\), \(H(t,Y)\) is continuous with respect to \((t,Y)\). By Peano’s theorem, we know that (3.2) admits a \(C^{1}\) solution on \([0, \tau_{n}]\), that is, (3.1) has a \(C^{2}\) solution on \([0,\tau_{n}]\) denoted by \(\eta_{n}^{1}(t)\). Let \(\eta(\tau_{n})\), \(\frac{\partial\eta (\tau_{n})}{\partial t}\) be the initial value of problem (3.1), then we can repeat the above process and get a \(C^{2}\) solution \(\eta_{n}^{2}(t)\) on \([\tau_{n}, 2\tau_{n}]\). Without loss of generality, we assume that \(T=[\frac{T}{\tau_{n}}]\tau_{n}+(\frac{T}{\tau_{n}})\tau_{n}\), \(0<(\frac{T}{\tau _{n}})<1\), where \([\frac{T}{\tau_{n}}]\) is the integer part of \(\frac {T}{\tau_{n}}\), \((\frac{T}{\tau_{n}})\) is the decimal part of \(\frac{T}{\tau _{n}}\). We can divide \([0, T]\) into \([(i-1)\tau_{n},i\tau_{n}]\), \(i=1,\ldots,L\), and \([L\tau_{n},T]\) where \(L=[\frac{T}{\tau_{n}}]\), then there exists a \(C^{2}\) solution \(\eta^{i}_{n}(t)\) in \([(i-1)\tau_{n},i\tau_{n}]\), \(i=1,\ldots,L\), and \(\eta^{L+1}_{n}(t)\) in \([L\tau_{n},T]\). Therefore, we get a solution \(\eta_{n}(t) \in C^{2}([0, T])\) defined by
$$\begin{aligned} \eta_{n}(t)= \textstyle\begin{cases} \eta^{1}_{n}(t), &\mbox{if } t\in[0,\tau_{n}], \\ \eta^{2}_{n}(t), &\mbox{if } t\in(\tau_{n},2 \tau_{n}], \\ \cdots \\ \eta^{L}_{n}(t), &\mbox{if } t\in((L-1) \tau_{n},L\tau_{n}], \\ \eta^{L+1}_{n}(t), &\mbox{if } t\in(L\tau_{n},T]. \end{cases}\displaystyle \end{aligned}$$

Lemma 3.2

(A priori estimates)

The estimates
$$\begin{aligned} &\int_{\Omega}\biggl|\frac{\partial u_{n}(x,t)}{\partial t}\biggr|^{2}\,dx+\int _{\Omega}\bigl|\Delta u_{n}(x,t)\bigr|^{p(x)}+\bigl|u_{n}(x,t)\bigr|^{q(x)}\,dx+ \varepsilon\int_{\Omega}\bigl|\Delta u_{n}(x,t)\bigr|^{2}\,dx \leq C,\\ &\quad \forall t\in[0,T], \\ &\int_{Q_{T}}\biggl|\frac{\partial u_{n}}{\partial t}\biggr|^{2}\,dx\,dt+\int _{Q_{T}}|\Delta u_{n}|^{p(x)}+|u_{n}|^{q(x)}\,dx\,dt+ \varepsilon\int_{Q_{T}}\biggl|\Delta\frac{\partial u_{n}}{\partial t}\biggr|^{2}\,dx\,dt \leq C \end{aligned}$$
hold uniformly with respect to n.

Proof

By (3.4), we have
$$\begin{aligned} &\int_{\Omega}\bigl|u_{n}(x,t)\bigr|^{2}+\biggl| \frac{\partial u_{n}(x,t)}{\partial t}\biggr|^{2}\,dx+\int_{\Omega}\bigl|\Delta u_{n}(x,t)\bigr|^{p(x)}+\bigl|u_{n}(x,t)\bigr|^{q(x)}\,dx\leq C, \\ &\quad\mbox{for } t\in[0,T]. \end{aligned}$$
Further, integrating the inequality (3.3) with respect to t over \([0,T]\), we obtain
$$\begin{aligned} \int_{Q_{T}}\varepsilon\biggl|\Delta\frac{\partial u_{n}}{\partial t}\biggr|^{2}+\biggl| \frac {\partial u_{n}}{\partial t}\biggr|^{2}\,dx\,dt\leq C. \end{aligned}$$
Moreover, for each \(t\in[0,T]\),
$$\begin{aligned} \int_{\Omega}\bigl|\Delta u_{n}(x,t)\bigr|^{2}\,dx\leq 2T\int_{\Omega}\int_{0}^{T}\biggl| \Delta\frac {\partial u_{n}}{\partial t}\biggr|^{2}\,dx\,dt+2\int_{\Omega}\bigl|\Delta u_{n}(x,0)\bigr|^{2}\,dx \leq\frac{C}{\varepsilon}. \end{aligned}$$
Thus, this lemma is proved □

By Lemma 3.2, we have the following.

Lemma 3.3

The estimate
$$\begin{aligned} &\|u_{n}\|_{L^{\infty}(0,T;W_{0}^{2,p(x)}(\Omega))}+ \bigl\| |\Delta u_{n}|^{p(x)-2} \Delta u_{n} \bigr\| _{L^{p^{\prime}(x)}(Q_{T})} \\ &\quad{}+ \bigl\| |u_{n}|^{q(x)-2}u_{n} \bigr\| _{L^{p^{\prime}(x)}(Q_{T})}+ \biggl\| a \biggl(\int_{\Omega}|u_{n}|^{q(x)}\,dx \biggr) \frac{\partial u_{n}}{\partial t} \biggr\| _{L^{2}(Q_{T})}\leq C \end{aligned}$$
holds uniformly with respect to n and ε.

Proof

By Theorem 2.3, we have
$$\begin{aligned} \|u_{n}\|_{W_{0}^{2,p(x)}(\Omega)}\leq C\|\Delta u_{n} \|_{L^{p(x)}(\Omega )}\leq C. \end{aligned}$$
Thus, \(\|u_{n}\|_{L^{\infty}(0,T;W_{0}^{2,p(x)}(\Omega))}\leq C\). By Lemma 3.2, we obtain
$$\begin{aligned} \int_{Q_{T}} \bigl|\bigl|\Delta u_{n}\bigr|^{p(x)-2}\nabla u_{n} \bigr|^{p^{\prime}(x)}\,dx\,dt \leq\int_{Q_{T}} |\nabla u_{n}|^{p(x)} \,dx\,dt\leq C. \end{aligned}$$
Thus,
$$\begin{aligned} & \bigl\| |\Delta u_{n}|^{p(x)-2}\nabla u_{n} \bigr\| _{L^{p^{\prime}(x)}(Q_{T})} \\ &\quad\leq \max \biggl\{ \biggl(\int_{\Omega}|\Delta u_{n}|^{p(x)}\,dx \biggr)^{\frac {p^{-}-1}{p^{-}}}, \biggl(\int _{\Omega}|\Delta u_{n}|^{p(x)}\,dx \biggr)^{\frac {p^{+}-1}{p^{+}}} \biggr\} \leq C. \end{aligned}$$
Similarly, \(\||u_{n}|^{q(x)-2}u_{n} \|_{L^{q^{\prime}(x)}(Q_{T})}\leq C\). Since \(a\in C([0,\infty))\) and \(\int_{\Omega}|u_{n}(x,t)|^{q(x)}\,dx\leq C\), we have
$$\biggl\| a \biggl(\int_{\Omega}|u_{n}|^{q(x)}\,dx \biggr)\frac{\partial u_{n}}{\partial t} \biggr\| _{L^{2}(Q_{T})}\leq C. $$
This lemma is proved. □

Theorem 3.1

Assume (H1)-(H3). Then for each \(\varepsilon\in(0,1)\) problem (1.2) has a weak solution.

Proof

By Lemma 3.2 and Lemma 3.3, there exist a subsequence of \(\{u_{n}\}\) (still denoted by \(\{u_{n}\}\)), \(u_{\varepsilon}\), ξ, η, and ζ such that
$$\begin{aligned} \textstyle\begin{cases} \frac{\partial u_{n}}{\partial t}\rightharpoonup \frac{\partial u_{\varepsilon}}{\partial t} & \mbox{weakly }* \mbox{ in } L^{\infty}(0, T; L^{2}(\Omega) ),\\ u_{n}\rightharpoonup u_{\varepsilon}&\mbox{weakly}* \mbox{ in } L^{\infty}(0,T;W_{0}^{2,p(x)}(\Omega)\cap L^{q(x)}(\Omega))\\ &\cap L^{\infty}(0,T;W_{0}^{2,2}(\Omega)),\\ \frac{\partial u_{n}}{\partial t}\rightharpoonup\frac {\partial u_{\varepsilon}}{\partial t} &\mbox{weakly in } L^{2}(0,T;W_{0}^{2,2}(\Omega)),\\ |\Delta u_{n}|^{p(x)-2}\Delta u_{n}\rightharpoonup\xi &\mbox{weakly in } L^{p'(x)}(Q_{T}),\\ |u_{n}|^{q(x)-2} u_{n}\rightharpoonup\eta & \mbox{ weakly in } L^{q'(x)}(Q_{T}),\\ a (\int_{\Omega}|u_{n}|^{q(x)}\,dx ) \frac{\partial u_{n}}{\partial t}\rightharpoonup\zeta &\mbox{weakly in } L^{2}(Q_{T}). \end{cases}\displaystyle \end{aligned}$$
Since \(u_{n}\in L^{\infty}(0,T;W_{0}^{1,2}(\Omega))\) and \(\frac{\partial u_{n}}{\partial t}\in L^{2}(Q_{T})\), by the Lions-Aubin lemma, there exists a subsequence of \(\{u_{n}\}\) still denoted by \(\{u_{n}\}\) such that \(u_{n}\rightarrow u_{\varepsilon}\) strongly in \(L^{2}(Q_{T})\) and a.e. on \(Q_{T}\). Further, \(|u_{n}|^{q(x)-2}u_{n}\rightarrow|u_{\varepsilon}|^{q(x)-2}u_{\varepsilon}\) a.e. on \(Q_{T}\). In view of Theorem 2.4, we obtain \(\eta=|u_{\varepsilon}|^{q(x)-2}u_{\varepsilon}\).

