\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2016 (2016), No. 209, pp. 1--15.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2016 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2016/209\hfil Parabolic equations with $p(x)$-growth] {Weak solutions for parabolic equations with $p(x)$-growth} \author[N. Pan, B. Zhang, J. Cao \hfil EJDE-2016/209\hfilneg] {Ning Pan, Binlin Zhang, Jun Cao} \address{Ning Pan \newline Department of Mathematics, Northeast Forestry University, Harbin 150040, China} \email{hljpning@163.com} \address{Binlin Zhang (corresponding author)\newline Department of Mathematics, Heilongjiang Institute of Technology, Harbin 150050, China} \email{zhangbinlin2012@163.com} \address{Jun Cao \newline College of Mechanical and Electrical Engineering, Northeast Forestry University, Harbin 150040, China} \email{zdhcaojun@163.com} \thanks{Submitted April 2, 2016. Published August 2, 2016.} \subjclass{35K15, 35K20, 35K55} \keywords{Parabolic equation; $W^{1,x}L^{p(x)}(Q)$ space; $p(x)$-growth condition} \begin{abstract} In this article we study nonlinear parabolic equations with $p(x)$-growth in the space $W^{1,x}L^{p(x)}(Q)\cap L^\infty(0,T; L^2(\Omega))$. By using the method of parabolic regularization, we prove the existence and uniqueness of weak solutions for the equation $$\frac{\partial u}{\partial t}=\operatorname{div}(a(u) |\nabla u|^{p(x)-2}\nabla u)+f(x,t).$$ Also, we study the localization property of weak solutions for the above equation. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \newtheorem{corollary}[theorem]{Corollary} \allowdisplaybreaks \section{Introduction and statement of main results} Let $N\geq2$ be an integer and $\Omega$ be a bounded simply connected domain in $\mathbb{R}^N$. Let $Q$ be $\Omega\times(0,T)$ where $T>0$ is given. We consider the parabolic initial boundary-value problem \begin{equation} \begin{gathered} \frac{\partial u}{\partial t}=\operatorname{div}(a(u) |\nabla u|^{p(x)-2}\nabla u)+f(x,t),\quad (x,t)\in Q,\\ u(x,t)=0, \quad (x,t)\in \Gamma,\\ u(x,0)=u_0(x),\quad x\in \Omega. \end{gathered} \label{e1.1} \end{equation} where $\Gamma$ denotes the lateral boundary of the cylinder $Q$, and $a(u)=u^\sigma+d_0$ with $\sigma$ and $d_0$ two positive constants to be defined later. For the case $p$ constant, there are many results about the existence, uniqueness and the qualitative properties of the solutions, we refer the reader to \cite{2, 4, 1, 3}. In recent years, the research of variational problems with nonstandard growth conditions has been an interesting topic, see for examples \cite{7, 8, 13, 5, 19, 11, 17, 6, 12, 9, 16, 15, 18, 10} and the references therein. In \cite{5}, the authors studied the nonlinear parabolic equations with nonstandard anisotropic growth conditions: \begin{align}\label{eq1} u_t-\sum_i\frac{d}{dx_i}[a_i(z,u)|D_iu|^{p_i(z)-2}D_iu+b_i(z,u)]+d(z,u)=0 \end{align} where $z=(x,t)$. They proved the existence and uniqueness of weak solutions by applying Galerkin's method in the Orlicz-Sobolev spaces $W(Q)$ with the norm $\|u\|_{W(Q)}=\sum_i\|D_iu\|_{p_i(z),Q}+\|u\|_{2,Q}$. Note that the coefficient of nonlinearity in \cite{5} is allowed to depend on $x$ and $t$ and is assumed to be the Caratheodory function, and so problem \eqref{eq1} is called the evolutional $p(x, t)$-Laplacian. In \cite{6}, the authors considered the quasilinear degenerate parabolic problem with nonstandard growth: \begin{equation} \label{eq2} \begin{gathered} \frac{\partial u}{\partial t}=\operatorname{div}(a(u)|\nabla u|^{p(x,t)-2}\nabla u) +f(x,t),\quad (x,t)\in Q_T,\\ u(x,t)=0,\quad (x,t)\in \Gamma_T,\\ u(x,0)=u_0(x),\quad x\in \Omega. \end{gathered} \end{equation} and studied the existence, uniqueness and localization property of weak solutions for \eqref{eq2}. It is worthy pointing