\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2016 (2016), No. 81, pp. 1--13.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2016 Texas State University.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2016/81\hfil Reaction diffusion equations] {Reaction diffusion equations with boundary degeneracy} \author[H. Zhan \hfil EJDE-2016/81\hfilneg] {Huashui Zhan} \address{Huashui Zhan \newline School of Applied Mathematics, Xiamen University of Technology, Xiamen, Fujian 361024, China} \email{2012111007@xmut.edu.cn} \thanks{Submitted May 14, 2015. Published March 23, 2016.} \subjclass[2010]{35L65, 35K85, 35R35} \keywords{Reaction diffusion equation; Fichera-Ole\v{i}nik theory; \hfill\break\indent boundary condition; degeneracy} \begin{abstract} In this article, we consider the reaction diffusion equation $$ \frac{\partial u}{\partial t} = \Delta A(u),\quad (x,t)\in \Omega \times (0,T), $$ with the homogeneous boundary condition. Inspired by the Fichera-Ole\v{i}nik theory, if the equation is not only strongly degenerate in the interior of $\Omega$, but also degenerate on the boundary, we show that the solution of the equation is free from any limitation 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 equation \begin{equation} \frac{\partial u}{\partial t} =\Delta A(u), \quad (x,t)\in \Omega \times (0,T),\label{e1.1} \end{equation} with the homogeneous boundary condition, where $\Omega \subset \mathbb{R}^{N}$ is an open bounded domain with the appropriately smooth boundary $\partial \Omega$, and \begin{equation} A(u)=\int_0^{u}a(s)ds,\quad a(s)\geq 0, \quad a(0)=0. \label{e1.2} \end{equation} One of particular cases of equation \eqref{e1.1} is \begin{equation} \frac{\partial u}{\partial t} =\Delta u^{m}.\label{e1.3} \end{equation} According to the degenerate parabolic equation theory, if there is no interior point in the set $\{s\in \mathbb{R}: a(s)=0\}$, as usual we say that equation \eqref{e1.1} is weakly degenerate; otherwise, we say that equation \eqref{e1.1} is strongly degenerate. For the Cauchy problem of equation \eqref{e1.1}, Vol'pert and Hudjave \cite{VH} investigated its solvability. Thereafter, much attention has dedicated to the study of its well-posedness \cite{ BK, CB, CG, K1, WZYL, Z, ZZ}. When we consider the initial-boundary value problem of equation \eqref{e1.1}, usually one needs the initial condition as \begin{equation} u(x,0)=u_0(x), \quad x\in \Omega.\label{e1.4} \end{equation} However, can we impose the Dirichlet homogeneous boundary condition \begin{equation} u(x,t)=0,\quad (x,t)\in \partial \Omega \times (0,T),\label{e1.5} \end{equation} into the problem? Obviously, when both \eqref{e1.2} and \eqref{e1.5} hold, equation \eqref{e1.1} is not only degenerate in the interior of $\Omega$, but also on the boundary $\partial \Omega$. If it is weakly degenerate, we will show that equation \eqref{e1.1} can be imposed by the boundary condition \eqref{e1.5} actually. While, if it is in the strongly degenerate case, we will show that the solution of equation \eqref{e1.1} is