Next, we prove that there exists a subsequence of \(\{u_{n}\}\) (still denoted by \(\{u_{n}\}\)) such that \(\frac{\partial u_{n}}{\partial t}\rightarrow\frac{\partial u_{\varepsilon}}{\partial t}\) strongly in \(L^{2}(Q_{T})\).

Since \((\eta^{\prime}_{n}(t))_{j}=\int_{\Omega}\frac{\partial u_{n}}{\partial t}\omega_{j}\,dx\), by Lemma 3.2, \((\eta^{\prime}_{n}(t))_{j}\) is uniformly bounded on \([0,T]\). For \(\forall0\leq t_{1}< t_{2}\leq T\), integrating (3.1) with respect to t from \(t_{1}\) to \(t_{2}\), we have
$$\begin{aligned} &\int_{\Omega}\frac{\partial u_{n}(x,t_{1})}{\partial t}\omega_{j}\,dx-\int _{\Omega}\frac{\partial u_{n}(x,t_{2})}{\partial t}\omega_{j}\,dx+\int _{t_{1}}^{t_{2}}\int_{\Omega}|\Delta u_{n}|^{p(x)-2}\Delta u_{n}\Delta\omega _{j}+|u_{n}|^{q(x)-2}u_{n} \omega_{j} \\ &\quad{}+\varepsilon\Delta\frac{\partial u_{n}}{\partial t}\Delta\omega_{j}+a \biggl(\int _{\Omega}|u_{n}|^{q(x)}\,dx \biggr) \frac{\partial u_{n}}{\partial t}\omega _{j}\,dx\,dt=\int_{t_{1}}^{t_{2}} \int_{\Omega}f_{n}\omega_{j}\,dx\,dt. \end{aligned}$$
Hölder’s inequality, Lemma 3.2, and Lemma 3.3 imply
$$\begin{aligned} &\bigl|\bigl(\eta_{n}(t_{1})\bigr)_{j}-\bigl( \eta_{n}(t_{2})\bigr)_{j}\bigr| \\ &\quad\leq 2 \biggl(\bigl\| |\Delta u_{n}|^{p(x)-2}\Delta u_{n} \bigr\| _{L^{p^{\prime}(x)}(Q_{T})}\| \Delta\omega_{j}\|_{L^{p(x)}(Q^{t_{2}}_{t_{1}})} + \bigl\| |u_{n}|^{q(x)-2}u_{n}\bigr\| _{L^{q^{\prime}(x)}(Q_{T})}\| \omega_{j}\| _{L^{q(x)}(Q^{t_{2}}_{t_{1}})} \\ &\qquad{}+\biggl\| \Delta\frac{\partial u_{n}}{\partial t}\biggr\| _{L^{2}(Q_{T})} \|\Delta\omega_{j}\| _{L^{2}(Q^{t_{2}}_{t_{1}})}+ \biggl\| a\biggl(\int_{\Omega}|u_{n}|^{q(x)}\,dx \biggr)\frac{\partial u_{n}}{\partial t} \biggr\| _{L^{2}(Q_{T})}\|\omega_{j}\| _{L^{2}(Q^{t_{2}}_{t_{1}})} \biggr) \\ &\quad\leq C \bigl(\|\Delta\omega_{j}\|_{L^{p(x)}(Q^{t_{2}}_{t_{1}})}+\|\Delta\omega _{j}\|_{L^{2}(Q^{t_{2}}_{t_{1}})} +\|\omega_{j}\|_{L^{2}(Q^{t_{2}}_{t_{1}})}+\| \omega_{j}\| _{L^{q(x)}(Q^{t_{2}}_{t_{1}})} \bigr) \\ &\quad\leq\max \bigl\{ |t_{1}-t_{2}|^{\frac{1}{p^{-}}},|t_{1}-t_{2}|^{\frac{1}{p^{+}}}, |t_{1}-t_{2}|^{\frac{1}{2}},|t_{1}-t_{2}|^{\frac{1}{q^{-}}},|t_{1}-t_{2}|^{\frac {1}{q^{+}}} \bigr\} \\ &\qquad{}\times \biggl( \biggl(\int_{\Omega}|\Delta \omega_{j}|^{p(x)}\,dx \biggr)^{\frac {1}{p^{-}}}+ \biggl(\int _{\Omega}|\Delta\omega_{j}|^{p(x)}\,dx \biggr)^{\frac{1}{p^{+}}} + \biggl(\int_{\Omega}|\Delta \omega_{j}|^{2}\,dx \biggr)^{\frac{1}{2}}\\ &\qquad{}+ \biggl(\int _{\Omega}|\omega_{j}|^{q(x)}\,dx \biggr)^{\frac{1}{q^{-}}} + \biggl(\int_{\Omega}|\omega_{j}|^{q(x)}\,dx \biggr)^{\frac{1}{q^{+}}}+ \biggl(\int_{\Omega}| \omega_{j}|^{2}\,dx \biggr)^{\frac {1}{2}} \biggr) \\ &\quad\leq C(j)\max \bigl\{ |t_{1}-t_{2}|^{\frac{1}{p^{-}}},|t_{1}-t_{2}|^{\frac{1}{p^{+}}}, |t_{1}-t_{2}|^{\frac{1}{2}},|t_{1}-t_{2}|^{\frac{1}{q^{-}}},|t_{1}-t_{2}|^{\frac {1}{q^{+}}} \bigr\} , \end{aligned}$$
where \(Q^{t_{2}}_{t_{1}}=\Omega\times(t_{1},t_{2})\). Thus, the sequence \(\{(\eta _{n}(t))_{j}\}^{\infty}_{n=1}\) is uniformly bounded and equi-continuous for fixed j and arbitrary \(n\geq j\). By the Ascoli-Arzela theorem and the usual diagonal procedure, there exists a subsequence of \(\{(\eta_{n})_{j}\} \) still denoted by \(\{(\eta_{n})_{j}\}\) such that \((\eta_{n}(t))_{j}\) converges uniformly on \([0,T]\) to some continuous function \(\lambda^{\varepsilon}_{j}(t)\) for each fixed \(j=1,2,\ldots \) .
For \(r\leq n\) with \(r\in\mathbb{N}\), by Lemma 3.2, we have
$$\begin{aligned} \sum_{j=1}^{r}\bigl|\bigl(\eta_{n}^{\prime}(t) \bigr)_{j}\bigr|^{2}\leq\int_{\Omega}\biggl| \frac{\partial u_{n}}{\partial t}\biggr|^{2}\,dx\leq C,\quad \forall t\in[0,T]. \end{aligned}$$
Letting \(n\rightarrow\infty\), we get
$$\begin{aligned} \sum_{j=1}^{r}\bigl|\lambda^{\varepsilon}_{j}(t)\bigr|^{2} \leq C,\quad \forall t\in[0,T]. \end{aligned}$$
Then letting \(r\rightarrow\infty\), we obtain
$$\begin{aligned} \sum_{j=1}^{\infty}\bigl|\lambda^{\varepsilon}_{j}(t)\bigr|^{2} \leq C, \quad\forall t\in[0,T]. \end{aligned}$$
Set \(\overline{u}_{\varepsilon}(x,t)=\sum_{j=1}^{\infty}\lambda ^{\varepsilon}_{j}(t)\omega_{j}(x)\), then \(\sup_{0\leq t\leq T}\| \overline{u}_{\varepsilon}(x,t)\|_{L^{2}(\Omega)}\leq C(T)\) and, for each \(j\in\mathbb{N}\), we have
$$\begin{aligned} \lim_{n\rightarrow\infty}\int_{\Omega}\frac{\partial u_{n}}{\partial t} \omega_{j}\,dx=\int_{\Omega}\overline{u}_{\varepsilon}\omega_{j}\,dx \end{aligned}$$
(3.5)
uniformly on \([0,T]\). For each \(\delta_{1}>0\) and \(\phi\in L^{2}(\Omega)\), by the completeness of \(\{\omega_{j}\}\), there exists a \(m_{0}>0\) such that \(\|\phi-\sum_{i=1}^{m_{0}}(\int_{\Omega}\phi\omega_{i}\,dx)\omega_{i}\| _{L^{2}(\Omega)}\leq\delta_{1}\). Thus,
$$\begin{aligned} \biggl\vert \int_{\Omega}\biggl( \frac{\partial u_{n}}{\partial t}-\bar{u}_{\varepsilon}\biggr)\phi \,dx\biggr\vert \leq{}&\biggl\| \frac{\partial u_{n}}{\partial t}-\bar {u}_{\varepsilon}\biggr\| _{L^{2}(\Omega)}\Biggl\| \phi-\sum _{i=1}^{m_{0}}\biggl(\int_{\Omega}\phi \omega_{i}\,dx\biggr)\omega_{i}\Biggr\| _{L^{2}(\Omega)} \\ &{}+\Biggl\vert \int_{\Omega}\biggl(\frac{\partial u}{\partial t}-\bar {u}_{\varepsilon}\biggr)\sum_{i=1}^{m_{0}} \biggl(\int_{\Omega}\phi\omega_{i} \,dx\biggr)\omega _{i}\,dx\Biggr\vert \\ \leq{}& C\delta_{1}+\Biggl\vert \int_{\Omega}\biggl(\frac{\partial u_{n}}{\partial t}-\bar {u}_{\varepsilon}\biggr)\sum _{i=1}^{m_{0}}\biggl(\int_{\Omega}\phi \omega_{i} \,dx\biggr)\omega _{i}\,dx\Biggr\vert . \end{aligned}$$
(3.6)
For \(\delta_{1}>0\), by (3.5), there exists a \(M>0\) such that
$$\begin{aligned} \biggl\vert \int_{\Omega}\biggl(\frac{\partial u_{n}}{\partial t}- \bar{u}_{\varepsilon}\biggr)\omega_{i}\,dx\biggr\vert \leq \frac{\delta_{1}}{m_{0}}, \quad\mbox{for } n>M \mbox{ and } i=1,\ldots,m_{0}. \end{aligned}$$
By (3.6) and the Hölder inequality, we have
$$\begin{aligned} \biggl\vert \int_{\Omega}\biggl( \frac{\partial u_{n}}{\partial t}-\bar{u}_{\varepsilon}\biggr)\phi \,dx\biggr\vert \leq{}&\biggl\| \frac{\partial u_{n}}{\partial t}-\bar {u}_{\varepsilon}\biggr\| _{L^{2}(\Omega)}\Biggl\| \phi-\sum _{i=1}^{m_{0}}\biggl(\int_{\Omega}\phi \omega_{i}\,dx\biggr)\omega_{i}\Biggr\| _{L^{2}(\Omega)} \\ &{}+\biggl|\int_{\Omega}\biggl(\frac{\partial u}{\partial t}- \bar{u}_{\varepsilon}\biggr)\sum_{i=1}^{m_{0}} \biggl(\int_{\Omega}\phi\omega_{i} \,dx\biggr) \omega_{i}\,dx\biggr| \\ \leq{}& C\delta_{1}+\sum_{i=1}^{m_{0}}\biggl| \int_{\Omega}\phi\omega_{i} \,dx\biggr|\biggl|\int _{\Omega}\biggl(\frac{\partial u_{n}}{\partial t}-\bar{u}_{\varepsilon}\biggr) \omega _{i}\,dx\biggr| \\ \leq{}&\bigl(C+\|\phi\|_{L^{2}(\Omega)}\bigr)\delta_{1}, \quad\mbox{for } n>M. \end{aligned}$$
(3.7)
It follows from (3.7) and the arbitrariness of \(\delta_{1}\) that
$$\begin{aligned} \frac{\partial u_{n}}{\partial t}\rightharpoonup\overline {u}_{\varepsilon}\quad\mbox{weakly in } L^{2}(\Omega). \end{aligned}$$
(3.8)