out that they used the Banach spaces $L^{p(x,t)}(Q_T)$ and $W(Q_T)$ which appeared in \cite{6} as solution space. Indeed, many authors dedicated to studying the variable exponent problems, in which $p(x,t)$ depends on $x$ and $t$, see for instance \cite{5, 6, 13, 15}. But for some important problems, the solution spaces only depending on variable $x$ for parabolic equations are needed. Note that $p(x)$-growth problems can be regarded as a kind of problems with nonstandard growth, which appear in nonlinear elastic, electrorheological fluids and other physics phenomena. For a recent overview of variable exponent spaces with applications to nonlinear partial differential equations we refer to \cite{9} and the references therein. To illustrate the significance of variable exponent spaces independent of the time variable $t$, we would like to mention a paper \cite{20}, which has been an excellent reference as the applications of variable exponent spaces. More precisely, the authors in \cite{20} studied the Dirichlet problem \begin{equation} \label{eq3} \frac{\partial u}{\partial t}-\operatorname{div}(\phi_r(x,Du)) +\lambda(u-I)=0,\quad (x,t)\in\Omega\times[0,T], \end{equation} which is a model for image denoising, enhancement, and restoration, where $\lambda\geq0$ is a constant, $$\phi(x,r)=\begin{cases} \frac{1}{q(x)}|r|^{q(x)},& |r|\leq\beta,\\ |r|-\frac{\beta q(x)-\beta^{q(x)}}{q(x)}, & |r|>\beta, \end{cases}$$ where $q(x)$ satisfies $1\leq q(x)\leq2$. They proved the existence and uniqueness of weak solutions and also discussed the behavior of weak solutions for \eqref{eq3} as $t\to \infty$. Notice that the direction and speed of diffusion at each location depend on the local behavior, hence $q(x)$ only depends on the location $x$ in the image. Thanks to this fact, the authors gave the above model which can study the denoising, enhancement, and restoration for the image well. Based on the above reason, we thus seek for a kind of space in which the variable exponent only depend on $x$ for problem \eqref{eq1}. Considering that the space $W^{1,x}L^{p(x)}(Q)$, which is different from the space $W(Q_T)$ in \cite{5, 6}, can provide a suitable framework to discuss the similar physical problems in \cite{20}, which was introduced and discussed in \cite{21, 22}, so we take this space as our working space to discuss the problem \eqref{eq1}, where $p(x)$ only depends on the space variable $x$, not on the time variable $t$. In this article, we will the existence, uniqueness and localization property of solutions for \eqref{eq1} in the space $W^{1,x}L^{p(x)}(Q)$. Throughout this paper, unless special statement, we always suppose that the exponent $p(x)$ is continuous on $\overline{\Omega}$ with logarithmic module of continuity \begin{gather} 1 < p^-=\inf_{x\in\Omega}p(x) \leq p(x)\leq \sup_{x\in\Omega} p(x) =p^+<\infty.\label{e1.4}\\ \forall x\in\Omega,\; y\in\Omega, |x-y|<1, \quad |p(x)-p(y)|\leq\omega(|x-y|),\label{e1.5} \end{gather} where $$\limsup_{\tau\to0^+}\omega(\tau)\ln\frac{1}{\tau}=C<+\infty.$$ First we give the definition of (weak) solutions for problem \eqref{e1.1}. \begin{definition} \label{def1.1} \rm A function $u(x,t)\in W^{1,x}L^{p(x)}(Q)\cap L^\infty(0,T; L^2(\Omega))$ is called a (weak) solution of \eqref{e1.1} if $$-\int_Q u\frac{\partial \varphi}{\partial t}\,dx\,dt +\int_\Omega u\varphi dx|^T_0 +\int_Q (u^\sigma+d_0)|\nabla u|^{p(x)-2}\nabla u\nabla \varphi \,dx\,dt =\int_Q f(x,t)\varphi \,dx\,dt$$ for all $\varphi\in C^1(0,T;C_0^\infty(\Omega))$. \end{definition} Now we are in a position to give results about the existence and uniqueness of solutions for problem \eqref{eq1}. \begin{theorem}\label{thm1} Let $p(x)$ satisfy \eqref{e1.4}--\eqref{e1.5}. If the following conditions hold \begin{itemize} \item[(H1)] $\max\{1, \frac{2N}{N+2}\}0\big\}}, $$where