\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{amssymb} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2016 (2016), No. 143, pp. 1--13.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2016 Texas State University.} \vspace{7mm}} \begin{document} \title[\hfilneg EJDE-2016/143\hfil Evolutionary $p(x)$-Laplacian equation] {Evolutionary $p(x)$-Laplacian equation free from the limitation of the boundary value} \author[H. Zhan, J. Wen \hfil EJDE-2016/143\hfilneg] {Huashui Zhan, Jie Wen} \address{Huashui Zhan \newline School of Applied Mathematics, Xiamen University of Technology, Xiamen, Fujian 361024, China} \email{2012111007@xmut.edu.cn} \address{Jie Wen \newline School of Sciences, Jimei University, Xiamen, Fujian 361021, China} \email{1195103523@qq.com} \thanks{Submitted June 2, 2015. Published June 14, 2016.} \subjclass[2010]{35L65, 35K85, 35R35} \keywords{$p(x)$-Laplacian equation; diffusion coefficient; Fichera-Oleinik theory; \hfill\break\indent boundary condition; Fichera function} \begin{abstract} In this article we consider the evolutionary $p(x)$-Laplacian equation $$ {u_t}= \operatorname{div} ({\rho^\alpha}{| {\nabla u} |^{p(x) - 2}}\nabla u),\quad (x,t) \in \Omega \times (0,T), $$ where $\rho(x) = \operatorname{dist} (x,\partial \Omega )$. If the diffusion coefficient degenerates on the boundary, then and the solution may be free from any limitations of the boundary condition. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction} Consider the usual evolutionary $p$-Laplacian equation \begin{equation} u_t = \operatorname{div}({| {\nabla u} |^{p - 2}}\nabla u),\quad (x,t)\in {Q_T} = \Omega\times (0,T), \label{e1.1} \end{equation} where $\Omega\subset R^{N}$ is a bounded domain with appropriately smooth boundary. The equation arises in the fields of mechanics, physics and biology \cite{D1, K, WGS, WZYL, YY,Z1}. In the theory of non-Newtonian fluids, the quantity $p$ is the characteristic of the medium. The media with $p > 2$ is called dilatant fluids, those with $p=2$ are Newtonian fluids, and those with $p < 2$ are called pseudoplastics. Note that if $p=2$, equation \eqref{e1.1} is known as the classical heat conduction equation, and the solution of equation has infinite propagation speed of disturbance. This property seems unreasonable. So, when $p\neq2$, equation \eqref{e1.1} can be better to reflect the actual physical situation of the heat conduction. In particular, when $p>2$ the solution of equation has finite propagation speed of disturbance, see \cite{K}. Much attention was dedicated to its well-posedness \cite{D2, LPV,N2,N1,SZ,XZ,Z2,Z3,Z4,Z5,Z6,Z7,Z8,ZG,ZX} and the references therein. Yin-Wang \cite{YW} considered \begin{equation} u_t = \operatorname{div}({\rho^\alpha}{| {\nabla u} |^{p - 2}}\nabla u), \quad (x,t)\in Q_T, \label{e1.2} \end{equation} where $\rho(x)=\operatorname{dist} (x,\partial \Omega)$ is the distance function from the boundary. An obvious character of the equation is that, the diffusion coefficient $\rho^{\alpha}(x)$ depends on the distance to the boundary. Since $\rho^{\alpha}(x)$ vanishes on the boundary $\partial \Omega$, it seems that there is no heat flux across the boundary. However, Yin-Wang showed that the fact might not coincide with what we image. In fact, the exponent $\alpha$ which characterizes the vanishing ratio of the diffusion coefficient, does determine the behavior of the heat transfer near the boundary. If $p(x)$ is only a measurable function, a new kind of fluids of prominent technological interest has recently emerged: the so-called electrorheological fluids. This model includes parabolic equations which are nonlinear