uniformly on \([0,T]\) as \(n\rightarrow\infty\). For each \(\varphi\in C_{0}^{\infty}(Q_{T})\), by Lebesgue’s dominated convergence theorem, we obtain
$$\begin{aligned} \lim_{n\rightarrow\infty}\int_{Q_{T}}\biggl( \frac{\partial u_{n}}{\partial t}-\overline{u}_{\varepsilon}\biggr)\varphi \,dx\,dt=0. \end{aligned}$$
Hence,
$$\begin{aligned} \lim_{n\rightarrow\infty}\int_{Q_{T}}\frac{\partial u_{n}}{\partial t} \varphi \,dx\,dt=\int_{Q_{T}}\overline{u}_{\varepsilon}\varphi \,dx\,dt. \end{aligned}$$
On the other hand, by integration by parts, we get
$$\begin{aligned} \int_{Q_{T}}\frac{\partial u_{n}}{\partial t}\varphi \,dx\,dt=-\int _{Q_{T}}u_{n}\frac{\partial\varphi}{\partial t}\,dx\,dt. \end{aligned}$$
Letting \(n\rightarrow\infty\) in above equality, we have
$$\begin{aligned} \int_{Q_{T}}\overline{u}_{\varepsilon}\varphi \,dx\,dt=-\int _{Q_{T}}u_{\varepsilon}\frac{\partial\varphi}{\partial t}\,dx\,dt, \quad\mbox{for } \varphi\in C_{0}^{\infty}(Q_{T}). \end{aligned}$$
Thus, we obtain \(\overline{u}= \frac{\partial u_{\varepsilon}}{\partial t}\). Moreover, for each \(j\in\mathbb{N}\), Lemma 3.2, and Lebesgue’s dominated convergence theorem yield
$$\begin{aligned} \lim_{n\rightarrow\infty}\int_{0}^{T} \biggl( \int_{\Omega}\biggl(\frac{\partial u_{n}}{\partial t}-\frac{\partial u_{\varepsilon}}{\partial t}\biggr) \omega _{j}\,dx \biggr)^{2}\,dt=0. \end{aligned}$$
Thus, for \(\delta>0\), by Proposition 2.1, there exists a positive number \(N_{\delta}\) independent of n such that
$$\begin{aligned} &\biggl\| \frac{\partial u_{n}}{\partial t}-\frac{\partial u_{\varepsilon}}{\partial t} \biggr\| _{L^{2}(Q_{T})}\\ &\quad\leq2\sum _{i=1}^{N_{\delta}}\int_{0}^{T} \biggl(\int_{\Omega}\biggl( \frac{\partial u_{n}}{\partial t}- \frac{\partial u_{\varepsilon}}{\partial t}\biggr)\omega_{i}\,dx \biggr)^{2}\,dt+2 \delta^{2}\int_{0}^{T}\biggl\| \frac {\partial u_{n}}{\partial t}-\frac{\partial u_{\varepsilon}}{\partial t}\biggr\| ^{2}_{W_{0}^{1,2}(\Omega)}\,dt. \end{aligned}$$
A similar discussion to (3.7) shows that there is a \(\widetilde {M}(\delta)>0\) such that
$$\begin{aligned} \biggl\| \frac{\partial u_{n}}{\partial t}-\frac{\partial u_{\varepsilon}}{\partial t} \biggr\| _{L^{2}(Q_{T})}\leq C \delta^{2},\quad \mbox{for } n>\widetilde{M}(\delta). \end{aligned}$$
Thus, \(\frac{\partial u_{n}}{\partial t}\rightarrow\frac{\partial u_{\varepsilon}}{\partial t}\) strongly in \(L^{2}(Q_{T})\). Further, there exists a subsequence of \(\{u_{n}\}\) still denoted by \(\{u_{n}\}\) such that \(\frac{\partial u_{n}}{\partial t}\rightarrow\frac{\partial u_{\varepsilon}}{\partial t}\) a.e. on \(Q_{T}\).
For \(\forall\varphi\in C^{1}(0,T;C_{0}^{\infty}(\Omega))\), we can choose a sequence \(\varphi_{k}\in C^{1}(0,T;V_{k})\) such that \(\varphi_{k}\rightarrow \varphi\) in \(C^{1,2}(Q_{T})\). Here for \(v\in C^{1,2}(Q_{T})\) equipped with the norm \(\|v\|=\sup_{|\alpha|\leq2,(x,t)\in\overline {Q_{T}}} \{| D^{\alpha}v|, |\frac{\partial v}{\partial t}| \}\). For \(\forall\tau\in(0,T]\), we have
$$\begin{aligned} &\lim_{k\rightarrow\infty}\lim_{n\rightarrow\infty}\int _{Q_{\tau}}\frac {\partial^{2} u_{n}}{\partial t^{2}}\varphi_{k}\,dx\,dt \\ &\quad=\lim_{k\rightarrow\infty}\lim_{n\rightarrow\infty} \biggl(\int _{\Omega}\frac{\partial u_{n}(x,\tau)}{\partial t}\varphi_{k}(x,\tau)\,dx-\int _{\Omega}\frac{\partial u_{n}(x,0)}{\partial t}\varphi_{k}(x,0)\,dx\\ &\qquad{}-\int _{Q_{\tau}}\frac {\partial u_{n}}{\partial t}\frac{\partial\varphi_{k}}{\partial t}\,dx\,dt \biggr) \\ &\quad=\lim_{k\rightarrow\infty} \biggl(\int_{\Omega}\frac{\partial u_{\varepsilon}(x,\tau)}{\partial t}\varphi_{k}(x,\tau)\,dx-\int_{\Omega}u_{1}\varphi (x,0)\,dx-\int_{Q_{\tau}}\frac{\partial u_{\varepsilon}}{\partial t} \frac {\partial\varphi_{k} }{\partial t}\,dx\,dt \biggr) \\ &\quad=\int_{\Omega}\frac{\partial u_{\varepsilon}(x,\tau)}{\partial t}\varphi (x,\tau)\,dx-\int _{\Omega}u_{1}\varphi(x,0)\,dx-\int_{Q_{\tau}} \frac{\partial u_{\varepsilon}}{\partial t}\frac{\partial\varphi}{\partial t}\,dx\,dt \\ &\quad=\lim_{n\rightarrow\infty}\int_{Q_{\tau}}\frac{\partial^{2} u_{n}}{\partial t^{2}} \varphi \,dx\,dt, \end{aligned}$$
where \(Q_{\tau}=\Omega\times(0,\tau)\). Replacing \(\omega_{i}\) in (3.1) by \(\varphi_{k}\), we obtain
$$\begin{aligned} &\int_{Q_{\tau}}\frac{\partial^{2} u_{n}}{\partial t^{2}}\varphi_{k}\,dx\,dt+\int _{Q_{\tau}}|\Delta u_{n}|^{p(x)-2}\Delta u_{n}\Delta\varphi _{k}+|u_{n}|^{q(x)-2}u_{n} \varphi_{k}+\varepsilon\Delta\frac{\partial u_{n}}{\partial t}\Delta \varphi_{k} \\ &\quad{}+a\biggl(t,\int_{\Omega}|u_{n}|^{q(x)}\,dx \biggr)\frac{\partial u_{n}}{\partial t}\varphi _{k}\,dx\,dt=\int_{Q_{\tau}}f_{n} \varphi_{k}\,dx\,dt. \end{aligned}$$
Thus, we have
$$\begin{aligned} \lim_{n\rightarrow\infty}\int_{Q_{\tau}}\frac{\partial^{2} u_{n}}{\partial t^{2}} \varphi \,dx\,dt= \int_{Q_{\tau}}f\varphi-\xi\Delta \varphi-|u_{\varepsilon}|^{q(x)-2}u_{\varepsilon}\varphi-\varepsilon \Delta\frac{\partial u_{\varepsilon}}{\partial t}\Delta\varphi-\zeta\varphi \,dx\,dt. \end{aligned}$$
(3.9)
Furthermore, for any \(\psi(x)\in C_{0}^{\infty}(\Omega)\), we get
$$\begin{aligned} &\int_{\Omega}\biggl(\frac{\partial u_{\varepsilon}(x,\tau)}{\partial t}-u_{1}\biggr) \psi \,dx \\ &\quad=\lim_{n\rightarrow\infty}\int_{\Omega}\biggl( \frac{\partial u_{n}(x,\tau )}{\partial t}-\frac{\partial u_{n}(x,0)}{\partial t} \biggr)\psi(x)\,dx \\ &\quad=\lim_{n\rightarrow\infty}\int_{0}^{\tau}\int _{\Omega}\frac{\partial^{2} u_{n}}{\partial t^{2}}\psi(x)\,dx\,dt \\ &\quad=\int_{Q_{\tau}}f\varphi-\xi\Delta\varphi-|u_{\varepsilon}|^{q(x)-2}u_{\varepsilon}\varphi-\varepsilon\Delta\frac{\partial u_{\varepsilon}}{\partial t}\Delta \varphi-\zeta\varphi \,dx\,dt\rightarrow0 \end{aligned}$$
as \(\tau\rightarrow0\). Similarly, for \(t_{0}\in[0,T]\), we have
$$\begin{aligned} \lim_{\tau\rightarrow0}\int_{\Omega}\biggl( \frac{\partial u_{\varepsilon}(x,\tau )}{\partial t}-\frac{\partial u_{\varepsilon}(x,t_{0})}{\partial t}\biggr)\psi \,dx=0, \quad\mbox{for } \psi\in C_{0}^{\infty}(\Omega). \end{aligned}$$
Furthermore, we obtain \(\frac{\partial u_{\varepsilon}(x,0)}{\partial t}=u_{1}\). Since \(u_{\varepsilon}\in L^{\infty}(0,T; W_{0}^{2,2}(\Omega))\) and \(\frac{\partial u_{\varepsilon}}{\partial t}\in L^{2}(0,T; W_{0}^{2,2}(\Omega ))\), we can assume that \(u_{\varepsilon}\in C(0,T;W_{0}^{2,2}(\Omega))\). Lemma 3.3 and the embedding \(W_{0}^{1,2}(\Omega)\hookrightarrow L^{2}(\Omega )\) imply that \(\int_{\Omega}u_{n}^{2}(x,T)\,dx\leq C(T)\). Thus, there exist a subsequence of \(\{u_{n}\}\) still denoted by \(\{u_{n}\}\) and a function \(\widehat{u}\) such that \(u_{n}(x,T)\rightharpoonup\widehat{u}\) weakly in \(L^{2}(\Omega)\). For each \(\varphi\in C_{0}^{\infty}(\Omega)\) and \(\eta\in C^{1}([0,T])\), we have
$$\begin{aligned} \int_{Q_{T}}\frac{\partial u_{n}}{\partial t}\varphi\eta \,dx\,dt=\int _{\Omega}u_{n}(x,T)\varphi\eta(T)-u_{n}(x,0) \varphi\eta(0)\,dx-\int_{Q_{T}}u_{n}\varphi\eta ^{\prime}(t)\,dx\,dt. \end{aligned}$$