G=\{x\in\Omega: \omega>0\}, B_\rho(x)=\{y\in\Omega: |x-y|<\rho\}. Hence we can present the localization property of solutions. \begin{theorem}\label{thm3} Assume that the hypotheses of Theorem \ref{thm2} are satisfied and 2<\sigma<\frac{2(p^+-p^-)}{p^-(p^+-1)}, \operatorname{supp}u_0\in\Omega. If u is a nonnegative solution of problem \eqref{eq1} and f\equiv 0, then \operatorname{supp}u \subset \operatorname{supp}u_0 a.e. in Q. \end{theorem} This paper is organized as follows. In Section 2, we shall introduce the space W^{m,x}L^{p(x)}(Q) and the necessary properties, which will be needed later. Section 3 and Section 4 are devoted to proving the existence and uniqueness of solutions for problem \eqref{eq1} respectively. In Section 5, we will discuss the localization property of solutions to problem \eqref{eq1}. \section{Preliminaries} In this section we recall the basic knowledge of the general spaces L^{p(x)}(\Omega), W^{m,p(x)}(\Omega) and W^{m,x}L^{p(x)}(Q) and the necessary results which will be useful in the sequel, we refer to \cite{21, 22, 23, 24} for more details. Denote$$ E=\{\omega: \omega \text{ is a measurable function on } \Omega\}, $$where \Omega\subset \mathbb{R}^N is an open subset. Let p(x):\Omega\to[1, \infty] be an element in E. Denote \Omega_\infty=\{x\in\Omega: p(x)=\infty\}. For u\in E, we define$$ \rho(u)=\int_{\Omega\setminus \Omega_\infty} |u(x)|^{p(x)}dx+ \operatorname{ess, sup}_{x\in\Omega_\infty}|u(x)|. $$The space L^{p(x)}(\Omega)=\{u\in E: \exists\lambda>0, \rho(\lambda u)<\infty\} endowed with the norm$$ \|u\|_{L^{p(x)}(\Omega)}=\inf\{\lambda>0: \rho(\frac{u}{\lambda})\leq1\}. $$We define the conjugate function p'(x) of p(x) by$$ p'(x)=\begin{cases} \infty, &\text{if } p(x)=1;\\ 1,&\text{if } p(x)=\infty;\\ \frac{p(x)}{p(x)-1}, &\text{if } 11)$ $\Leftrightarrow \rho(u)<1(=1,>1)$. \item If $\|u\|_{L^{p(x)}(\Omega)}\geq1$, then $\|u\|_{L^{p(x)}(\Omega)}^{p^-}\leq\rho(u)\leq\|u\|_{L^{p(x)}(\Omega)}^{p^+}$. \item If $\|u\|_{L^{p(x)}(\Omega)}\leq1$, then $\|u\|_{L^{p(x)}(\Omega)}^{p^+}\leq\rho(u)\leq\|u\|_{L^{p(x)}(\Omega)}^{p^-}$. \end{enumerate} \end{lemma} Let $m>0$ be an integer. For each $\alpha=(\alpha_1, \alpha_2,\cdots, \alpha_n)$, $\alpha_i$ are nonnegative integers and $|\alpha|=\Sigma^n_{i=1}\alpha_i$, and denote by $D^\alpha$ the distributional derivative of order $\alpha$ with respect to the variable $x$. We now introduce the generalized Lebesgue-Sobolev space $W^{m,p(x)}(\Omega)$ which is defined as $$W^{m,p(x)}(\Omega)=\{u\in L^{p(x)}(\Omega): D^\alpha u\in L^{p(x)}(\Omega), |\alpha|\leq m\}.$$ Here $W^{m,p(x)}(\Omega)$ is a Banach space endowed with the norm $$\|u\|=\sum_{|\alpha|\leq m}\|D^\alpha u\|_{L^{p(x)}(\Omega)}.$$ The space $W^{m,p(x)}_0(\Omega)$ is defined as the closure of $C^\infty_0(\Omega)$ in $W^{m,p(x)}(\Omega)$. The dual space $(W^{m,p(x)}_0(\Omega))^*$ is denoted by $W^{-m,p'(x)}(\Omega)$ equipped with the norm $$\|f\|_{W^{-m,p'(x)}(\Omega)}=\inf\Sigma_{|\alpha|\leq m}\|f_\alpha\|_{L^{p'(x)}(\Omega)},$$ where infimum is taken on all possible decompositions $$f=\sum_{|\alpha|\leq m}(-1)^{|\alpha|}D^\alpha f_\alpha, \quad f_\alpha\in L^{p'(x)}(\Omega).$$ \begin{lemma}[\cite{21}] \label{lem2.5} (i) $W^{m,p(x)}(\Omega)$ and $W^{m,p(x)}_0(\Omega)$ are separable if $1\leq p(x)<\infty$. (ii) $W^{m,p(x)}(\Omega)$ and $W^{m,p(x)}_0(\Omega)$ are reflexive if \eqref{e1.4} holds. \end{lemma} We define the space $$W^{m,x}L^{p(x)}(Q)=\{u\in L^{p(x)}(Q): D^\alpha u\in L^{p(x)}(Q), |\alpha|\leq m\}.