free from any limitation of the boundary condition. Let us give a brief review on the corresponding problems. The memoir by Tricomi \cite{T}, as well as subsequent investigations of equations of mixed type, elicited interest in the general study of elliptic equations degenerating on the boundary of the domain. The paper by Keldy\v{s} \cite{K2} plays a significant role in the development of the theory. It was brought to light that in the case of elliptic equations degenerating on the boundary, under definite assumptions, a portion of the boundary may be free from the prescription of boundary conditions. Later, Fichera\cite{F1,F2} and Ole\v{i}nik \cite{O1,O2} developed the general theory of second order equations with a nonnegative characteristic form, which, in particular contains those degenerating assumptions on the boundary. We can call the theory as the Fichera-Ole\v{i}nik theory. To study the boundary value problem of a linear degenerate elliptic equation: \begin{equation} \sum_{r,s=1}^{N+1}a^{rs}(x)\frac{\partial^2 u}{\partial x_{r}\partial x_s}+\sum_{r=1}^{N+1} b_{r}(x)\frac{\partial u}{\partial x_{r}}+c(x)u=f(x), x\in \widetilde{\Omega}\subset\mathbb{R}^{N+1},\label{e1.6} \end{equation} it needs and only needs the part boundary condition. In detail, let $\{n_s\}$ be the unit inner normal vector of $\partial\widetilde{\Omega}$ and denote \begin{equation} \begin{gathered} \Sigma_2=\{x\in \partial \widetilde{\Omega}: a^{rs}n_{r}n_s=0, (b_{r}-a^{rs}_{x_s})n_{r}<0\},\\ \Sigma_3 =\{x\in \partial \widetilde{\Omega}: a^{rs}n_sn_{r}>0\}. \end{gathered} \label{e1.7} \end{equation} Then, to ensure the well-posedness of equation \eqref{e1.7}, according to the Fichera-Oleinik theory, the suitable boundary condition is \begin{equation} u|_{\Sigma_2\cup\Sigma_3}=g(x).\label{e1.8} \end{equation} In particular, if the matrix $(a^{rs})$ is definite positive, \eqref{e1.8} is the regular Dirichlet boundary condition. If $A^{-1}$ exists, in other words, equation \eqref{e1.1} is weakly degenerate, let $v=A(u)$ and $u=A^{-1}(v)$. Then it has \begin{equation} \Delta v-(A^{-1}(v))_{t}=0.\label{e1.9} \end{equation} According to the Fichera-Oleinik theory, one can impose the Dirichlet homogeneous boundary condition \eqref{e1.5}. But, if equation \eqref{e1.1} is strongly degenerate, then $A^{-1}$ does not exist, we can not deal with it as equation \eqref{e1.9}. We rewrite equation \eqref{e1.1} as \begin{equation} \frac{\partial u}{\partial t} =a(u)\Delta u+a'(u)|\nabla u|^2,\quad (x,t)\in\Omega \times (0,T),\label{e1.10} \end{equation} and let $t=x_{N+1}$. We regard the strongly degenerate parabolic equation \eqref{e1.10} as the form of a ``linear" degenerate elliptic equation as follows: when $i,j=1, 2, \dots, N$, $a^{ii}(x,t)=a(u(x,t))$, $a^{ij}(x,t)=0$, $i\neq j$, then it has $$ (\widetilde{a}^{rs})_{(N+1)\times (N+1)} =\begin{pmatrix} a^{ij} & 0 \\ 0 & 0 \end{pmatrix}. $$ If $a(0)=0$, then equation \eqref{e1.10} is not only strongly degenerate in the interior of $\Omega$, but also degenerate on the boundary $\partial \Omega$. We can see that $\Sigma_3$ is an empty set, while $$ \widetilde{b}_s(x,t)=\begin{cases} a'(u)\frac{\partial u}{\partial x_i}, & 1\leq s\leq N,\\ -1, & s=N+1. \end{cases} $$ Under this observation, according to the Fichera-Oleinik theory, the initial condition \eqref{e1.4} is always required. But on the lateral boundary $\partial \Omega\times (0,T)$, by $a(0)=0$, the part of boundary in which we should give the boundary value is \begin{equation} \Sigma_{p}=\big\{x\in \partial \Omega: \Big(a'(0)\frac{\partial u}{\partial x_i} \big|_{x\in\partial \Omega}-a'(0)\frac{\partial u}{\partial x_i} \big|_{x\in\partial \Omega} \Big)n_i<0 \big\}=\emptyset,\label{e1.11} \end{equation} where $\{n_i\}$ is the unit inner normal vector of $\partial{\Omega}$. This implies that no any boundary condition is necessary. In other words, the initial-boundary problem of equation \eqref{e1.1} is actually free from the limitation of the boundary condition. Certainly, the above discussion is based on the assumption that there is a classical solution of equation \eqref{e1.1}. In fact, due to the strongly degenerate properties of $A(u)$, equation \eqref{e1.1} generally only has a weak solution. So it remains to be clarified whether the solution of the equation is actually free from the limitation of the boundary condition or not? \section{Main results} For small $\eta>0$, let \begin{equation} S_{\eta}(s)=\int_0^{s}h_{\eta}(\tau)d\tau,\quad h_{\eta}(s)=\frac{2}{\eta}(1-\frac{| s|}{\eta })_{+}.\label{e2.1} \end{equation} Obviously, $h_{\eta}(s)\in C(\mathbb{R})$, and \begin{equation} \begin{gathered} h_{\eta}(s)\geq 0,\quad | sh_{\eta}(s)| \leq 1,\quad | S_{\eta}(s)| \leq 1,\\ \lim_{\eta \to 0} S_{\eta}(s)=\operatorname{sign}s,\quad \lim_{\eta \to0} sS_{\eta}'(s)=0. \end{gathered} \label{e2.2} \end{equation} \begin{definition}\label{def1}\rm A function $u$ is said to be the entropy solution of \eqref{e1.1} with the initial condition \eqref{e1.4}, if 1. $u$ satisfies \begin{equation} u\in BV(Q_T)\cap L^{\infty}(Q_T),\quad \frac{\partial}{\partial x_i}\int_0^{u}\sqrt{a(s)}ds\in L^2(Q_T).\label{e2.3} \end{equation} 2. For any $\varphi\in C_0^2(Q_T)$, $\varphi\geq 0$, $k\in \mathbb{R}$, with a small $\eta >0$, $u$ satisfies \begin{equation} \iint_{Q_T}\Big[I_{\eta}(u-k)\varphi_{t}+A_{\eta}(u,k)\Delta \varphi -S_{\eta}'(u-k)| \nabla \int_0^{u}\sqrt{a(s)}ds| ^2\varphi\Big]\,dx\,dt \geq 0.\label{e2.4} \end{equation} 3. The initial condition is true in the sense that \begin{equation} \lim_{t\to 0}\int_{\Omega }| u(x,t)-u_0(x)| dx=0.\label{e2.5} \end{equation} \end{definition} One can see that if \eqref{e1.1} has a classical solution $u$, by multiplying \eqref{e1.1} by $\varphi_1 S_{\eta }(u-k)$ and integrating it over $ Q_T$, we are able to show that $u$ satisfies Definition \ref{def1}. On the other hand, letting $\eta\to 0$ in \eqref{e2.4}, we have $$ \iint_{Q_T}[|u-k|\varphi _{t}+\operatorname{sign}(u-k)(A(u)-A(k))\Delta \varphi]\,dx\,dt\geq 0. $$ Thus if $u$ is the entropy solution as in Definition \ref{def1}, then $u$ is a entropy solution as defined in \cite{K1,VH} et al. \begin{theorem}\label{thm2.2} Suppose that $A(s)$ is $C^{3}$ and $u_0(x)\in L^{\infty}(\Omega)$. Suppose that \begin{equation} A'(0)=a(0)=0.