Letting \(n\rightarrow\infty\), we get
$$\begin{aligned} \int_{Q_{T}}\frac{\partial u_{\varepsilon}}{\partial t}\varphi\eta \,dx\,dt=\int _{\Omega}\widehat{u}\varphi\eta(T)-u_{0}\varphi \eta(0)\,dx-\int_{Q_{T}}u_{\varepsilon}\varphi \eta^{\prime}(t)\,dx\,dt. \end{aligned}$$
By integration by parts, we have
$$\begin{aligned} \int_{\Omega}\bigl(u_{\varepsilon}(x,T)-\widehat{u}\bigr) \varphi\eta(T)\,dx=\int_{\Omega}\bigl(u_{\varepsilon}(x,0)-u_{0} \bigr)\varphi\eta(0)\,dx. \end{aligned}$$
Choosing \(\eta(T)=1\), \(\eta(0)=0\) or \(\eta(T)=0\), \(\eta(0)=1\), we obtain \(\widehat{u}=u_{\varepsilon}(x,T)\) and \(u_{\varepsilon}(x,0)=u_{0}(x)\) for \(x\in\Omega\). Similarly, we can prove that \(\Delta u_{\varepsilon}(x,0)=\Delta u_{0}\), \(\Delta u_{n}(x,T)\rightharpoonup\Delta u_{\varepsilon}(x,T)\) weakly in \(L^{2}(\Omega)\) (up to a subsequence) and
$$\begin{aligned} \int_{\Omega}\bigl|\Delta u_{\varepsilon}(x,T)\bigr|^{2}\,dx\leq \liminf_{n\rightarrow \infty}\int_{\Omega}\bigl|\Delta u_{n}(x,T)\bigr|^{2}\,dx. \end{aligned}$$
(3.10)
Further, by the compact embedding \(W_{0}^{1,2}(\Omega)\hookrightarrow \hookrightarrow L^{2}(\Omega)\), we get \(u_{n}(x,T)\rightarrow u_{\varepsilon}(x,T)\) strongly in \(L^{2}(\Omega)\).
Taking \(\varphi=u_{k}\) in (3.9) and letting \(k\rightarrow\infty\), we get
$$\begin{aligned} &\int_{\Omega}\frac{\partial u_{\varepsilon}(x,T)}{\partial t}u_{\varepsilon}(x,T)\,dx-\int_{\Omega}u_{1} u_{0}\,dx-\int _{Q_{T}}\biggl|\frac{\partial u_{\varepsilon}}{\partial t}\biggr|^{2} \,dx\,dt+\int _{Q_{T}}\xi\Delta u_{\varepsilon}+|u_{\varepsilon}|^{q(x)} \\ &\quad{}+\varepsilon\Delta\frac{\partial u_{\varepsilon}}{\partial t}\Delta u_{\varepsilon}+\zeta u_{\varepsilon}\,dx\,dt =\int_{Q_{T}}fu_{\varepsilon}\,dx\,dt. \end{aligned}$$
(3.11)
Multiplying (3.1) by \((\eta_{n})_{j}\) and summing up j from 1 to n, then integrating with respect to t over \([0,T]\), we have
$$\begin{aligned} &\int_{0}^{T}\int _{\Omega}\frac{\partial^{2} u_{n}}{\partial t^{2}}u_{n}\,dx\,dt+\int _{0}^{T}\int_{\Omega}|\Delta u_{n}|^{p(x)}+|u_{n}|^{q(x)}+\varepsilon \Delta\frac {\partial u_{n}}{\partial t}\Delta u_{n} \\ &\quad{}+a \biggl(\int_{\Omega}|u_{n}|^{q(x)}\,dx \biggr)\frac{\partial u_{n}}{\partial t}u_{n}\,dx\,dt=\int_{0}^{T} \int_{\Omega}f_{n}u_{n}\,dx\,dt. \end{aligned}$$
(3.12)
Thus,
$$\begin{aligned} 0\leq&\int_{0}^{T}\int_{\Omega}\bigl(|\Delta u_{n}|^{p(x)-2}\Delta u_{n}-|\Delta u_{\varepsilon}|^{p(x)-2}\Delta u_{\varepsilon}\bigr) (\Delta u_{n}-\Delta u_{\varepsilon})\,dx\,dt \\ ={}&\int_{0}^{T}\int_{\Omega}f_{n} u_{n}-|u_{n}|^{q(x)}-a \biggl(\int _{\Omega}|u_{n}|^{p(x)}\,dx \biggr) \frac{\partial u_{n}}{\partial t}u_{n}-\varepsilon\Delta \frac{\partial u_{n}}{\partial t}\Delta u_{n}\,dx\,dt \\ &{}-\int_{\Omega}\frac {\partial u_{n}(x,T)}{\partial t}u_{n}(x,T)\,dx+\int _{\Omega}\frac{\partial u_{n}(x,0)}{\partial t}u_{n}(x,0)\,dx +\int _{0}^{T}\int_{\Omega}\biggl| \frac{\partial u_{n}}{\partial t}\biggr|^{2}\,dx\,dt \\ &{}-\int_{0}^{T}\int_{\Omega}| \Delta u_{n}|^{p(x)-2}\Delta u_{n}\Delta u_{\varepsilon}+|\Delta u_{\varepsilon}|^{p(x)-2}\Delta u_{\varepsilon}(\Delta u_{n}-\Delta u_{\varepsilon})\,dx\,dt. \end{aligned}$$
By (3.10) and (3.11), we get
$$\begin{aligned} &\limsup_{n\rightarrow\infty}\int_{0}^{T}\int _{\Omega}\bigl(|\Delta u_{n}|^{p(x)-2}\Delta u_{n}-|\Delta u_{\varepsilon}|^{p(x)-2}\Delta u_{\varepsilon}\bigr) (\Delta u_{n}-\Delta u_{\varepsilon})\,dx\,dt \\ &\quad\leq \int_{0}^{T}\int_{\Omega}fu_{\varepsilon}-|u_{\varepsilon}|^{p(x)}-a\biggl(\int _{\Omega}|u_{\varepsilon}|^{q(x)}\,dx\biggr) \frac{\partial u_{\varepsilon}}{\partial t}u_{\varepsilon}-\xi\Delta u_{\varepsilon}\,dx\,dt\\ &\qquad{}- \frac{\varepsilon}{2}\int_{\Omega}\bigl|\Delta u_{\varepsilon}(x,T)\bigr|^{2}\,dx +\frac{\varepsilon}{2}\int_{\Omega}\bigl|\Delta u_{\varepsilon}(x,0)\bigr|^{2}\,dx- \int_{\Omega}\frac{u_{\varepsilon}(x,T)}{\partial t}u_{\varepsilon}(x,T)\,dx \\ &\qquad{}+\int _{\Omega}u_{1}u_{0}\,dx+\int _{0}^{T}\int_{\Omega}\biggl| \frac{\partial u_{\varepsilon}}{\partial t}\biggr|^{2}\,dx\,dt \\ &\quad=0. \end{aligned}$$
It follows that
$$\begin{aligned} \lim_{n\rightarrow\infty}\int_{0}^{T}\int _{\Omega}\bigl(|\Delta u_{n}|^{p(x)-2}\Delta u_{n}-|\Delta u_{\varepsilon}|^{p(x)-2}\Delta u_{\varepsilon}\bigr) (\Delta u_{n}-\Delta u_{\varepsilon})\,dx\,dt=0. \end{aligned}$$
Following the ideas of [4], we set \(Q_{1}=\{(x,t)\in Q_{T}: p(x)\geq2\}\) and \(Q_{2}=\{(x,t)\in Q_{T}: 1< p(x)<2\}\), then, as \(n\rightarrow\infty\),
$$\begin{aligned} &\int_{Q_{1}}|\Delta u_{n}-\Delta u_{\varepsilon}|^{p(x)}\,dx\,dt \\ &\quad\leq C\int_{Q_{1}}\bigl(|\Delta u_{n}|^{p(x)-2} \Delta u_{n}-|\Delta u_{\varepsilon}|^{p(x)-2}\Delta u_{\varepsilon}\bigr) (\Delta u_{n}-\Delta u_{\varepsilon})\,dx\,dt \\ &\quad\rightarrow 0 \end{aligned}$$
and
$$\begin{aligned} &\int_{Q_{2}}|\Delta u_{n}-\Delta u_{\varepsilon}|^{p(x)}\,dx\,dt \\ &\quad\leq C \bigl\| \bigl[\bigl(|\Delta u_{n}|^{p(x)-2}\Delta u_{n}-|\Delta u_{\varepsilon}|^{p(x)-2}\Delta u_{\varepsilon}\bigr) (\Delta u_{n}-\Delta u_{\varepsilon})\bigr]^{\frac {p(x)}{2}} \bigr\| _{L^{\frac{2}{p(x)}}(Q_{2})}\\ &\qquad{}\times \bigl\| \bigl(|\Delta u_{n}|^{p(x)}+|\Delta u_{\varepsilon}|^{p(x)}\bigr)^{\frac {2-p(x)}{2}} \bigr\| _{L^{\frac{2}{2-p(x)}}(Q_{2})}\\ &\quad\rightarrow 0. \end{aligned}$$
Therefore, we obtain \(\Delta u_{n}\rightarrow\Delta u_{\varepsilon}\) strongly in \(L^{p(x)}(Q_{\tau})\). Thus, there exists a subsequence of \(\{u_{n}\}\) still denoted by \(\{u_{n}\} \) such that \(\Delta u_{n}\rightarrow\Delta u_{\varepsilon}\) a.e. on \(Q_{T}\). Further,
$$\begin{aligned} |\Delta u_{n}|^{p(x)-2}\Delta u_{n}\rightarrow|\Delta u_{\varepsilon}|^{p(x)-2}\Delta u_{\varepsilon}, \quad\mbox{for a.e. } (x,t) \in Q_{T}. \end{aligned}$$
In view of Theorem 2.4, we get \(\xi=|\Delta u_{\varepsilon}|^{p(x)-2}\Delta u_{\varepsilon}\). Similarly, we can prove that \(u_{n}\rightarrow u_{\varepsilon}\) strongly in \(L^{q(x)}(Q_{T})\). Thus, there exists a subsequence of \(\{u_{n}\}\) still denoted \(\{u_{n}\}\) such that
$$\begin{aligned} \lim_{n\rightarrow0}\int_{\Omega}\bigl| u_{n}(x,t)-u_{\varepsilon}(x,t)\bigr|^{q(x)}\,dx=0, \quad\mbox{for a.e. } t\in[0,T]. \end{aligned}$$
Furthermore, we have
$$\begin{aligned} \lim_{n\rightarrow0}\int_{\Omega}\bigl| u_{n}(x,t)\bigr|^{q(x)}\,dx=\int_{\Omega}\bigl|u_{\varepsilon}(x,t)\bigr|^{q(x)}\,dx, \quad\mbox{for a.e. } t\in[0,T]. \end{aligned}$$
Thus, \(a(\int_{\Omega}|u_{n}|^{q(x)}\,dx)\frac{\partial u_{n}}{\partial t}\rightarrow a(\int_{\Omega}|u_{\varepsilon}|^{q(x)}\,dx)\frac{\partial u_{\varepsilon}}{\partial t}\) a.e. on \(Q_{T}\). By Theorem 2.4, we obtain \(\zeta=a(\int_{\Omega}|u_{\varepsilon}|^{q(x)}\,dx)\frac{\partial u_{\varepsilon}}{\partial t}\). It follows from (3.9) that the theorem is proved. □