with respect to the gradient of the thought solution, and with variable exponents of nonlinearity \cite{AM,R} \begin{equation} u_{t} =\operatorname{div}(|\nabla u|^{p(x)-2}\nabla u),\quad (x,t) \in Q_T. \label{e1.3} \end{equation} A natural functional space used to study the well-posedness of the solutions of equation \eqref{e1.3} is $W^{1,p(x)}(\Omega)$. In what follows, we denote \begin{equation*} p^+ = \operatorname{ess\,sup}_{\bar{\Omega}}p(x), p^{-} = \operatorname{ess\,inf}_{\bar{\Omega}}p(x). \end{equation*} In particular, we assume that \begin{equation*} 10: \int_{\Omega}\bigl|\frac{u(x)}{\lambda}\bigl|^{p(x)}dx\leq 1\}, $$ and is a separable, uniformly convex Banach space. $$ W^{1,p(x)}(\Omega) =\{ u \in L^{p(x)}(\Omega): |\nabla u|\in L^{p(x)}(\Omega)\}, $$ which is endowed with the norm \begin{equation*} |u|_{W^{1,p(x)}}=|u|_{L^{p(x)}(\Omega)} + |\nabla u|_{L^{p(x)}(\Omega)}, \quad \forall u\in W^{1,p(x)}(\Omega). \end{equation*} $W_0^{1,p(x)}(\Omega)$ is the closure of $C^{\infty}_0(\Omega)$ in $W^{1,p(x)}$. Let us recall some properties of these function spaces, see \cite{FZ,KR}. \begin{lemma}\label{lem1.1} (i) The spaces $\left(L^{p(x)}(\Omega), | \cdot |_{L^{p(x)}(\Omega)}\right)$, $\left(W^{1,p(x)}(\Omega), |\cdot |_{W^{1,p(x)}(\Omega)}\right)$ and $W^{1,p(x)}_0(\Omega)$ are reflexive Banach spaces. (ii) $p(x)$-H\"{o}lder's inequality. Let $q_{1}(x)$ and $q_{2}(x)$ be real functions with $\frac{1}{q_1(x)}+\frac{1}{q_2(x)} = 1$ and $q_1(x) > 1$. Then, the conjugate space of $L^{q_1(x)}(\Omega)$ is $L^{q_2(x)}(\Omega)$. And for any $u \in L^{q_1(x)}(\Omega)$ and $ v \in L^{q_2(x)}(\Omega)$, we have $$ \bigl|\int_{\Omega}uv dx\bigl|\leq 2|u|_{L^{q_1(x)} (\Omega)}|v|_{L^{q_2(x)}(\Omega)}. $$ (iii) We have \begin{gather*} \text{If } |u|_{L^{p(x)}(\Omega)} = 1, \text{ then } \int_{\Omega}|u|^{p(x)} dx = 1;\\ \text{If } |u|_{L^{p(x)}}(\Omega) > 1, \text{ then } |u|^{p^{-}}_{L^{p(x)}}\leq\int_{\Omega}|u|^{p(x)} dx\leq|u|^{p^{+}}_{L^{p(x)}}; \\ \text{If } |u|_{L^{p(x)}}(\Omega) < 1, \text{then } |u|^{p^{+}}_{L^{p(x)}}\leq\int_{\Omega}|u|^{p(x)} dx\leq|u|^{p^{-}}_{L^{p(x)}}. \end{gather*} (iv) If $p_{1}(x)\leq p_{2}(x)$, then $L^{p_{1}(x)}(\Omega)\supset L^{p_{2}(x)}(\Omega)$. (v) If $p_{1}(x)\leq p_{2}(x)$, then $W^{1,p_{1}(x)}(\Omega)\hookrightarrow W^{1,p_{2}(x)}(\Omega)$. (vi) $p(x)$-Poincar\'{e} inequality: if $p(x)\in C(\Omega)$, then there is a constant $C>0$ such that $$ |u|_{L^{p(x)}}(\Omega) \leq C|\nabla u|_{L^{p(x)}(\Omega)},\quad \forall u\in W^{1,p(x)}_0(\Omega). $$ This implies that $|\nabla u|_{L^{p(x)}}(\Omega)$ and $|u|_{W^{1,p(x)}(\Omega)}$ are equivalent norms of $W^{1,p(x)}_0(\Omega)$. \end{lemma} Zhikov \cite{Z9} showed that \[ W^{1,p(x)}_0 (\Omega) \neq \{v\in W^{1,p(x)}_0 (\Omega)\mid v|_{\partial \Omega} = 0\}=\mathring{W} ^{1,p(x)}(\Omega). \] This fact provides us a good idea which may be used to study the well-posedness of evolutionary $p$-Laplacian equation \cite{D2,LPV, N2,SZ,XZ,N1, Z2,Z3,Z4,Z5,Z6,Z7,Z8,ZG,ZX}. However, if $p(x)$ satisfies the so-called logarithmic H\"{o}lder continuity condition \begin{equation} |p(x)-p(y)|\leq \omega(|x-y|), \forall x,y\in Q_{T},\quad |x-y| <\frac{1}{2},\label{e1.4} \end{equation} with $$ \limsup_{s \to {0^+ }} \omega (s)\ln \big(\frac{1} {s}\big) = C < \infty, $$ then (see \cite{Z10}) $$ W_0^{1,p(x)}(\Omega)= \mathring{W} ^{1,p(x)}(\Omega). $$ Using this fact, Antontsev-Shmarev \cite{AS} established existence and uniqueness results for \eqref{e1.3}. Later, many researchers have been interested in studying \eqref{e1.3}, see \cite{AS, P, LGYC,ZZX,Y}. In this article, we consider the equation \begin{equation} u_t =\operatorname{div} ({\rho^\alpha }{| {\nabla u} |^{p(x) - 2}}\nabla u) , \quad (x,t) \in Q_T,\label{e1.5} \end{equation} with $\alpha>0$. If we want to consider its initial boundary value problem, usually we need the initial value condition \begin{equation} u|_{t=0} = u_0(x),\quad x\in\Omega.