\label{e2.6} \end{equation} Then \eqref{e1.1} with the initial condition \eqref{e1.4} has a entropy solution in the sense of Definition \ref{def1}. \end{theorem} \begin{theorem}\label{thm2.3} Suppose that $A(s)$ is $C^2$. Let $u$ and $v$ be solutions of \eqref{e1.1} with the different initial values $u_0(x)$, $v_0(x)\in L^{\infty}(\Omega)$ respectively. Suppose that the distance function $d(x)=\operatorname{dist}(x,\Sigma)<\lambda$ satisfies \begin{equation} |\Delta d|\leq c, \ \frac{1}{\lambda}\int_{\Omega_{\lambda}}dx\leq c, \label{e2.7} \end{equation} where $\lambda$ is a sufficiently small constant, and $\Omega_{\lambda}=\{x\in \Omega, d(x,\partial \Omega)<\lambda\}$. Then \begin{equation} \int_{\Omega}| u(x,t)-v(x,t)| dx\leq \int_{\Omega}| u_0-v_0| dx+\operatorname{ess\,sup}_{(x,t)\in \partial \Omega\times(0,T)} |u(x,t)-v(x,t)|.\label{e2.8} \end{equation} \end{theorem} \section{Proof of Theorem \ref{thm2.2}} Let $\Gamma_u$ be the set of all jump points of $u\in BV(Q_T),v$ be the normal of $\Gamma_u$ at $X=(x,t)$, $u^{+}(X)$ and $u^{-}(X)$ be the approximate limit of $u$ at $X\in \Gamma_u$ with respect to $(v,Y-X)>0$ and $(v,Y-X)<0$ respectively. For the continuous function $p(u,x,t)$ and $u\in BV(Q_T)$, we define \begin{equation} \widehat{p}(u,x,t)=\int_0^{1}p(\tau u^{+}+(1-\tau )u^{-},x,t)d\tau,\label{e3.1} \end{equation} which is called the composite mean value of $p$. For a given $t$, we denote $\Gamma_u^t,\;H^t,(v_{1}^t,\dots ,v_N^t)$ and $u_{\pm}^t$ as all jump points of $u(\cdot,t)$, Housdorff measure of $\Gamma_u^t$, the unit normal vector of $\Gamma_u^t$, and the asymptotic limit of $u(\cdot,t)$ respectively. Moreover, if $f(s)\in C^{1}( \mathbb{R})$ and $u\in BV(Q_T)$, then $f(u)\in BV(Q_T)$ and \begin{equation} \frac{\partial f(u)}{\partial x_i}=\widehat{f^{\prime }}(u)\frac{\partial u}{\partial x_i},\;i=1, 2, \dots, N, N+1, \label{e3.2} \end{equation} holds, where $x_{N+1}=t$. \begin{lemma}\label{lem1} Let $u$ be a solution of \eqref{e1.1}. Then \begin{equation} a(s)=0,\quad s\in I(u^{+}(x,t),u^{-}(x,t))\quad \text{a.e. on } \Gamma_u,\label{e3.3} \end{equation} where $I(\alpha,\beta)$ denote the closed interval with endpoints $\alpha $ and $\beta$, and \eqref{e3.3} is in the sense of Hausdorff measure $H_N(\Gamma_u)$. \end{lemma} The proof of the above lemma is similar to the one in \cite{ZZ}, so we omit it. \begin{lemma}[\cite{E}] \label{lem2} Assume that $\Omega \subset \mathbb{R}^{N}$ is an open bounded set and let $f_k,f\in L^{q}(\Omega)$, as $k\to \infty $, $f_k\rightharpoonup f$ weakly in $L^{q}(\Omega)$ and $1\leq q<\infty$. Then \begin{equation} \liminf_{k\to \infty} \| f_k\|_{L^{q}(\Omega)}^{q} \geq \| f\|_{L^{q}(\Omega )}^{q}.