Remark 3.1

Obviously, in this section, the two inequalities in (H2) can be replaced by \(1< p^{-}\leq p^{+}<\infty\) and \(1< q^{-}\leq q^{+}<\infty\), respectively.

4 Existence of Young measure solutions for problem (1.2)

In this section, from the sequence of approximate solutions \(\{ u_{\varepsilon}\}_{0<\varepsilon<1}\) of problem (1.2), we shall prove that the limit of \(u_{\varepsilon}\) (as \(\varepsilon\rightarrow0^{+}\)) is a Young measure solution of problem (1.1).

Definition 4.1

A pair \((u,\nu)\) is called a Young measure solution of problem (1.1) if
$$\begin{aligned} &u\in L^{\infty}\bigl(0,T;W_{0}^{2,p(x)}(\Omega)\bigr) \cap L^{\infty}\bigl(0,T;L^{q(x)}(\Omega)\bigr)\cap W^{1,\infty}\bigl(0,T;L^{2}(\Omega)\bigr), \\ &\nu=\{\nu_{x,t}\}_{x,t}\mbox{ is a probability measure}, \end{aligned}$$
and
$$\begin{aligned} &\int_{\Omega}\frac{\partial u(x,\tau)}{\partial t}\varphi \,dx-\int _{\Omega}u_{1}\varphi(x,0)\,dx-\int_{Q_{T}} \frac{\partial u}{\partial t}\frac{\partial \varphi}{\partial t}\,dx\,dt+\int_{Q_{T}}\int _{\mathbb{R}}|A|^{p(x)-2}A\,d\nu (A)\Delta\varphi \\ &\quad{}+|u|^{q(x)-2}u\varphi \,dx\,dt+\int_{Q_{T}}a\biggl(\int _{\Omega}|u|^{q(x)}\,dx\biggr)\frac {\partial u}{\partial t}\varphi \,dx\,dt=\int_{Q_{T}}f\varphi \,dx\,dt, \end{aligned}$$
for all \(\varphi\in C^{1}(0,T;C_{0}^{\infty}(\Omega))\) and \(\tau\in(0,T]\).

Theorem 4.1

Under conditions (H1)-(H3), problem (1.1) has a Young measure solution.