\label{e1.6} \end{equation} Note that equation \eqref{e1.5} is degenerate on the boundary. Can we impose the general homogeneous boundary value condition as follows? \begin{equation} u|_{\Gamma_{T}} = 0,\ (x,t)\in \Gamma_{T} =\partial \Omega \times (0,T ).\label{e1.7} \end{equation} In this study, we introduce the Fichera-Ole\v{i}nik theory to study how to propose the boundary condition of equation \eqref{e1.5} with $p^{-}>1$. \begin{definition}\label{def1.2} \rm A function $u(x,t)$ is said to be a solution of \eqref{e1.5} with initial condition \eqref{e1.6}, if $u \in L^{\infty}(Q_T)$, $u_t\in L^2(Q_T)$, $\rho^\alpha|\nabla u |^{p(x)} \in L^1(Q_T)$ and \begin{equation} \iint_{{Q_T}} (-u\varphi_t + \rho ^\alpha | \nabla u |^{p(x)- 2}\nabla u \cdot \nabla \varphi )\,dx\,dt = 0, \label{e1.8} \end{equation} for any function $\varphi \in C_0^\infty ({Q_T})$. \end{definition} \begin{definition}\label{def1.3}\rm A function $u(x,t)$ is said to be a solution of equation \eqref{e1.5} with the initial-boundary conditions \eqref{e1.6}-\eqref{e1.7}, if $u$ satisfies Definition \ref{def1.2} and the boundary condition \eqref{e1.7} is satisfied in the sense of the trace. \end{definition} We summarize our main result as follows. \begin{theorem}\label{thm1.4} Suppose $p^{-}>1$, ${u_0} \in {L^\infty }(\Omega )$, and ${\rho ^\alpha }| {\nabla {u_0}} |^{p^{+}} \in {L^1}(\Omega)$. If $\alphap^{+}-1$, then there exists a unique solution of \eqref{e1.5} with the initial value \eqref{e1.6}. \end{theorem} Theorem \ref{thm1.4} implies that, the solution of \eqref{e1.5} is free from any limitations of the boundary condition provided that $\alpha>p^{+}-1$. When $p^{+}-1\geq\alpha\geq p^{-}-1$, the problem of the well-posedness of equation \eqref{e1.5} remains open. \section{Fichera-Oleinik theory and its applications} If for any real vector $\xi = ({\xi _1},{\xi _2}, \cdots ,{\xi _m})$ and any point $x\in\Omega$, \begin{equation} {a^{rs}}{\xi _r}{\xi _s} \geqslant 0,\label{e2.1} \end{equation} the second-order equation of the form \begin{equation} L(u) = {a^{rs}}(x){u_{{x_r}{x_s}}} + {b^r}(x){u_{{x_r}}} + c(x)u = f(x), \label{e2.2} \end{equation} is called the second-order equation with nonnegative characteristic form in $\Omega$. Obviously, it contains elliptic equation, parabolic equation, first-order equation (the case ${a^{rs}}{\xi _r}{\xi _s} \equiv 0$), ultra parabolic equation, Brown motion equation, and Tricomi equation on the half-plane and so on. Consider the first-boundary value problem of equation \eqref{e2.2} in $\Omega$, Fichera once dealt with this problem in \cite{F}. Suppose that on $\bar \Omega = \Omega \cup \sum$, all the points $x$ and all $\xi\in R^n$ satisfy the condition \eqref{e2.1}, and ${a^{rs}} \in {C^2}(\Omega ),{b^r} \in {C^1}(\Omega ),c \in {C^0}(\Omega )$. Let $\{n_{s}\}$ be the unit inner normal vector of $\partial\widetilde{\Omega}$. The Fichera function is defined as \begin{equation} b(x) \equiv ({b^r} - a_{{x_s}}^{rs}){n_r}.