$$ It is easy to see that $W^{m,x}L^{p(x)}(Q)$ is a Banach space with the norm $\|u\|=\sum_{|\alpha|\leq m}\|D^\alpha u\|_{L^{p(x)}(Q)}$, where $p(x)$ is independent of $t$, see \cite{19} for further discussions. The space $W^{m,x}_0L^{p(x)}(Q)$ is defined as the closure of $C^\infty_0(Q)$ in $W^{m,x}L^{p(x)}(Q)$ and $W^{m,x}_0L^{p(x)}(Q)\hookrightarrow L^{p(x)}(Q)$ is continuous embedding. Let $\bar{M}$ be the number of multiindexes $\alpha$ which satisfies $0\leq|\alpha|\leq m$, then the space $W^{m,x}_0L^{p(x)}(Q)$ can be considered as a close subspace of the product space $\Pi_{i=1}^{\bar{M}} L^{p(x)}(Q)$. So if 1| =\inf\sum_{|\alpha|\leq m}\|f_\alpha\|_{L^{p'(x)}(Q)}, $$where the infimum is taken on all possible decompositions$$ f=\sum_{|\alpha|\leq m}(-1)^{|\alpha|}D_x^\alpha f_\alpha, \quad f_\alpha\in L^{p'(x)}(Q). In what follows, we denote \|u(x,t)\|_{k,\Omega}=\big(\int_\Omega |u(x,t)|^k dx\big)^{1/k}, \|u(x,t)\|_{\infty,Q}=\sup_{(x,t)\in Q}|u(x,t)|. \section{Existence of solutions} Let us consider the auxiliary parabolic problem \begin{equation} \begin{gathered} \frac{\partial u}{\partial t}=\operatorname{div}(a_{n,H}(u) |\nabla u|^{p(x)-2}\nabla u)+f(x,t),\quad (x,t)\in Q,\\ u(x,t)=0,\quad (x,t)\in \Gamma,\\ u(x,0)=u_0(x),\quad x\in \Omega, \end{gathered} \label{e3.1} \end{equation} here H stands for a positive parameter to be chosen later and notice that $0H;\\ u_n,&\text{if } |u_n|\leq H;\\ -H,&\text{if } u_n<-H. \end{cases}$ We choose the function u_{nH}^{2k-1} as a test function in \eqref{e3.2} with k\in N. In \eqref{e3.2}, let t_2=t+h, t_1=t, with t,t+h\in(0,T). Then \begin{align*} &\int_{t}^{t+h}\int_\Omega\Big[(u_{nH})_tu_{nH}^{2k-1} +a_{n,H}(u_{nH})|\nabla u_{nH}|^{p(x)-2}\nabla u_{nH} \nabla u_{nH}^{2k-1}\\ &-f(x,t)u_{nH}^{2k-1}\Big]\,dx\,dt=0, \end{align*} i.e. \begin{equation} \begin{aligned} &\frac{1}{2k}\int_{t}^{t+h}\frac{d}{dt}\Big(\int_\Omega u_{nH}^{2k}dx\Big)dt\\ &+\int_{t}^{t+h}\int_\Omega (2k-1)a_{n,H}(u_{nH})u_{nH}^{2(k-1)} |\nabla u_{nH}|^{p(x)}\,dx\,dt \\ &=\int_{t}^{t+h}\int_\Omega f(x,t)u_{nH}^{2k-1}\,dx\,dt. \end{aligned} \label{e3.3} \end{equation} Dividing by h, letting h\to0, and applying Lebesgue's dominated convergence theorem, we have that for all t\in (0,T), \begin{equation} \begin{aligned} &\frac{1}{2k}\frac{d}{dt}\int_\Omega u_{nH}^{2k}dx +\int_\Omega (2k-1)a_{n,H}(u_{nH})u_{nH}^{2(k-1)}|\nabla u_{nH}|^{p(x)}dx \\ &= \int_\Omega f(x,t)u_{nH}^{2k-1}dx.\label{e3.4} \end{aligned} \end{equation} By Lemma \ref{lem2.3}, the right-hand side of the above equality can be rewritten as \big|\int_\Omega f(x,t)u_{nH}^{2k-1}dx\big| \leq \|u_{nH}\|^{2k-1}_{2k,\Omega}\|f\|_{2k,\Omega},\quad k=1,2,3,\dots, whence \begin{equation} \begin{aligned} &\| u_{nH}\|^{2k-1}_{2k,\Omega}\frac{d}{dt}(\| u_{nH}\|_{2k,\Omega})+(2k-1) \int_\Omega a_{n,H}(u_{nH})u_{nH}^{2(k-1)}|\nabla u_{nH}|^{p(x)}dx \\ &\leq \|u_{nH}\|^{2k-1}_{2k,\Omega}\|f\|_{2k,\Omega},\quad k=1,2,3,\dots \end{aligned}\label{e3.5} \end{equation} Integrating over (0,t) for the above inequality for all t, we obtain \|u_{nH}(\cdot,t)\|_{2k,\Omega}\leq \|u_{nH}(\cdot,0)\|_{2k,\Omega} +\int_0^T\|f\|_{2k,\Omega}dt,\quad \forall k\in N. $$Letting k\to\infty, one gets$$ \|u_{nH}(\cdot,t)\|_{\infty,\Omega} \leq \|u_{nH}(\cdot,0)\|_{\infty,\Omega} +\int_0^T\|f\|_{\infty,\Omega}dt\leq \|u_{0}\|_{\infty,\Omega} +\int_0^T\|f\|_{\infty,\Omega}dt. $$If we choose H>K(T), then u_{nH}(\cdot,t)\leq \sup |u_{nH}(\cdot,t)|\leq K(T)K(T), one gets a_{n,H}(u_{n})\geq u_n^\sigma, hence we obtain \eqref{e3.6}. \end{proof} \begin{lemma} \label{lem3.3} The solution of \eqref{e3.1} satisfies the estimate$$ \|u_{nt}\|_{W^{-1,x}L^{p(x)}(Q)}\leq C(H,\sigma, p^\pm,K(T),|\Omega|). \end{lemma} \begin{proof} From \eqref{e3.2}, for \xi\in W_0^{1,x}L^{p(x)}(Q) we have \begin{align*} &\int_Qu_{nt}\xi \,dx\,dt\\ &=-\int_Q\Big[\Big(u_n^2+\frac{1}{n^2}\Big)^{\sigma/2} +d_0\Big]|\nabla u_n|^{p(x)-2}\nabla u_n\nabla\xi \,dx\,dt+\int_Qf\xi \,dx\,dt\\ &\leq \int_Q\Big[\Big(u_n^2+\frac{1}{n^2}\Big)^{\sigma/2} +d_0\Big]|\nabla u_n|^{p(x)-1}|\nabla\xi| \,dx\,dt+\int_Q|f||\xi| \,dx\,dt\\ &\leq2\|[(u_n^2+\frac{1}{n^2})^{\sigma/2}+d_0]|\nabla u_n|^{p(x)-1} \|_{p'(x)}\|\nabla\xi\|_{p(x)} +2\|f\|_{p'(x)}\|\xi \|_{p(x)}\\ &\leq2\max \Big\{\Big(\int_Q \Big\{\Big[(u_n^2+\frac{1}{n^2})^{\sigma/2} +d_0\Big]|\nabla u_n|^{p(x)-1}\Big\}^{\frac{p(x)}{p(x)-1}}\,dx\,dt\Big)^{\frac{1}{p'^+}}, \\ &\Big(\int_Q\Big\{\Big[\Big(u_n^2+\frac{1}{n^2}\Big)^{\sigma/2} +d_0\Big]|\nabla u_n|^{p(x)-1}\Big\}^{\frac{p(x)}{p(x)-1}}\,dx\,dt \Big)^{\frac{1}{p'^-}}\Big\}\|\nabla\xi\|_{p(x)}\\ &+2\max\Big\{\Big(\int_Q|f|^{p'(x)}\,dx\,dt\Big)^{\frac{1}{p'^+}}, \Big(\int_Q|f|^{p'(x)}\,dx\,dt\Big)^{\frac{1}{p'^-}}\Big\}\|\xi\|_{p(x)}\\ &\leq (2((K^2(T)+1)^{\sigma/2}+d_0)^{\frac{1}{p^{\pm}-1}}K(T) |\Omega|H+2|f|_\infty|T|)\|\xi\|_{W^{1,x}L^{p(x)}(Q)}, \end{align*} which yields the desired conclusion. \end{proof} From the above conclusion and the uniform estimates in n, we obtain a subsequence, still denoted \{u_n\}_n, such that \begin{equation} \begin{gathered} u_n\to u \quad\text{a.e. in } Q;\\ \nabla u_n\rightharpoonup \nabla u \quad \text{weakly in } L^{p(x)}(Q);\\ u_n^\sigma|\nabla u_n|^{p(x)-2}D_iu_n\rightharpoonup A_i(x,t) \quad \text{weakly in } L^{p'(x)}(Q);\\ |\nabla u_n|^{p(x)-2}D_iu_n\rightharpoonup W_i(x,t) \quad \text{weakly in } L^{p'(x)}(Q), \end{gathered} \label{e3.9} \end{equation} for u\in W^{1,x}L^{p(x)}(Q), A_i(x,t)\in L^{p'(x)}(Q), W_i(x,t)\in L^{p'(x)}(Q). \begin{lemma} \label{lemma3.3} For almost all (x,t)\in Q, \lim_{n\to \infty}\int_{Q}\Big(\Big(u_n^2+\frac{1}{n^2}\Big)^{\sigma/2} -u^\sigma_n\Big)|\nabla u_n|^{p(x)-2}\nabla u_n\nabla \xi \,dx\,dt=0, \quad \forall\xi\in W_0^{1,x}L^{p(x)}(Q). \end{lemma} \begin{proof} By Young's inequality, we have \begin{align*} I &:= \int_Q\Big(\Big(u_n^2+\frac{1}{n^2}\Big)^{\sigma/2} -u^\sigma_n\Big)|\nabla u_n|^{p(x)-2}\nabla u_n\nabla \xi \,dx\,dt\\ &=\frac{\sigma}{2}\frac{1}{n^2}\int_Q\Big(\int_0^1 \Big(u_n^2+s\frac{1}{n^2}\Big)^{\frac{\sigma-2}{2}}ds\Big) |\nabla u_n|^{p(x)-2}\nabla u_n\nabla \xi \,dx\,dt\\ &\leq \sigma \frac{1}{n^2}\Big(K^2(T)+1\Big)^{\frac{\sigma-2}{2}}\| |\nabla u_n|^{p(x)-1}\|_{p'(x)}\|\nabla \xi\|_{p(x)}\\ &\leq C \frac{1}{n^2}\Big\{\Big(\int_Q|\nabla u_n|^{p(x)}\,dx\,dt \Big)^{\frac{p^+-1}{p^+}},\Big(\int_Q|\nabla u_n|^{p(x)}\,dx\,dt \Big)^{\frac{p^--1}{p^-}}\Big\}\|\nabla \xi\|_{p(x)}. \end{align*} By \eqref{e3.7}, we obtain I\leq C H \big(\frac{1}{n}\big)^{2-\sigma\frac{p^+-1}{p^+}}\|\nabla\xi\|_{p(x)}. $$Letting n\to\infty, we obtain the desired conclusion. \end{proof} \begin{lemma} \label{lemma3.4} For almost all (x,t)\in Q,$$ A_i(x,t)=u^\sigma W_i(x,t),\quad i=1,2,\dots , N. \end{lemma} \begin{proof} In \eqref{e3.9}, letting n\to\infty, we have \begin{gather} \int_Qu_n^\sigma|\nabla u_n|^{p(x)-2}\nabla u_n\nabla \xi \,dx\,dt \to \sum_{i=1}^N \int_QA_i(x,t)D_i\xi \,dx\,dt;\label{e3.10} \\ \int_Q|\nabla u_n|^{p(x)-2}\nabla u_n\nabla \xi \,dx\,dt\to \sum_{i=1}^N \int_QW_i(x,t)D_i\xi \,dx\,dt.\label{e3.11} \end{gather} By Lebesgue's dominated convergence theorem we have \begin{equation} \lim_{n\to\infty}\sum_{i=1}^N\int_Q(u_n^\sigma-u^\sigma)A_i(x,t)D_i\xi \,dx\,dt=0. \label{e3.12} \end{equation} From \eqref{e3.9} it follows that \begin{align*} &\lim_{n\to\infty}\sum_{i=1}^N\int_Q[u_n^\sigma|\nabla u_n|^{p(x)-2} D_iu_n-u^\sigma W_i(x,t)]D_i\xi \,dx\,dt \\ &=\lim_{n\to\infty}\sum_{i=1}^N\int_Q[(u_n^\sigma-u^\sigma) |\nabla u_n|^{p(x)-2}D_iu_n \\ &\quad +u^\sigma(|\nabla u_n|^{p(x)-2}D_iu_n-W_i(x,t))]D_i\xi \,dx\,dt =0. \end{align*} By \eqref{e3.10}--\eqref{e3.12} and the above equalities, we complete the proof. \end{proof} \begin{lemma} \label{lem3.6} For almost all (x,t)\in Q, W_i(x,t)=|\nabla u|^{p(x)-2} D_i(u),\quad i=1,2,\dots N. \end{lemma} \begin{proof} In \eqref{e3.2}, choosing \xi=(u_n-u)\Phi with \Phi\in W_0^{1,x}L^{p(x)}(Q), \Phi\geq0, we have \begin{align*} &\int_Q[u_{nt}(u_n-u)\Phi+\Phi(u_n^\sigma+d_0)|\nabla u_n|^{p(x)-2} \nabla u_n\nabla(u_n-u)] \,dx\,dt \\ +&\int_Q[(u_n-u)(u_n^\sigma+d_0)|\nabla u_n|^{p(x)-2} \nabla u_n\nabla\Phi-f(x,t)(u_n-u)\Phi]\,dx\,dt \\ +&\int_Q ((u_n^2+\frac{1}{n^2})^{\sigma/2}-u_n^\sigma) |\nabla u_n|^{p(x)-2}\nabla u_n\nabla\xi \,dx\,dt=0. \end{align*} It follows that \begin{equation} \int_Q\Phi(u_n^\sigma+d_0)|\nabla u_n|^{p(x)-2}\nabla u_n\nabla(u_n-u)] \,dx\,dt=0. \label{e3.13} \end{equation} On the other hand, by the fact that u_n,\ u\in L^\infty(Q) and |\nabla u|\in L^{p(x)}(Q), we have \begin{gather} \lim_{n\to\infty}\int_Q\Phi(u^\sigma+d_0)|\nabla u|^{p(x)-2} \nabla u\nabla(u_n-u)\,dx\,dt=0.\label{e3.14} \\ \lim_{n\to\infty}\int_Q\Phi(u^\sigma_n-u^\sigma) |\nabla u|^{p(x)-2}\nabla u\nabla(u_n-u)\,dx\,dt=0.\label{e3.15} \end{gather} Note that \begin{equation} \begin{aligned} 0&\leq(|\nabla u|^{p(x)-2}\nabla u_n-|\nabla u|^{p(x)-2}\nabla u)\nabla(u_n-u) \\ &\leq\frac{1}{d_0}[(u_n^\sigma+d_0)|\nabla u_n|^{p(x)-2} \nabla u_n-(u_n^\sigma-u^\sigma)|\nabla u|^{p(x)-2}\nabla u]\nabla(u_n-u) \\ &\quad -\frac{1}{d_0}(u^\sigma+d_0)|\nabla u|^{p(x)-2} \nabla u\nabla (u_n-u). \end{aligned}\label{e3.16} \end{equation} Bring \eqref{e3.13}--\eqref{e3.15} into \eqref{e3.16}, we obtain \lim_{n\to\infty}\int_Q \Phi(|\nabla u_n|^{p(x)-2} \nabla u_n-|\nabla u|^{p(x)-2}\nabla u)\nabla(u_n-u)\,dx\,dt=0. The rest arguments are the same as those of \cite[Theorem 2.1]{25}. Thus the existence of weak solutions for problem \eqref{eq1} is obtained by a standard limiting process. \end{proof} \section{Uniqueness of solutions} In this section, we study the uniqueness of the solutions to \eqref{e1.1}. To obtain the main conclusion of this section, we need the following lemma. \begin{lemma} \label{lem4.1} Let M(s)=|s|^{p(x)-2}s, then for all \xi, \eta \in \mathbb{R}^N, \begin{align*} &(M(\xi)-M(\eta))(\xi-\eta)\\ &\geq \begin{cases} 2^{-p(x)}|\xi-\eta|^{p(x)}, &\text{if } 2\leq p(x)<\infty;\\ (p(x)-1)|\xi-\eta|^2(|\xi|^{p(x)}+|\eta|^{p(x)})^{\frac{p(x)-2}{p(x)}}, &\text{if } 10 such that for some 0<\tau\leq T, w=u-v>\delta on the set \Omega_\delta=\Omega \cap \{x:w(x,t)>\delta\} and \mu(\Omega_\delta)>0. Let F_\varepsilon(\xi)=\begin{cases} \frac{1}{\alpha-1}\varepsilon^{1-\alpha}-\frac{1}{\alpha-1}\xi^{1-\alpha}, &\text{if } \xi>\varepsilon;\\ 0,&\text{if } \xi\leq\varepsilon. \end{cases} where \delta>2\varepsilon>0 and \alpha=\sigma/2. By the definition of weak solution, we take a test-function \xi=F_\varepsilon(w), \begin{equation} \begin{aligned} 0&=\int_{Q_\tau}[w_tF_\varepsilon(w)+(v^\sigma+d_0)(|\nabla u|^{p(x)-2} \nabla u-|\nabla v|^{p(x)-2}\nabla v)\nabla F_\varepsilon(w)]\,dx\,dt \\ &\quad +\int_{Q_\tau}(u^\sigma-v^\sigma)|\nabla u|^{p(x)-2} \nabla u\nabla F_\varepsilon(w)\,dx\,dt \\ &=\int_{Q_{\varepsilon,\tau}}w_tF_\varepsilon(w)\,dx\,dt \\ &\quad +\int_{Q_{\varepsilon,\tau}}(v^\sigma+d_0) w^{-\alpha}(|\nabla u|^{p(x)-2}\nabla u-|\nabla v|^{p(x)-2}\nabla v) \nabla w\,dx\,dt \\ &\quad +\int_{Q_{\varepsilon,\tau}}(u^\sigma-v^\sigma)w^{-\alpha} |\nabla u|^{p(x)-2}\nabla u\nabla w\,dx\,dt \\ &=J_1+J_2+J_3, \end{aligned} \label{e4.1} \end{equation} with Q_{\varepsilon,\tau}=Q_\tau\cap\{(x,t)\in Q_\tau:w>\varepsilon\}. Now, let t_0=\inf\{t\in(0,\tau]:w>\varepsilon\}, then we estimate J_1, J_2, J_3. \begin{equation} \begin{aligned} J_1&=\int_{Q_{\varepsilon,\tau}}w_tF_\varepsilon(w)\,dx\,dt\\ &=\int_{\Omega}\Big(\int_0^{t_0}w_tF_\varepsilon(w)dt +\int_{t_0}^{\tau}w_tF_\varepsilon(w)dt\Big)dx \\ &\geq \int_\Omega\int_{\varepsilon}^{w(x,\tau)}F_\varepsilon(s)\,ds\,dx \\ &\geq \int_{\Omega_\delta}\int_\varepsilon^{w(x,\tau)}F_\varepsilon(s)\,ds\,dx\\ &\geq\int_{\Omega_\delta} (w-2\varepsilon)F_\varepsilon(\varepsilon)dx \\ &\geq(\delta-2\varepsilon)F_\varepsilon(\varepsilon)\mu(\Omega_\delta), \end{aligned}\label{e4.2} \end{equation} Let