Proof

For each \(\tau\in(0,T]\) and \(\varphi\in C^{1}(0,T;C_{0}^{\infty}(\Omega))\), we have by (3.9)
$$\begin{aligned} &\int_{\Omega}\frac{\partial u_{\varepsilon}(x,\tau)}{\partial t}\varphi (x, \tau)\,dx-\int_{\Omega}u_{1}\varphi(x,0)\,dx-\int _{Q_{\tau}}\frac{\partial u_{\varepsilon}}{\partial t}\frac{\partial\varphi}{\partial t}\,dx\,dt+\int _{Q_{\tau}}|\Delta u_{\varepsilon}|^{p(x)-2}\Delta u_{\varepsilon}\Delta \varphi \\ &\quad{}+|u_{\varepsilon}|^{q(x)-2}u_{\varepsilon}\varphi +a\biggl(\int _{\Omega}|u_{\varepsilon}|^{q(x)}\,dx\biggr) \frac{\partial u_{\varepsilon}}{\partial t}\varphi+\varepsilon\Delta\frac{\partial u_{\varepsilon}}{\partial t}\Delta\varphi \,dx\,dt= \int_{Q_{\tau}} f\varphi \,dx\,dt. \end{aligned}$$
(4.1)
Since the constant in Lemma 3.2 is independent of n and ε, by the convergence of \(u_{n}\) and \(\frac{\partial u_{n}}{\partial t}\) in Section 3, we have
$$\begin{aligned} &\int_{\Omega}\biggl|\frac{\partial u_{\varepsilon}(x,t)}{\partial t}\biggr|^{2}\,dx+\int _{\Omega}\bigl|\Delta u_{\varepsilon}(x,t)\bigr|^{p(x)}+\bigl|u_{\varepsilon}(x,t)\bigr|^{q(x)}\,dx+\varepsilon\int_{\Omega}\bigl|\Delta u_{\varepsilon}(x,t)\bigr|^{2}\,dx\leq C \end{aligned}$$
for a.e \(t\in[0,T]\) and
$$\begin{aligned} \int_{Q_{T}}\biggl|\frac{\partial u_{\varepsilon}}{\partial t}\biggr|^{2}\,dx\,dt+\int _{Q_{T}}|\Delta u_{\varepsilon}|^{p(x)}+|u_{\varepsilon}|^{q(x)}\,dx\,dt+\varepsilon\int_{Q_{T}}\biggl|\Delta \frac{\partial u_{\varepsilon}}{\partial t}\biggr|^{2}\,dx\,dt\leq C. \end{aligned}$$
Similarly, by Lemma 3.3, we have
$$\begin{aligned} &\|u_{\varepsilon}\|_{L^{\infty}(0,T;W_{0}^{2,p(x)}(\Omega))}+ \bigl\| |\Delta u_{\varepsilon}|^{p(x)-2}\Delta u_{\varepsilon}\bigr\| _{L^{p^{\prime}(x)}(Q_{T})} \\ &\quad{}+ \bigl\| |u_{\varepsilon}|^{q(x)-2} u_{\varepsilon}\bigr\| _{L^{q^{\prime}(x)}(Q_{T})}+ \biggl\| a \biggl(\int_{\Omega}|u_{\varepsilon}|^{q(x)}\,dx \biggr)\frac{\partial u_{\varepsilon}}{\partial t} \biggr\| _{L^{2}(Q_{T})}\leq C. \end{aligned}$$
(4.2)
Thus, there exists a subsequence of \(\{u_{\varepsilon}\}_{0<\varepsilon <1}\) still denoted by \(\{u_{\varepsilon}\}_{0<\varepsilon<1}\) such that
$$\begin{aligned} \textstyle\begin{cases} u_{\varepsilon}\rightharpoonup u &\mbox{weakly}* \mbox{ in } L^{\infty}(0,T;W_{0}^{2,p(x)}(\Omega) \cap L^{q(x)}(\Omega)), \\ \Delta u_{\varepsilon}\rightharpoonup\Delta u &\mbox{weakly in } L^{p(x)}(Q_{T}), \\ u_{\varepsilon}\rightharpoonup u &\mbox{weakly in }L^{q(x)}(Q_{T}), \\ \frac{\partial u_{\varepsilon}}{\partial t}\rightharpoonup\frac{\partial u}{\partial t} &\mbox{weakly}* \mbox{ in } L^{\infty}(0,T;L^{2}(\Omega )), \\ |u_{\varepsilon}|^{q(x)-2}u_{\varepsilon}\rightharpoonup\alpha & \mbox{weakly in} L^{q^{\prime}(x)}(Q_{T}), \\ a(\int_{\Omega}|u_{\varepsilon}|^{q(x)}\,dx) \frac{\partial u_{\varepsilon}}{\partial t}\rightharpoonup\beta &\mbox{weakly in} L^{2}(Q_{T}). \end{cases}\displaystyle \end{aligned}$$
Since \(p^{-}>\max\{1,\frac{2N}{N+4}\}\), the embedding \(W_{0}^{2,p(x)}(\Omega )\hookrightarrow L^{2}(\Omega)\) is compact. Further, as \(u_{\varepsilon}\in L^{\infty}(0,T;W_{0}^{2,p(x)}(\Omega))\) and \(\frac{\partial u_{\varepsilon}}{\partial t}\in L^{\infty}(0,T;L^{2}(\Omega))\), by the Lions-Aubin lemma, there exists a subsequence of \(\{u_{\varepsilon}\}_{0<\varepsilon<1}\) still denoted by \(\{u_{\varepsilon}\}_{0<\varepsilon<1}\) such that \(u_{\varepsilon}\rightarrow u\) strongly in \(L^{2}(Q_{T})\) and a.e. on \(Q_{T}\). Thus, \(|u_{\varepsilon}|^{q(x)-2}u_{\varepsilon}\rightarrow|u|^{q(x)-2}u\) a.e. on \(Q_{T}\). In view of Theorem 2.4, we obtain \(\alpha =|u|^{q(x)-2}u\). By assumption (H1), we have \(\mu=\inf_{\overline {\Omega}}(\frac{Np(x)}{N-2p(x)}-q(x))>0\). For each measurable subset \(U\subset Q_{T}\) with \(|U|\leq1\), by Hölder’s inequality, Theorem 2.2, and Theorem 2.3, we obtain
$$\begin{aligned} \int_{\Omega}|u_{\varepsilon}|^{q(x)}&\leq2 \bigl\| |u_{\varepsilon}|^{q(x)}\bigr\| _{L^{\frac{Np(x)}{(N-2p(x))q(x)}}(U)}\|1\|_{L^{\frac {Np(x)}{Np(x)-Nq(x)+2p(x)q(x)}}(U)}\leq C|U|^{\frac{\mu(N-2p^{+})}{Np^{+}}}. \end{aligned}$$
Thus, the sequence \(\{|u_{\varepsilon}-u|^{q(x)}\}_{0<\varepsilon<1}\) is equi-integrable on \(L^{1}(Q_{T})\). The Vitali convergence theorem implies that \(\int_{Q_{T}}|u_{\varepsilon}-u|^{q(x)}\,dx\,dt\rightarrow0\), that is to say, we obtain \(u_{\varepsilon}\rightarrow u\) strongly in \(L^{q(x)}(Q_{T})\). Thus, there exists a subsequence of \(\{u_{\varepsilon}\} _{0<\varepsilon<1}\) still labeled by \(\{u_{\varepsilon}\}_{0<\varepsilon <1}\) such that
$$\begin{aligned} \lim_{\varepsilon\rightarrow0}\int_{\Omega}|u_{\varepsilon}-u|^{q(x)}\,dx=0, \quad \mbox{for a.e. } t\in[0,T]. \end{aligned}$$
Furthermore,
$$\begin{aligned} \lim_{\varepsilon\rightarrow0}\int_{\Omega}\bigl||u_{\varepsilon}|^{q(x)}-|u|^{q(x)}\bigr|\,dx=0, \quad\mbox{for a.e. } t\in[0,T]. \end{aligned}$$
Hence, we find by the continuity of a that \(a(\int_{\Omega}|u_{\varepsilon}|^{q(x)}\,dx)\rightarrow a(\int_{\Omega}|u|^{q(x)}\,dx)\) for a.e. \(t\in[0,T]\). Since \(\int_{\Omega}|u_{\varepsilon}|^{q(x)}\,dx\leq C\) for a.e. \(t\in[0,T]\) and \(a\in C([0,\infty))\), for each \(\varphi\in L^{2}(Q_{T})\), by Lebesgue’s dominated convergence theorem, we have
$$\begin{aligned} a \biggl(\int_{\Omega}|u_{\varepsilon}|^{q(x)}\,dx \biggr)\varphi\rightarrow a \biggl(\int_{\Omega}|u|^{q(x)}\,dx \biggr)\varphi \quad\mbox{strongly in } L^{2}(Q_{T}). \end{aligned}$$
Further, by the weak convergence of \(\frac{\partial u_{\varepsilon}}{\partial t}\) in \(L^{2}(Q_{T})\), we get
$$\begin{aligned} \lim_{\varepsilon\rightarrow0}\int_{Q_{T}}a\biggl(\int _{\Omega}|u_{\varepsilon}|^{q(x)}\,dx\biggr) \frac{\partial u_{\varepsilon}}{\partial t}\varphi \,dx\,dt &=\lim_{\varepsilon\rightarrow0}\int_{Q_{T}}\frac{\partial u_{\varepsilon}}{\partial t} \biggl(a \biggl(\int_{\Omega}|u_{\varepsilon}|^{q(x)}\,dx \biggr) \varphi \biggr)\,dx\,dt \\ &=\int_{Q_{T}}a\biggl(\int_{\Omega}|u|^{q(x)}\,dx \biggr)\frac{\partial u}{\partial t}\varphi \,dx\,dt. \end{aligned}$$
Thus, \(a(\int_{\Omega}|u_{\varepsilon}|^{q(x)}\,dx)\frac{\partial u_{\varepsilon}}{\partial t}\rightharpoonup a(\int_{\Omega}|u|^{q(x)}\,dx)\frac{\partial u}{\partial t}\) weakly in \(L^{2}(Q_{T})\). The uniqueness of the limit implies that \(\beta=a(\int_{\Omega}|u|^{q(x)}\,dx)\frac{\partial u}{\partial t}\).
Finally, we prove that the sequence \(\{\Delta u_{\varepsilon}\} _{0<\varepsilon<1}\) generates a Young measure \(\{\nu_{x,t}\}_{x,t}\) such that
$$\begin{aligned} |\Delta u_{\varepsilon}|^{p(x)-2}\Delta u_{\varepsilon}\rightharpoonup \int_{\mathbb{R}}|A|^{p(x)-2}A\,d\nu_{x,t}(A) \quad\mbox{weakly in } L^{1}(Q_{T}). \end{aligned}$$
(4.3)
Following Theorem 2.5, we first verify that the Young measure \(\nu _{x,t}\) generated by the sequence \(\{\Delta u_{\varepsilon}\} _{0<\varepsilon<1}\) is a probability measure for a.e. \((x,t)\in Q_{T}\). Indeed, for \(s\leq p^{-}\), we have
$$\begin{aligned} \int_{Q_{T}}|\Delta u_{\varepsilon}|^{s}\,dx\leq| \Omega|T+\int_{Q_{T}}|\Delta u_{\varepsilon}|^{p(x)}\,dx \leq C. \end{aligned}$$
It follows from (iv) in Theorem 2.5 that \(\nu_{x,t}\) is a probability measure. Set \(H(x,A)=|A|^{p(x)-2}A\). Next, we prove that the sequence \(\{ H(x,\Delta u_{\varepsilon})\}_{\varepsilon}\) is weakly relatively compact in \(L^{1}(Q_{T})\). It is clear that \(\{H(x,\Delta u_{\varepsilon})\} _{\varepsilon}\) will be weakly relatively compact in \(L^{1}(Q_{T})\), if we prove that \(\{H(x,\Delta u_{\varepsilon})\}_{\varepsilon}\) is uniformly bounded and equi-integrable on \(L^{1}(Q_{T})\); see [42], Proposition 1.3. Indeed, for each measurable subset \(U\subset Q_{T}\) with \(|U|\leq1\), by (4.2) and Hölder’s inequality, we have
$$\begin{aligned} \int_{U}\bigl|H(x,\Delta u_{\varepsilon})\bigr|\,dx\,dt=\int _{U}|\Delta u_{\varepsilon}|^{p(x)-1}\,dx\,dt&\leq2\| \Delta u_{\varepsilon}\|_{L^{p^{\prime}(x)}(U)}\|1\| _{L^{p(x)}(U)}\leq C|U|^{\frac{1}{p^{+}}}. \end{aligned}$$
Thus, the sequence \(\{H(x,\Delta u_{\varepsilon})\}_{\varepsilon}\) is equi-integrable. Similarly, the sequence \(\{H(x,\Delta u_{\varepsilon})\} _{\varepsilon}\) is uniformly bounded on \(L^{1}(Q_{T})\). Therefore, the convergence property (4.3) holds.
The estimate (4.2) implies
$$\begin{aligned} \varepsilon\Delta\frac{\partial u_{\varepsilon}}{\partial t}\rightarrow0 \quad\mbox{strongly in } L^{2}(Q_{T}). \end{aligned}$$
From the same procedures as in Section 3, we can prove there exists a subsequence of \(\{u_{\varepsilon}\}_{0<\varepsilon<1}\) still denoted by \(\{u_{\varepsilon}\}_{0<\varepsilon<1}\) such that \(\frac{\partial u_{\varepsilon}(x,t)}{\partial t}\rightharpoonup\frac{\partial u(x,t)}{\partial t}\) weakly in \(L^{2}(\Omega)\) uniformly on \([0,T]\). Taking \(\varepsilon\rightarrow0\) in Definition 4.1, we obtain
$$\begin{aligned} &\int_{\Omega}\frac{\partial u(x,\tau)}{\partial t}\varphi(x,\tau)\,dx-\int _{\Omega}u_{1}\varphi(x,0)\,dx-\int_{Q_{T}} \frac{\partial u}{\partial t}\frac {\partial\varphi}{\partial t}\,dx\,dt\\ &\quad{}+\int_{Q_{T}}\int _{\mathbb {R}}|A|^{p(x)-2}A\,d\nu_{x,t}(A)\Delta\varphi +|u|^{q(x)-2}u\varphi +a \biggl(\int_{\Omega}|u|^{q(x)}\,dx \biggr)\frac{\partial u}{\partial t}\varphi \,dx\,dt\\ &\qquad{}=\int_{Q_{T}} f\varphi \,dx\,dt, \end{aligned}$$
for all \(\varphi\in C^{1}(0,T;C_{0}^{\infty}(\Omega))\) and \(\tau\in(0,T]\). □