\label{e2.3} \end{equation} We denote \begin{gather*} \Sigma^{0}=\{x\in \Sigma: a^{rs}n_{r}n_{s}=0\}, \\ \Sigma_{1}=\{x\in \Sigma^{0}: (b_{r}-a^{rs}_{x_{s}})n_{r}>0\}, \\ \Sigma_{2}=\{x\in \Sigma^{0}: (b_{r}-a^{rs}_{x_{s}})n_{r}<0\}, \\ \Sigma_0=\{x\in \Sigma^{0}: (b_{r}-a^{rs}_{x_{s}})n_{r}=0\}, \\ \Sigma_3=\Sigma {\backslash \Sigma^0}. \end{gather*} The first boundary value problem of equation \eqref{e2.2} is quoted as follows, in $\bar \Omega = \Omega \cup \sum $, to find a function $u$ such that \begin{gather} L(u) = f(x),x\in\Omega,\label{e2.4}\\ u(x)=g(x),x\in \Sigma_2 \cup \Sigma_3,\label{e2.5} \end{gather} where $f$ is a given function in $\Omega$, and $g$ is a given function on $\Sigma_2 \cup \Sigma_3$. Clearly, if \eqref{e2.2} is an elliptic equation, then \eqref{e2.4}-\eqref{e2.5} is the usual Dirichlet problem. For the cylindrical region, \eqref{e2.4}-\eqref{e2.5} consists of the mixed problem, also known as parabolic equations with the initial boundary values. Consider the evolutionary $p(x)$-Laplacian equation \begin{equation} {u_t} = \operatorname{div} ({\rho ^\alpha }{| {\nabla u} |^{p(x) - 2}} \nabla u),\quad (x,t) \in {Q_T}.\label{e2.6} \end{equation} Since \begin{align*} &\operatorname{div}(\rho^{\alpha}|\nabla u|^{p(x)-2}\nabla u)\\ &=\alpha\rho^{\alpha-1}|\nabla u|^{p(x)-2}\nabla u\cdot\nabla \rho +\rho ^\alpha | {\nabla u} |^{p(x) - 2}\nabla p\cdot\nabla u \\ &\quad +{\rho ^\alpha }{| {\nabla u} |^{p(x) - 4}} [(p(x)-2)u_{x_ix_j}u_{x_i}u_{x_j}+|\nabla u|^2u_{x_ix_i}], \end{align*} we rewrite \eqref{e2.6} as \begin{equation} u_{t}=a^{ij}(x,t)\frac{\partial^2 u}{\partial x_{i}\partial x_{j}} +{\beta _i}(x,t)\frac{{\partial u}}{{\partial {x_i}}},\label{e2.7} \end{equation} where \begin{gather*} a^{ij}(x,t)=\rho^{\alpha}| {\nabla u} |^{p(x) - 2} [\delta_{ij}+(p(x)-2)|\nabla u|^{-2}u_{x_i}u_{x_j}], \\ \beta_i=\alpha\rho^{\alpha-1}|\nabla u|^{p(x)-2}\rho_{x_i} +\rho ^\alpha | {\nabla u} |^{p(x) - 2}p_{x_i}. \end{gather*} Then \begin{align*} a_{x_j}^{ij} &=\alpha {\rho ^{\alpha - 1}}{\rho _{{x_j}}}{| {\nabla u} |^{p(x) - 4}}[{\delta _{ij}}{| {\nabla u} |^2} + (p(x) - 2){u_{{x_i}}}{u_{{x_j}}}] \\ &\quad +\rho^{\alpha}| {\nabla u} |^{p(x) - 4}[p_{x_j}|\nabla u|^2 \log|\nabla u|+(p(x)-2)u_{x_k}u_{x_kx_j}+p_{x_j}u_{x_j}u_{x_i}] \\ &\quad +(p(x)-2)\rho^{\alpha}| {\nabla u} |^{p(x) - 6}u_{x_j} u_{x_i}[p_{x_j}| {\nabla u} |^2\log|\nabla u|+(p(x)-4)u_{x_k}u_{x_kx_j}] \\ &\quad +(p(x)-2)| {\nabla u} |^{p(x)- 4}(u_{x_ix_j}u_{x_j}+u_{x_i}u_{x_jx_j}). \end{align*} If we compare \eqref{e2.7} with \eqref{e2.5}, according to Fichera-Oleinik theory, the initial value condition is necessary. As for the boundary condition of equation \eqref{e2.6}, let us discuss it as follows. \smallskip \noindent\textbf{Case 1:} When $\alpha>1$, by $ \rho |_{\partial \Omega} = 0$, then $({\beta_i} - \alpha _{{x_i}}^{ij}){n_i}\equiv 0,x\in \partial \Omega$, $$ \Sigma_2 \cup \Sigma_3= \emptyset. $$ \noindent\textbf{Case 2:} When $\alpha\leq1$, by the fact of that $\rho_{x_j}=n_j$, it has \begin{equation} \begin{aligned} &({\beta ^i} - \alpha _{{x_j}}^{ij}){n_i}\\ &= \alpha {\rho ^{\alpha - 1}}{| {\nabla u} |^{p(x)- 4}} [{\rho _{{x_i}}}{| {\nabla u} |^2} - {\rho _{{x_j}}}({\delta _{ij}} {| {\nabla u} |^2} + (p(x)- 2){u_{{x_i}}}{u_{{x_j}}})]{n_i} \\ &= - (p(x)- 2)\alpha {\rho ^{\alpha - 1}} ({| {\nabla u} |^{p(x)- 4}}{u_{{x_i}}}{u_{{x_j}}}){n_i}{n_j}. \end{aligned} \label{e2.8} \end{equation} In this case, it is difficult to know that the Fichera function $({\beta ^i} - \alpha _{{x_j}}^{ij}){n_i}$ is negative or not, except the dimension of spatial variable, $N=1$. When $\alpha<1$ and $p(x)\leq p^{+}<2$, \eqref{e2.8} is transformed into \begin{equation} (\beta - {a_x})n = - (p(x)- 2)\alpha {\rho ^{\alpha - 1}}|u_{x}|^{p(x)-2} = + \infty > 0.\label{e2.9} \end{equation} When $\alpha<1$ and $p(x)\geq p^{-}>2$, \eqref{e2.8} is transformed into \begin{equation} (\beta - {a_x})n = - (p(x)- 2)\alpha {\rho ^{\alpha - 1}}|u_{x}|^{p(x)-2} = -\infty < 0.