us first consider the case p^-\geq 2. By the first inequality of Lemma \ref{lem4.1}, we obtain \begin{equation} \begin{aligned} J_2&=\int_{Q_{\varepsilon,\tau}}(v^\sigma+d_0)w^{-\alpha}(|\nabla u|^{p(x)-2} \nabla u-|\nabla v|^{p(x)-2}\nabla v)\nabla w\,dx\,dt \\ &\geq \int_{Q_{\varepsilon,\tau}}(v^\sigma+d_0)w^{-\alpha}2^{-p(x)} |\nabla w|^{p(x)}\,dx\,dt \\ &\geq 2^{-p^+}\int_{Q_{\varepsilon,\tau}}(v^\sigma+d_0)w^{-\alpha} |\nabla w|^{p(x)}\,dx\,dt\geq 0, \end{aligned} \label{e4.3} \end{equation} Noting that \frac{p(x)}{p(x)-1}\geq \frac{p^+}{p^+-1}=\alpha>1 and applying Young's inequality, we estimate integrand of J_3 in the following way \begin{equation} \begin{aligned} &|(u^\sigma-v^\sigma)w^{-\alpha}|\nabla u|^{p(x)-2}\nabla u\nabla w| \\ &=\big|\sigma w\int_0^1(\theta u+(1-\theta)v)^{\sigma-1} d\theta w^{-\alpha}|\nabla u|^{p(x)-2}\nabla u\nabla w\big| \\ &\leq \frac{C}{W^\alpha}\Big[\frac{v^\sigma+d_0}{C}|\nabla w|^{p(x)} +C_1(\sigma,d_0,K(T),p^\pm)|w|^{p'(x)}|\nabla u|^{p(x)}\Big] \\ &\leq \frac{v^\sigma+d_0}{2^{p^++1}w^\alpha}|\nabla w|^{p(x)} +C_1(\sigma,d_0,K(T),p^\pm)|w|^{p'(x)-\alpha}|\nabla u|^{p(x)} \\ &\leq \frac{v^\sigma+d_0}{2^{p^++1}w^\alpha}|\nabla w|^{p(x)} +C_1(\sigma,d_0,K(T),p^\pm)|\nabla u|^{p(x)}. \end{aligned}\label{e4.4} \end{equation} Substituting \eqref{e4.4} into J_3, we obtain \begin{equation} J_3\leq \frac{1}{2}J_2+C\int_{Q_{\varepsilon,\tau}}|\nabla u|^{p(x)}\,dx\,dt. \label{e4.5} \end{equation} Next we consider the case 12. According to the second inequality of Lemma \ref{lem4.1}, it is easy to see that the following inequalities hold \begin{equation} \begin{aligned} J_2&=\int_{Q_{\varepsilon,\tau}}(v^\sigma+d_0)w^{-\alpha} (|\nabla u|^{p(x)-2}\nabla u-|\nabla v|^{p(x)-2}\nabla v)\nabla w\,dx\,dt \\ &\geq (p^--1)\int_{Q_{\varepsilon,\tau}}(v^\sigma+d_0) w^{-\alpha}(|\nabla u|+|\nabla v|)^{p(x)-2}|\nabla w|^{2}\,dx\,dt\geq 0. \end{aligned}\label{e4.6} \end{equation} Using Young's inequality and the fact that 1<\alpha\leq\frac{p^+}{p^+-1}\leq 2, we evaluate integrand of J_3 as follows: \begin{equation} \begin{aligned} &|(u^\sigma-v^\sigma)w^{-\alpha}|\nabla u|^{p(x)-2}\nabla u\nabla w| \\ &= \big|\sigma w\int_0^1(\theta u+(1-\theta)v)^{\sigma-1} d\theta w^{-\alpha}|\nabla u|^{p(x)-2}\nabla u\nabla w\big| \\ &\leq \frac{(v^\sigma+d_0)(p^--1)}{2w^\alpha}(|\nabla u|+|\nabla v|)^{p(x)-2} |\nabla w|^2 \\ &\quad +C_1(\sigma,d_0,K(T),p^\pm)|w|^{2-\alpha}(|\nabla u|+|\nabla v|)^{p(x)} \\ &\leq \frac{(v^\sigma+d_0)(p^--1)}{2w^\alpha}(|\nabla u|+|\nabla v|)^{p(x)-2} |\nabla w|^2 \\ &\quad +C_1(\sigma,d_0,K(T),p^\pm)(|\nabla u|+|\nabla v|)^{p(x)}. \end{aligned}\label{e4.7} \end{equation} Inserting \eqref{e4.7} into J_3, we obtain J_3\leq \frac{1}{2}J_2+C\int_{Q_{\varepsilon,\tau}} (|\nabla u|+|\nabla v|)^{p(x)}\,dx\,dt. $$Plugging the estimates \eqref{e4.2}, \eqref{e4.3}, \eqref{e4.5} and \eqref{e4.2}, \eqref{e4.6}, \eqref{e4.7} into \eqref{e4.1} and dropping the nonnegative terms, we arrive at the inequality$$ (\delta-2\varepsilon)(1-2^{1-\alpha})\varepsilon^{1-\alpha} \mu(\Omega_\delta)\leq\tilde{C}, $$with a constant \tilde{C} independent of \varepsilon. Notice that \lim_{\varepsilon\to 0}(\delta-2\varepsilon)(1-2^{1-\alpha}) \varepsilon^{1-\alpha}\mu(\Omega_\delta)=+\infty, we obtain a contradiction. This means \mu(\Omega_\delta)=0 and w\leq 0, a.e. in Q_\tau. Thus the proof is complete. \section{Localization property of solutions} In this section, we shall focus on the study of localization of solutions to problem \eqref{e1.1}. The proof is similar to that of \cite[Theorem 4.1]{6}, we would like to give the detailed treatment, just for the reader's convenience. In fact, by Definition \ref{def1.1}, it follows easily that \begin{equation} \label{e5.1} \int_{Q}u_\tau\xi+(u^\sigma+d_0)|\nabla