By Theorem 4.1, we have the following corollary.

Corollary 4.1

Suppose that \(f(x,t)\equiv0\) and (H1) and (H2) are satisfied. Then for a given \(u_{0}\in W^{2,p(x)}(\Omega)\cap W_{0}^{2,2}(\Omega)\), there exist a function \(u:\Omega\times(0,\infty)\rightarrow\mathbb{R}\) and a Young measure \(\nu_{x,t}\) such that for \(\forall T>0\),
$$\begin{aligned} u\in L^{\infty}\bigl(0,T; W_{0}^{2,p(x)}(\Omega)\bigr) \cap L^{\infty}\bigl(0,T; L^{q(x)}(\Omega)\bigr)\cap W^{1,\infty}\bigl(0,T;L^{2}(\Omega)\bigr) \end{aligned}$$
and
$$\begin{aligned} &\int_{\Omega}\frac{\partial u(x,\tau)}{\partial t}\varphi(x,\tau)\,dx-\int _{\Omega}\frac{\partial u(x,0)}{\partial t}\varphi(x,0)\,dx-\int _{Q_{T}}\frac {\partial u}{\partial t}\frac{\partial\varphi}{\partial t}\,dx\,dt\\ &\quad{}+\int _{Q_{T}}\int_{\mathbb{R}}|A|^{p(x)-2}A d \nu_{x,t}(A)\Delta\varphi +|u|^{q(x)-2}u\varphi+a\biggl(\int_{\Omega}|u|^{q(x)}\,dx \biggr)\frac{\partial u}{\partial t}\varphi \,dx\,dt =0, \end{aligned}$$
for all \(\varphi\in C^{1}(0,T;C_{0}^{\infty}(\Omega))\) and \(\tau\in(0,T]\).

5 Energy decay of Young measure solutions

In this section, we give the decay estimates of weak solutions obtained by Corollary 4.1. First we give a lemma by Nakao [43]

Lemma 5.1

(see [43])

Let \(\Psi:(0,\infty)\rightarrow\mathbb{R}\) be a bounded nonnegative function. If there exist two constants \(\alpha>0\) and \(\beta\geq0\) such that
$$\begin{aligned} \sup_{t\leq s\leq t+1}\Psi^{1+\beta}(s)\leq\alpha\bigl(\Psi(t)- \Psi(t+1)\bigr),\quad \textit{for } \forall t\geq0, \end{aligned}$$
then there exist positive constants C and γ such that
$$\begin{aligned} \textstyle\begin{cases} \Psi(t)\leq C \mathrm{e}^{-\gamma t}, \quad\forall t \geq0, \textit{as } \beta =0,\\ \Psi(t)\leq C(t+1)^{-\frac{1}{\beta}}, \quad\forall t\geq0, \textit{as } \beta>0. \end{cases}\displaystyle \end{aligned}$$

Theorem 5.1

Let \(p^{-}\geq2\). Then there exist constants \(C, \gamma>0\) such that the weak solutions obtained by Corollary  4.1 have the following estimates: If \(p^{+}=2\), then
$$\begin{aligned} \int_{\Omega}\biggl|\frac{\partial u(x,t)}{\partial t}\biggr|^{2}\,dx+\int _{\Omega}\bigl|u(x,t)\bigr|^{q(x)}\,dx\leq C e^{-\gamma t},\quad \textit{for a.e. } t\geq0. \end{aligned}$$
If \(p^{+}>2\), then
$$\begin{aligned} \int_{\Omega}\biggl|\frac{\partial u(x,t)}{\partial t}\biggr|^{2}\,dx+\int _{\Omega}\bigl|u(x,t)\bigr|^{q(x)}\,dx\leq C (t+1)^{-\frac{p^{+}}{p^{+}-2}},\quad \textit{for a.e. } t\geq0. \end{aligned}$$