\label{e2.10} \end{equation} If $\alpha=1$ and $p^{+}<2$, then we have \begin{equation} (\beta - {a_x})n = - (p(x) - 2)\alpha {\rho ^{\alpha - 1}}|u_{x}|^{p(x)-2} = - (p(x)-2)\alpha> 0.\label{e2.11} \end{equation} If $\alpha=1$ and $p^{-}>2$, then we get \begin{equation} (\beta - {a_x})n = - (p(x) - 2)\alpha {\rho ^{\alpha - 1}}|u_{x}|^{p(x)-2} = - (p(x)-2)\alpha< 0.\label{e2.12} \end{equation} In a word, if $p^{+}<2$ and $N=1$, we may conjecture that $\Sigma_2 \cup \Sigma_3= \emptyset$. If $p^{-}>2$ and $N=1$, we may conjecture that $\Sigma_2 \cup \Sigma_3= \partial\Omega$. While, $N>1$, the partial boundary $\Sigma_2 \cup \Sigma_3\subseteq \partial \Omega$ may be very complicated. Certainly, equation\eqref{e1.5} is a degenerate parabolic equation, it only has a weak solution generally, so the above linearization is informal. We just give some ideas to the partial boundary condition to assure the well-posedness of the weak solutions. Whether these ideas are true or not remains to be verified. \section{Existence of solutions} In this section, we consider the initial value problem of equation \eqref{e1.5}. \begin{theorem}\label{thm3.1} If $p^{-}>1$, and \begin{equation} u_0\in L^{\infty}(\Omega), \quad {\rho ^\alpha }{| {\nabla {u_0}} |^{p^{+}}} \in {L^1}(\Omega ),\label{e3.1} \end{equation} then there exists a weak solution of \eqref{e1.5} with initial condition \eqref{e1.6}. Moreover, if $\alpha\geq 1$ and $u_0\in C_0^{\infty}(\Omega)$, then $u_{t}\in L^{\infty}(Q_T)$. \end{theorem} If $u_0\in C_0^{\infty}(\Omega)$, we consider the regularized problem \begin{gather} {u_{\varepsilon t}} - \operatorname{div} ({\rho ^\alpha _\varepsilon } (| {\nabla {u_\varepsilon }} |^2+\varepsilon)^{\frac{p(x)- 2}{2}} \nabla {u_\varepsilon }) = 0,\quad (x,t)\in {Q_T},\label{e3.2} \\ {u_\varepsilon }(x,t) = 0,\quad (x,t) \in \partial \Omega \times (0,T), \label{e3.3}\\ {u_\varepsilon }(x,0) = {u_0}(x),\quad x\in\Omega.\label{e3.4} \end{gather} where ${\rho_\varepsilon} = \rho + \varepsilon$, $\varepsilon >0$. Just as we had done on the usual evolutionary $p$-Laplacian equation, we are able to prove that the above problem has a unique weak solution $u_{\varepsilon}$, which satisfies \begin{equation} u_\varepsilon \in L^{\infty}(Q_T),\quad u_{\varepsilon t}\in L^2(Q_T), \quad u_\varepsilon\in L^{\infty}(0,T; W^{1,p(x)}_0(\Omega)).\label{e3.5} \end{equation} \begin{lemma}\label{lem3.2} If ${u_0} \in C_0^{\infty}(\Omega)$, then the solution ${u_\varepsilon}$ of initial boundary-value problem \eqref{e3.2}-\eqref{e3.4} is weakly star convergent and strongly convergent to $u \in L_{\rm loc}^{r}(Q_T)$, and the limit function $u$ is the solution of equation \eqref{e1.5} with the initial condition \eqref{e1.6}. Here, if $p^{-}<2$ and $1\lambda\}$ for any given $\lambda>0$, by that $\rho(x)>\lambda$ when $x\in \Omega_{\lambda}$, then \begin{equation} \int_0^{T}\int_{\Omega_{\lambda}}|\nabla u_\varepsilon|^{p^{-}}\,dx\,dt \leq\iint_{{Q_T}} \rho^\alpha (|\nabla u_\varepsilon|^2 +\varepsilon)^{\frac{p(x)-2}{2}}|\nabla u_\varepsilon|^2\,dx\,dt \leqslant c(\lambda).\label{e3.8} \end{equation} Multiplying \eqref{e3.2} by $u_{\varepsilon t}$, integrating it over $Q_T$, we have \begin{equation} \iint_{{Q_T}} (u_{\varepsilon t})^2\,dx\,dt = \iint_{{Q_T}} \operatorname{div} (\rho_\varepsilon^\alpha | \nabla u_\varepsilon |^{p(x) - 2}\nabla {u_\varepsilon }) \cdot u_{\varepsilon t}\,dx\,dt . \label{e3.9} \end{equation} Since $$ (|\nabla u_{\varepsilon}|^2+\varepsilon)^{\frac{p(x)-2}{2}} \nabla {u_\varepsilon} \cdot \nabla u_{\varepsilon t} = \frac{1}{2}\frac{d}{dt}\int_0^{| \nabla u_{\varepsilon}(x,t)|^2 +\varepsilon} s^{\frac{p(x) - 2}{2}}ds, $$ it follows that \begin{equation} \begin{aligned} &\iint_{{Q_T}} \operatorname{div} (\rho _\varepsilon^\alpha (|\nabla u_{\varepsilon}|^2+\varepsilon)^{\frac{p(x)-2}{2}}\nabla u_\varepsilon) \cdot u_{\varepsilon t})\,dx\,dt\\ &= - \iint_{{Q_T}} \rho_\varepsilon^\alpha (|\nabla u_{\varepsilon}|^2 +\varepsilon)^{\frac{p(x)-2}{2}}\nabla u_\varepsilon\nabla u_{\varepsilon t} \,dx\,dt \\ &= -\frac{1}{2}\iint_{{Q_T}} \rho_\varepsilon^\alpha \frac{d}{dt} \int_0^{| \nabla u_\varepsilon(x,t) |^2+\varepsilon} s^{\frac{p(x) - 2}{2}}ds \,dx\,dt. \end{aligned} \label{e3.10} \end{equation} By \eqref{e3.9}-\eqref{e3.10}, we have $$ \iint_{{Q_T}} (u_{\varepsilon t})^2\,dx\,dt + \iint_{{Q_T}} \rho_\varepsilon^\alpha \frac{d}{dt} \int_0^{| \nabla u_\varepsilon (x,t) |^2} s^{\frac{p(x) - 2}{2}}ds \,dx\,dt \leqslant c, $$ and \begin{equation} \iint_{{Q_T}} (u_{\varepsilon t})^2\,dx\,dt \leq c + c\int_\Omega \rho_\varepsilon^\alpha | \nabla u_{\varepsilon,0} |^{p(x)}dx \leq c.\label{e3.11} \end{equation} Differentiating \eqref{e3.2} with $t$, and denoting $w=u_{\varepsilon t}$, we obtain \begin{align*} \frac{\partial w}{\partial t} &=(\rho+\varepsilon)^{\alpha}(p(x)-2)(|\nabla u_\varepsilon|^2 +\varepsilon)^{\frac{p-4}{2}}u_{x_k}u_{x_i}w_{x_kx_i}\\ &\quad +(\rho+\varepsilon)^{\alpha}(|\nabla u_\varepsilon|^2 +\varepsilon)^{\frac{p(x)-2}{2}}w_{x_ix_i}\\ &\quad +(p(x)-2)\nabla[(\rho+\varepsilon)^{\alpha}] \cdot\nabla u_\varepsilon(|\nabla u_\varepsilon|^2 +\varepsilon)^{\frac{p-4}{2}}u_{x_k}w_{x_k} \\ &\quad +\nabla[(\rho+\varepsilon)^{\alpha}](|\nabla u_\varepsilon|^2 +\varepsilon)^{\frac{p(x)-2}{2}}\cdot\nabla w \\ &\quad +(\rho+\varepsilon)^{\alpha}(p(x)-2)(p(x)-4)(|\nabla u_\varepsilon|^2 +\varepsilon)^{\frac{p(x)-6}{2}}u_{x_j}u_{x_ix_j}u_{x_i}w_{x_k}u_{x_k} \\ &\quad +(\rho+\varepsilon)^{\alpha}(p(x)-2)(|\nabla u_\varepsilon|^2 +\varepsilon)^{\frac{p(x)-4}{2}}\big(u_{x_i}u_{x_ix_k}w_{x_k} +u_{x_k}u_{x_kx_i}w_{x_i} \\ &\quad +u_{x_k}u_{x_ix_i}w_{x_k}\big). \end{align*} We can rewrite it as $$ \frac{\partial w}{\partial t}=a_{ij}\frac{\partial^2w}{\partial x_i\partial x_j} +f_{i}(x,t,w) w_{x_i}, $$ where \begin{gather*} a_{ij}=(\rho+\varepsilon)^{\alpha}(|\nabla u_\varepsilon|^2 +\varepsilon)^{\frac{p(x)-2}{2}}(\delta_{ij}+(p(x)-2)(|\nabla u_\varepsilon|^2 +\varepsilon)^{-1}u_{x_{i}}u_{x_j}). \\ \begin{aligned} f_{i}(x,t,w) &= (p(x)-2)\nabla[(\rho+\varepsilon)^{\alpha}] \cdot\nabla u_\varepsilon(|\nabla u_\varepsilon|^2+\varepsilon)^{\frac{p(x)-4}{2}} u_{x_i} \\ &\quad +(|\nabla u_\varepsilon|^2+\varepsilon)^{\frac{p(x)-2}{2}} [(\rho+\varepsilon)^{\alpha}]_{x_{i}} \\ &\quad +(\rho+\varepsilon)^{\alpha}(p(x)-2)(p(x)-4)(|\nabla u_\varepsilon|^2 +\varepsilon)^{\frac{p(x)-6}{2}}u_{x_j}u_{x_kx_j}u_{x_k}u_{x_i} \\ &\quad +(\rho+\varepsilon)^{\alpha}(p(x)-2)(|\nabla u_\varepsilon|^2 +\varepsilon)^{\frac{p(x)-4}{2}}\big(u_{x_k}u_{x_kx_i}+u_{x_i}u_{x_ix_k}\\ &\quad +u_{x_i}u_{x_kx_k}\big). \end{aligned} \end{gather*} Clearly, $w$ satisfies \begin{gather*} w(x,t)=0,(x,t)\in\partial \Omega\times [0,T], \\ w(x,0)=\operatorname{div}((\rho+\varepsilon)^{\alpha}(|\nabla u_0|^2 +\varepsilon)^{\frac{p(x)-2}{2}}\nabla u_0), \quad x\in \Omega. \end{gather*} If we denote $$ a_0=(|\nabla u_\varepsilon|^2+\varepsilon)^{\frac{p(x)-2}{2}}, $$ then $$ \min\{p(x)-1,1\}a_0|\xi|^2\leq a_{ij}\xi_i\xi_j\leq \max\{p(x)-1,1\}a_0|\xi|^2. $$ By the maximum principle, we have \begin{equation} \sup_{\Omega\times(0,T)}|u_{\varepsilon t}|\leq\sup_{\Omega}| \operatorname{div}((\rho+\varepsilon)^{\alpha}(|\nabla u_0|^2 +\varepsilon)^{\frac{p(x)-2}{2}}\nabla u_0|\leq c,\label{e3.12} \end{equation} because $u_0(x)\in C_0^{\infty}(\Omega), \alpha\geq 1$. By \eqref{e3.6},\eqref{e3.8} and \eqref{e3.11}, we know that there exist a subsequence (still denoted as $u_{\varepsilon}$) of $u_{\varepsilon}$, a $n-$dimensional vector function $\overrightarrow \zeta = ({\zeta_1}, \cdots ,{\zeta_n})$, such that \begin{gather*} {u_{\varepsilon}} \rightharpoonup * u,\quad \text{in } {L^{\infty}(Q_T)}, \\ {u_{\varepsilon}} \to u,\quad \text{in } {L_{\rm loc}^{r}(Q_T)}, \\ {\nabla u_{\varepsilon}} \rightharpoonup \nabla u \quad \text{in } L_{\rm loc}^{p(x)}(Q_T), \\ \rho _\varepsilon^\alpha (|\nabla u_\varepsilon|^2+\varepsilon)^{\frac{p(x)-2}{2}} \rightharpoonup \overrightarrow \zeta \quad \text{in } {L^{\frac{p(x)}{p(x)-1}}}({Q_T}). \end{gather*} Here, if $p^{-}\geq 2$, $r=2$, while $p^{-}<2$, $1 0$. For any $u_{\varepsilon,0} \in C^\infty_0(\Omega)$, $\rho_\varepsilon ^\alpha | \nabla u_{\varepsilon,0} |^{p^{+}}\in L^1(\Omega)$, the above problem has a unique weak solution, and hence for any $\varphi \in C_0^\infty ({Q_T})$, $u_\varepsilon$ satisfies \begin{equation} \iint_{{Q_T}} (u_{\varepsilon t}\varphi + \rho_{\varepsilon}^\alpha | \nabla u_{\varepsilon} |^{p(x) - 2} \nabla u_{\varepsilon}\cdot \nabla \varphi)\,dx\,dt = 0.\label{e3.19} \end{equation} As in the proof of Lemma \ref{lem3.2}, except that the uniformly bounded estimate of $u_{\varepsilon t}$, we can prove the following lemma. \begin{lemma}\label{lem3.3} If $p^{-}>1$, $u_0$ satisfies \eqref{e3.1}. Then the solution ${u_\varepsilon}$ of initial boundary value problem \eqref{e3.16}-\eqref{e3.18} is convergent to $u$ weakly star, and strongly convergent to $u \in L_{\rm loc}^{r}(Q_T)$, and $u$ is the solution of equation \eqref{e1.5} with the initial condition \eqref{e1.6}. \end{lemma} Clearly, Theorem \ref{thm3.1} is a direct corollary of Lemmas \ref{lem3.2} and \ref{lem3.3}. \section{Uniqueness of the solution} \begin{lemma}\label{lem4.1} If $\alpha1$, such that \begin{equation} \iint_{{Q_T}} | \nabla u |^{\gamma}\,dx\,dt \leq c.\label{e4.1} \end{equation} \end{lemma} \begin{proof} Since $\frac{\alpha }{{p^{-} - 1}} < 1$ and $p^{-} - \alpha > 1$, there exists a constant $\beta \in (\frac{\alpha }{{p^{-} - 1}},1)$ such that $p^{-} -\frac{\alpha }{\beta} > 1$. Because of $\beta<1$ and $p^{-}- \frac{\alpha }{\beta} > 1$, there exists $\gamma \in (1, p^{-} - \frac{\alpha }{\beta})$ such that $\beta \gamma < 1$. Then we find \begin{align*} &\iint_{{Q_T}} {{{| {\nabla {u}} |}^\gamma }\,dx\,dt} \\ &= \iint_{\{ (x,t) \in {Q_T};\rho^\beta | {\nabla {u}} | \leqslant 1\} } {{{| {\nabla {u}} |}^\gamma}\,dx\,dt} + \iint_{\{ (x,t) \in {Q_T};\rho^\beta | {\nabla {u}} | > 1\} } {{{| {\nabla {u}} |}^\gamma}\,dx\,dt} \\ &\leqslant \iint_{{Q_T}} {\rho^{ - \beta \gamma }\,dx\,dt} + \iint_{{Q_T}} {\rho^\alpha {{| {\nabla {u}} |}^{\alpha /\beta + \gamma }}\,dx\,dt} \\ &\leqslant \iint_{{Q_T}} {\rho^{ - \beta \gamma }\,dx\,dt} + \iint_{{Q_T}} \rho^\alpha (1 + {{| {\nabla {u}} |}^{p^{-}}}) \,dx\,dt \leqslant c. \end{align*} \end{proof} Thus, if $\alpha 1/n, \\ n^2 s^2 e^{1 - n^2s^2}, & s \leqslant 1/n. \end{cases} $$ Choosing ${g_n}(u - v)$ as the test function in \eqref{e4.3}, we have \begin{equation} \begin{aligned} &\iint_{{Q_T}} g_n(u - v)\frac{\partial (u - v)}{\partial t}\,dx\,dt \\ &+ \iint_{{Q_T}} \rho^\alpha (| \nabla u |^{p(x) - 2}\nabla u - | \nabla v |^{p(x) - 2}\nabla v) \cdot \nabla (u - v)g'_n(u-v)=0. \end{aligned} \label{e4.4} \end{equation} Thus we further have \begin{gather*} \int_\Omega g_n(u - v)\frac{\partial (u - v)}{\partial t}dx = \frac{d}{dt}\| u - v \|_1,\\ \iint_{{Q_T}} \rho^\alpha (| \nabla u |^{p(x) - 2}\nabla u - | \nabla v |^{p(x) - 2}\nabla v) \cdot \nabla (u - v)g'_n(u-v)\,dx\,dt \geqslant 0, \end{gather*} Let $n\to\infty$ in \eqref{e4.4}. Then $ \frac{d}{dt}\| u - v \|_1 \leqslant 0$. This implies $$ \int_\Omega | u(x,t) - v(x,t) |dx \leqslant \int_\Omega | u_0 - v_0 |dx,\quad \forall t \in [0,T). $$ In addition, if $u_0(x)=v_0(x)$, by the arbitrariness of $t$, $$ u(x,t) = v(x,t)\quad \text{a.e. in } (x,t) \in {Q_T}; $$ then the solution of \eqref{e1.5} with the homogeneous boundary value is unique. \end{proof} \begin{theorem}\label{thm4.3} Suppose that ${u_0} \in {L^\infty}(\Omega )$ and ${\rho ^\alpha }| {\nabla {u_0}} |^{p^{+}} \in {L^1}(\Omega)$. If $\alpha>p^{+}-1$, then for any $t\in[0,T)$, we have \begin{equation} \int_{\Omega}[u(x,t)-v(x,t)]^2dx\leq\int_{\Omega}[u(x,0)-v(x,0)]^2dx. \label{e4.5} \end{equation} In other words, the solution of equation \eqref{e1.5} is free from any limitations of the boundary condition. \end{theorem} \begin{proof} Denote $\Omega_{\varepsilon}=\{x\in\Omega: dist(x,\partial \Omega)>\varepsilon\}$ as before, and let $\xi_{\varepsilon}\in C_0^{\infty}(\Omega_{\varepsilon})$ such that $\xi_{\varepsilon}=1$ on $\Omega_{2\varepsilon}$, $0\leq \xi_{\varepsilon}\leq 1$ and $|\nabla\xi_{\varepsilon}|\leq c/\varepsilon$. From the definition of the weak solution, we have \begin{equation} \begin{aligned} &\iint_{Q_T}\varphi\frac{\partial (u-v)}{\partial t}\,dx\,dt\\ &=-\iint_{Q_T}\rho^{\alpha}(|\nabla u|^{p(x)-2}\nabla u-|\nabla v|^{p(x)-2}\nabla v) \nabla\varphi \,dx\,dt, \quad \forall\varphi\in C^{\infty}_0(Q_T).\label{e4.6} \end{aligned} \end{equation} For any fixed $s\in [0,T)$, after an approximate procedure, we may choose $\chi_{[0,s]}(u-v)\xi_{\varepsilon}$ as a test function in the above equality, where $\chi_{[0,s]}$ is the characteristic function on $[0,s]$. Thus we have $$ \iint_{Q_s}\varphi\frac{\partial (u-v)}{\partial t}\,dx\,dt =-\iint_{Q_s}\rho^{\alpha}(|\nabla u|^{p(x)-2}\nabla u-|\nabla v|^{p(x)-2}\nabla v)\nabla\varphi \,dx\,dt, $$ and \begin{equation} \begin{aligned} &\int_{\Omega}[u(x,s)-v(x,s)]^2\xi_{\varepsilon}dx\\ &=\int_{\Omega}[u(x,0)-v(x,0)]^2\xi_{\varepsilon}dx \\ &\quad -2\iint_{Q_s}\xi_{\varepsilon}\rho^{\alpha}(|\nabla u|^{p(x)-2}\nabla u -|\nabla v|^{p(x)-2}\nabla v)\nabla(u-v)\,dx\,dt \\ &\quad -2\iint_{Q_s}(u-v)\rho^{\alpha}(|\nabla u|^{p(x)-2}\nabla u -|\nabla v|^{p(x)-2}\nabla v)\nabla\xi_{\varepsilon}\,dx\,dt\\ &\leq \int_{\Omega}[u(x,0)-v(x,0)]^2\xi_{\varepsilon}dx \\ &\quad -2\iint_{Q_s}(u-v)\rho^{\alpha}(|\nabla u|^{p(x)-2} \nabla u-|\nabla v|^{p(x)-2}\nabla v)\nabla\xi_{\varepsilon}\,dx\,dt. \end{aligned}\label{e4.7} \end{equation} Since \begin{equation} \begin{aligned} &\big|-2\iint_{Q_s}(u-v)\rho^{\alpha}(|\nabla u|^{p(x)-2}\nabla u -|\nabla v|^{p(x)-2}\nabla v)\nabla\xi_{\varepsilon}\,dx\,dt\big| \\ &\leq 2\iint_{Q_s}|u-v|\rho^{\alpha}(|\nabla u|^{p(x)-1} +|\nabla v|^{p(x)-1})|\nabla \xi_{\varepsilon}|\,dx\,dt \\ &\leq c\int_0^{T}\int_{\Omega_{\varepsilon}\setminus\Omega_{2\varepsilon}} \big[{\frac{p(x)-1}{p(x)}}\rho^{\alpha}(|\nabla u|^{p(x)}+|\nabla v|^{p(x)})+{\frac{1}{p(x)}}\rho^{\alpha}|\nabla\xi_{\varepsilon}|^{p(x)}\big]\,dx\,dt \\ &\leq c\int_0^{T}\int_{\Omega_{\varepsilon}\setminus\Omega_{2\varepsilon}} \big[{\frac{p(x)-1}{p(x)}}\rho^{\alpha}(|\nabla u|^{p(x)}+|\nabla v|^{p(x)}) +{\frac{1}{p(x)}}\varepsilon^{\alpha-p(x)}\big]\,dx\,dt, \end{aligned} \label{e4.8} \end{equation} by $\alpha> p^{+}-1$ and $\alpha-p(x)>-1$, using \eqref{e4.8} yields \begin{equation} \lim_{\varepsilon\to 0} \big|-2\iint_{Q_s}(u-v)\rho^{\alpha}(|\nabla u|^{p(x)-2}\nabla u -|\nabla v|^{p(x)-2}\nabla v)\nabla\xi_{\varepsilon}\,dx\,dt\big|=0.\label{e4.9} \end{equation} Let $\varepsilon\to 0$. 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