u|^{p(x)-2}\nabla u\nabla \xi=0, \end{equation} with \tau\in(0,T). Let$$ \Psi=\inf\{\text{dist}(x, \operatorname{supp}\ u_0\cup\partial\Omega)/\lambda, 1\}, where 0<\lambda<1, and F_\varepsilon(\xi) is mentioned in Section 4 with \alpha=\sigma /2. Taking \xi=\Psi F_\varepsilon(u) (0<\varepsilon<1) and substituting it into \eqref{e5.1}, we obtain \begin{equation} \begin{aligned} 0&=\int_{Q_{\varepsilon,\tau}}u_t\Psi F_\varepsilon(u)\,dx\,dt +\int_{Q_{\varepsilon,\tau}}\Psi(u^\sigma+d_0)|\nabla u|^{p(x)-2} \nabla u\nabla F_\varepsilon(u)\,dx\,dt\\ &+\int_{Q_{\varepsilon,\tau}} F_\varepsilon(u)(u^\sigma+d_0)|\nabla u|^{p(x)-2}\nabla u\nabla\Psi \,dx\,dt := I_1+I_2+I_3. \end{aligned}\label{e5.2} \end{equation} with Q_{\varepsilon,\tau}=Q_\tau\cap\{(x,t)\in Q_\tau: u>\varepsilon\}. Denote E=\{x\in\{\Psi=1\}:u(x,\tau)>\delta\} with \delta>2\varepsilon>0, then \begin{equation} \begin{aligned} I_1 &=\int_{Q_{\varepsilon,\tau}} u_t\Psi F_\varepsilon(u)\,dx\,dt \geq\int_{\Omega_\varepsilon}\chi_{\operatorname{supp}\Psi} \Psi\int^u_\varepsilon F_\varepsilon(s)\,ds\,dx\\ &\geq \int_{\Omega_\varepsilon}\chi_{\operatorname{supp}\Psi}\Psi(u-\varepsilon) F_\varepsilon(\delta)dx \\ &\geq \big(\delta-\frac{3}{2}\varepsilon\big)F_\varepsilon \big(\frac{3}{2}\varepsilon\big)\operatorname{meas}(E). \end{aligned}\label{e5.3} \end{equation} and \begin{equation} \begin{aligned} I_2&=\int_{Q_{\varepsilon,\tau}}\Psi(u^\sigma+d_0)|\nabla u|^{p(x)-2} \nabla u\nabla \frac{1}{\alpha-1}(-u^{1-\alpha})\,dx\,dt\\ &\geq \int_{Q_{\varepsilon,\tau}}\Psi(u^\sigma+d_0)|\nabla u|^{p(x)} u^{-\alpha}\,dx\,dt\geq0. \end{aligned} \label{e5.4} \end{equation} Applying Young's inequality with \eta and choosing \eta=(\varepsilon^\beta)^{1-p(x)}, we may estimate that \begin{equation} \begin{aligned} |I_3|&=\big|\int_{Q_{\varepsilon,\tau}}F_\varepsilon(u)(u^\sigma+d_0) |\nabla u|^{p(x)-2}\nabla u\nabla\Psi \,dx\,dt\big| \\ &\leq C \int_{Q_{\varepsilon,\tau}}\varepsilon^{1-\alpha}|\nabla u|^{p(x)-1} |\nabla\Psi|\,dx\,dt\\ &\leq C(\sigma,d_0,K(T),p^\pm)\varepsilon^{\beta+\frac{(1-\alpha)p^-}{p^--1}} \int_{Q_{\varepsilon,\tau}}|\nabla u|^{p(x)}\,dx\,dt \\ &\quad +\varepsilon^{\beta(1-p^+)}\int_{Q_{\varepsilon,\tau}}|\nabla \Psi|^{p(x)}\,dx\,dt, \end{aligned} \label{e5.5} \end{equation} where C>0 denote the various constants. Choosing \beta=\frac{\alpha p^--1}{p^--1}>0 and putting \eqref{e5.3}--\eqref{e5.5} into \eqref{e5.2}, we deduce \begin{equation} \begin{aligned} &\frac{1}{2}[1-(3/2)^{1-\alpha}] \varepsilon^{2-\alpha-\beta+\frac{(\alpha-1)p^-}{p^--1}}\operatorname{meas}(E)\\ &\leq (\delta-3\varepsilon/2)[1-(3/2)^{1-\alpha}]\varepsilon^{1-\alpha-\beta +\frac{(\alpha-1)p^-}{p^--1}}\operatorname{meas}(E)\\ &\leq \widetilde{C}\Big(1+\varepsilon^{\frac{(\alpha-1)p^-}{p^--1} -\beta p^+}\Big), \end{aligned}\label{e5.6} \end{equation} with the positive constant \widetilde{C} independent of \varepsilon. Noticing that 2<\sigma< \frac{2(p^+-p^-)}{p^-(p^+-1)}<\frac{2p^+}{p^+-1},$we have \begin{gather} 1<\alpha=\frac{\sigma}{2}<\frac{(p^+-p^-)}{p^-(p^+-1)},\quad 1-\beta+\frac{(\alpha-1)p^-}{p^--1}=0;\label{e5.7} \\ \frac{(\alpha-1)p^-}{p^--1}-\beta p^+=\frac{(p^+-p^-)-\alpha p^-(p^+-1)}{p^--1}>0. \label{e5.8} \end{gather} Assume that there exists the constant$\tau_0\in(0,T)$such that$\operatorname{meas}(E)\neq 0.$Thus, \eqref{e5.6}--\eqref{e5.8} yield a contradiction. Hence, we have \begin{equation} \label{eq5} \operatorname{meas}\{x\in\{\Psi=1\}:u(x,\tau)>\delta\}=0, %\label{e5.9} \end{equation} for all$\delta\in(0,1)$and a.e.$\tau\in(0,T)$. Then Theorem \ref{thm3} follows from \eqref{eq5} and the arbitrariness of$\lambda$. \subsection*{Acknowledgments} Ning Pan was supported by the Fundamental Research Funds for the Central Universities (DL11BB40). 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