Proof

We define
$$\begin{aligned} I_{n}(t)=\frac{1}{2}\int_{\Omega}\biggl| \frac{\partial u_{n}}{\partial t}\biggr|^{2}\,dx+\int_{\Omega}\frac{|\Delta u_{n}|^{p(x)}}{p(x)}+\frac{|u_{n}|^{q(x)}}{q(x)}\,dx. \end{aligned}$$
The definition of \(I_{n}(t)\) and equality (3.1) imply \(I_{n}(t)\) is nonnegative and uniformly bounded. We assume that \(I_{n}(t)\leq M\), \(M>0\) is a constant. For \(\forall t>0\) fixed, it follows from (3.9) and (H2) that
$$\begin{aligned} \frac{d}{dt}I_{n}(t)+\varepsilon\int_{\Omega}\biggl| \Delta\frac{\partial u_{n}}{\partial t}\biggr|^{2}\,dx+a_{0}\int _{\Omega}\biggl|\frac{\partial u_{n}}{\partial t}\biggr|^{2}\,dx \leq0. \end{aligned}$$
(5.1)
This implies \(I_{n}(t)\) is a nonincreasing function. Putting \(J^{2}_{n}(t)=I_{n}(t)-I_{n}(t+1)\) and integrating (5.1) over \((t,t+1)\), we get
$$\begin{aligned} J^{2}_{n}(t)&\geq\varepsilon\int _{t}^{t+1}\int_{\Omega}\biggl|\Delta \frac{\partial u_{n}(x,\tau)}{\partial t}\biggr|^{2}+a_{0}\biggl|\frac{\partial u_{n}(x,\tau)}{\partial t}\biggr|^{2}\,dx\,d\tau \\ &\geq a_{0}\int_{t}^{t+1}\int _{\Omega}\biggl|\frac{\partial u_{n}(x,\tau)}{\partial t}\biggr|^{2}\,dx\,d\tau. \end{aligned}$$
(5.2)
By the mean value theorem and (5.2), there exist \(t_{1}\in[t,t+\frac {1}{3}]\) and \(t_{2}\in[t+\frac{2}{3},t+1]\) such that
$$\begin{aligned} \int_{\Omega}\biggl|\frac{\partial u_{n}(x,t_{i})}{\partial t}\biggr|^{2}\,dx\leq \frac {1}{a_{0}}J^{2}_{n}(t),\quad i=1,2. \end{aligned}$$
(5.3)
From (3.1), we have
$$\begin{aligned} &\int_{\Omega}|\Delta u_{n}|^{p(x)}+|u_{n}|^{q(x)}\,dx \\ &\quad=- \int_{\Omega}\frac{\partial ^{2} u_{n}}{\partial t^{2}}u_{n}+\varepsilon\Delta \frac{\partial u_{n}}{\partial t}\Delta u_{n}+a\biggl(\int_{\Omega}|u_{n}|^{q(x)}\,dx \biggr)\frac{\partial u_{n}}{\partial t}u_{n}\,dx. \end{aligned}$$
(5.4)
Integrating (5.4) from \(t_{1}\) to \(t_{2}\), we obtain
$$\begin{aligned} &\int_{t_{1}}^{t_{2}}\int _{\Omega}\bigl|\Delta u_{n}(x,\tau)\bigr|^{p(x)}\,dx\,d\tau+\int_{t_{1}}^{t_{2}}\int_{\Omega}|u_{n}|^{q(x)}\, dx\, d\tau \\ &\quad=-\int_{\Omega}\frac{\partial u_{n}(x,t_{2})}{\partial t}u_{n}(x,t_{2})\,dx+ \int_{\Omega}\frac{\partial u_{n}(x,t_{1})}{\partial t}u_{n}(x,t_{1})\,dx+ \int_{t_{1}}^{t_{2}}\int_{\Omega}\biggl| \frac{\partial u_{n}(x,\tau)}{\partial t}\biggr|^{2}\,dx\,d\tau \\ &\qquad{}-\varepsilon\int_{t_{1}}^{t_{2}}\int _{\Omega}\Delta\frac{\partial u_{n}(x,\tau )}{\partial t}\Delta u_{n}(x, \tau)\,dx\,d\tau \\ &\qquad{}-\int_{t_{1}}^{t_{2}}\int _{\Omega}a\biggl(\int_{\Omega}|u_{n}|^{q(x)}\,dx \biggr)\frac{\partial u_{n}}{\partial t} u_{n}\,dx\,d\tau. \end{aligned}$$
(5.5)
The Hölder inequality, (5.3), Theorem 2.2, Theorem 2.3, and \(I_{n}(t)\) being decreasing imply
$$\begin{aligned} \biggl|\int_{\Omega}\frac{\partial u_{n}(x,t_{i})}{\partial t}u_{n}(x,t_{i})\,dx \biggr|& \leq\bigl\| u_{n}(x,t_{i})\bigr\| _{L^{2}(\Omega)} \biggl\| \frac{\partial u_{n}(x,t_{i})}{\partial t} \biggr\| _{L^{2}(\Omega)} \\ &\leq C_{1}\bigl\| \Delta u_{n}(x,t_{i}) \bigr\| _{L^{p(x)}(\Omega)}J_{n}(t) \\ &\leq C_{2} \biggl(\int_{\Omega}\frac{|\Delta u_{n}(x,t_{i})|^{p(x)}}{p(x)}\,dx \biggr)^{\frac{1}{p^{+}}}J_{n}(t) \\ &\leq C_{2} \bigl(I_{n}(t)\bigr)^{\frac{1}{p^{+}}}J_{n}(t),\quad i=1,2. \end{aligned}$$
(5.6)
Here the third inequality in (5.6) is obtained by
$$\begin{aligned} \bigl\| \Delta u_{n}(x,t_{i})\bigr\| _{L^{p(x)}(\Omega)}\leq{}&\bigl(p^{+} \bigr)^{\frac{1}{p^{-}}}\max \biggl\{ \biggl(\int_{\Omega}\frac{|\Delta u_{n}(x,t_{i})|^{p(x)}}{p(x)}\,dx\biggr)^{\frac {1}{p^{-}}},\\ &{}\biggl(\int_{\Omega}\frac{|\Delta u_{n}(x,t_{i})|^{p(x)}}{p(x)}\,dx\biggr)^{\frac {1}{p^{+}}} \biggr\} \\ \leq{}&\bigl(p^{+}\bigr)^{\frac{1}{p^{-}}} \max\bigl\{ M,M^{\frac{1}{p^{-}}-\frac{1}{p^{+}}}\bigr\} \biggl( \int_{\Omega}\frac{|\Delta u_{n}(x,t_{i})|^{p(x)}}{p(x)}\,dx\biggr)^{\frac{1}{p^{+}}}, \quad i=1,2. \end{aligned}$$
Similarly,
$$\begin{aligned} &\biggl|\varepsilon\int_{t_{1}}^{t_{2}}\int _{\Omega}\Delta\frac{\partial u_{n}}{\partial t}\Delta u_{n}\,dx\,d\tau\biggr| \\ &\quad\leq \int_{t_{1}}^{t_{2}}\biggl\| \varepsilon\Delta \frac{\partial u_{n}(x,\tau )}{\partial t}\biggr\| _{L^{2}(\Omega)} \sup_{t\leq\tau\leq t+1}\bigl\| \Delta u_{n}(x,\tau)\bigr\| _{L^{2}(\Omega)}\,d\tau \\ &\quad\leq C_{3}\biggl(\varepsilon\int_{t_{1}}^{t_{2}} \int_{\Omega}\Delta\frac{\partial u_{n}(x,\tau)}{\partial t}\,dx\,d\tau\biggr)^{\frac{1}{2}} \sup_{t\leq\tau\leq t+1}\bigl\| \Delta u_{n}(x,\tau)\bigr\| _{W^{2,p(x)}(\Omega)}\,d\tau \\ &\quad\leq C_{4}J_{n}(t) \bigl(I_{n}(t) \bigr)^{\frac{1}{p^{+}}}. \end{aligned}$$
(5.7)
From the assumption (H2), Hölder’s inequality, the second inequality in (5.2), Theorem 2.2, Theorem 2.3, and the boundedness of \(I_{n}\), we have
$$\begin{aligned} &\biggl|\int_{t_{1}}^{t_{2}}\int _{\Omega}a \biggl(\int_{\Omega}|u_{n}|^{q(x)}\,dx \biggr)\frac {\partial u_{n}(x,\tau)}{\partial t}u_{n}(x,\tau)\,dx\,d\tau\biggr| \\ &\quad\leq C_{5}\int_{t_{1}}^{t_{2}}\biggl\| \frac{\partial u_{n}}{\partial t}\biggr\| _{L^{2}(\Omega)} \|u_{n}\|_{L^{2}(\Omega)}\,d\tau \leq C_{6}J_{n}(t) \bigl(I_{n}(t) \bigr)^{\frac{1}{p^{+}}}. \end{aligned}$$
(5.8)
Gathering (5.5) with (5.6)-(5.8), we obtain
$$\begin{aligned} \int_{t_{1}}^{t_{2}}\int_{\Omega}\bigl| \Delta u_{n}(x,\tau)\bigr|^{p(x)}+\bigl|u_{n}(x,\tau )\bigr|^{q(x)}\,dx\,d\tau\leq\frac{1}{a_{0}}J_{n}^{2}(t)+ (2C_{2}+C_{4}+C_{6})J_{n}(t) \bigl(I_{n}(t)\bigr)^{\frac{1}{p^{+}}}. \end{aligned}$$
Thus,
$$\begin{aligned} \int_{t_{1}}^{t_{2}}I_{n}(\tau)\,d\tau\leq \frac{3}{a_{0}} J_{n}^{2}(t)+(2C_{2}+C_{4}+C_{6})J_{n}(t) \bigl(I_{n}(t)\bigr)^{\frac{1}{p^{+}}}. \end{aligned}$$
By \(I_{n}(t+1)\leq3\int_{t_{1}}^{t_{2}}I_{n}(\tau)\,d\tau\) and \(I_{n}(t+1)=I_{n}(t)-J_{n}^{2}(t)\), we have
$$\begin{aligned} I_{n}(t)\leq\biggl(1+\frac{9}{a_{0}}\biggr) J_{n}^{2}(t)+(6C_{2}+3C_{4}+3C_{6})J_{n}(t) \bigl(I_{n}(t)\bigr)^{\frac{1}{p^{+}}}. \end{aligned}$$
Further, Young’s inequality yields
$$\begin{aligned} I_{n}(t)\leq C_{7}J_{n}^{2}(t)+C_{8} \bigl(J_{n}(t)\bigr)^{\frac{p^{+}}{p^{+}-1}}. \end{aligned}$$
(5.9)
Now we divide the proof in two cases: \(p^{+}=2\) and \(p^{+}>2\). We consider the case \(p^{+}=2\) first. By the boundedness of \(J_{n}(t)\), we have \(I_{n}(t)\leq C_{9}J_{n}^{2}(t)\). Since \(I_{n}(t)\) is nonincreasing, by Lemma 5.1, there exist constants \(C>0\) and \(\gamma>0\) such that
$$\begin{aligned} I_{n}(t)\leq C \mathrm{e}^{-\gamma t}, \quad\forall t\geq0. \end{aligned}$$
Letting \(n\rightarrow\infty\) in the above inequality, we arrive at
$$\begin{aligned} \int_{\Omega}\biggl|\frac{\partial u_{\varepsilon}}{\partial t}\biggr|^{2}\,dx+\int _{\Omega}|\Delta u_{\varepsilon}|^{p(x)}+|u_{\varepsilon}|^{q(x)}\,dx \leq C \mathrm{e}^{-\gamma t},\quad \mbox{a.e. } t\geq0. \end{aligned}$$
Thus,
$$\begin{aligned} \int_{\Omega}\biggl|\frac{\partial u_{\varepsilon}}{\partial t}\biggr|^{2}\,dx+\int _{\Omega}|u_{\varepsilon}|^{q(x)}\,dx\leq C \mathrm{e}^{-\gamma t},\quad \mbox{a.e. } t\geq0. \end{aligned}$$
Since \(\frac{\partial u_{\varepsilon}(x,t)}{\partial t}\rightharpoonup \frac{\partial u(x,t)}{\partial t}\) weakly in \(L^{2}(\Omega)\) uniformly on \([0,T]\) (\(\forall T>0\)) and \(u_{\varepsilon}\rightarrow u\) strongly in \(L^{q(x)}(\Omega)\) for a.e. \(t\in[0,T]\), we obtain
$$\begin{aligned} \int_{\Omega}\biggl|\frac{\partial u}{\partial t}\biggr|^{2}\,dx+\int _{\Omega}|u|^{q(x)}\,dx\leq C \mathrm{e}^{-\gamma t}, \quad\mbox{for a.e. } t\geq0. \end{aligned}$$
It remains to consider the case \(p^{+}>2\). It follows from (5.9) that \(I_{n}(t)\leq C_{10}(J_{n}(t))^{\frac{p^{+}}{p^{+}-1}}\). Employing Lemma 5.1, we obtain
$$\begin{aligned} I_{n}(t)\leq C(t+1)^{-\frac{p^{+}}{p^{+}-2}}. \end{aligned}$$
Then letting \(n\rightarrow\infty\), we deduce
$$\begin{aligned} \int_{\Omega}\biggl|\frac{\partial u_{\varepsilon}}{\partial t}\biggr|^{2}\,dx+\int _{\Omega}|u_{\varepsilon}|^{q(x)}\,dx\leq C (t+1)^{-\frac{p^{+}}{p^{+}-2}}, \quad \mbox{a.e. } t\geq0. \end{aligned}$$
Finally, letting \(\varepsilon\rightarrow0\), we conclude
$$\begin{aligned} \int_{\Omega}\biggl|\frac{\partial u}{\partial t}\biggr|^{2}\,dx+\int _{\Omega}|u|^{q(x)}\,dx\leq C (t+1)^{-\frac{p^{+}}{p^{+}-2}}, \quad\mbox{a.e. } t\geq0. \end{aligned}$$
Hence the theorem is proved. □

Declarations

Acknowledgements

The author was supported by the Fundamental Research Funds for the Central Universities (No. 3122015L014).

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
College of Science, Civil Aviation University of China

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