\label{e3.4} \end{equation} \end{lemma} We now consider the regularized problem \begin{equation} \frac{\partial u}{\partial t}=\Delta A(u)+\varepsilon \Delta u,\; (x,t)\in\Omega \times (0,T),\label{e3.5} \end{equation} with the initial and boundary conditions \begin{gather} u(x,0)=u_0(x), \quad x\in \Omega,\label{e3.6}\\ u(x,t)=0,\quad (x,t)\in \partial \Omega \times (0,T).\label{e3.7} \end{gather} It is well known that there are classical solutions $u_{\varepsilon}\in C^2(\overline{Q_T})\cap C^{3}(Q_T)$ of this problem provided that $A(s)$ satisfies the assumptions in Theorem \ref{thm2.2}. One can refer to \cite{WZ} or the eighth chapter of \cite{G} for details. We need to make some estimates for $u_{\varepsilon}$ of \eqref{e3.5}. Firstly, since $u_0(x)\in L^{\infty}(\Omega)$ is sufficiently smooth, by the maximum principle we have \begin{equation} | u_{\varepsilon }| \leq \| u_0\|_{L^{\infty}}\leq M.\label{e3.8} \end{equation} Secondly, let us make the $BV$ estimates of $u_{\varepsilon }$. To the end, we begin with the local coordinates of the boundary $\partial \Omega$. Let $\delta_0>0$ be small enough. Denote $$ E^{\delta_0}=\{x\in\bar{\Omega}; \operatorname{dist}(x, \Sigma)\leq \delta_0\}\subset \cup_{\tau=1}^{n}V_{\tau}, $$ where $V_{\tau}$ is a region, and one can introduce local coordinates of $V_{\tau}$, \begin{equation} y_k=F^k_{\tau}(x)\quad (k=1, 2, \dots, N), \quad y_N|_{\Sigma}=0,\label{e3.9} \end{equation} with $F^k_{\tau}$ appropriately smooth and $F^{N}_{\tau}=F^{N}_{l}$, such that the $y_N-$axes coincides with the inner normal vector. \begin{lemma}[\cite{WZ}]\label{lem3} Let $u_{\varepsilon}$ be the solution of equation \eqref{e3.5} with \eqref{e3.6}, \eqref{e3.7}. If the assumptions of Theorem \ref{thm2.2} are true, then \begin{equation} \varepsilon \int_{\Sigma}|\frac{\partial u_{\varepsilon}}{\partial n} |d\sigma\leq c_{1}+c_2(|\nabla u_{\varepsilon}|_{L^{1}(\Omega)} +|\frac{\partial u_{\varepsilon}}{\partial t}|_{L^{1}(\Omega)}),\label{e3.10} \end{equation} with constants $c_i$, $i=1, 2$ independent of $\varepsilon$. \end{lemma} We have the following important estimates of the solutions $u_{\varepsilon}$ of equation \eqref{e3.5} with \eqref{e3.6}, \eqref{e3.7}. \begin{theorem}\label{thm3.4} Let $u_{\varepsilon}$ be the solution of equation \eqref{e3.5} with \eqref{e3.6}, \eqref{e3.7}. If the assumptions of Theorem \ref{thm2.2} are true, then \begin{equation} |\operatorname{grad}u_{\varepsilon}|_{L^{1}(\Omega)}\leq c, \label{e3.11} \end{equation} where $|\operatorname{grad}u|^2=\sum_{i=1}^{N}|\frac{\partial u}{\partial x_i}|^2+|\frac{\partial u}{\partial t}|^2$, and $c$ is independent of $\varepsilon$. \end{theorem} \begin{proof} Differentiate \eqref{e3.5} with respect to $x_s$, $s=1, 2, \cdot, N, N+1$, $x_{N+1}=t$, and sum up for $s$ after multiplying the resulting relation by $u_{\varepsilon x_s} \frac{S_{\eta}(|\operatorname{grad}u_{\varepsilon}|)}{|\operatorname{grad} u_{\varepsilon}|}$. In what follows, we simply denote $u_{\varepsilon}$ by $u$, denote $\partial \Omega$ by $\Sigma$, and denote $d\sigma$ by the surface integral unite on $\Sigma$. Integrating it over $\Omega$ yields $$ \int_{\Omega}\frac{\partial u_{x_s}}{\partial t}u_{ x_s}\frac{S_{\eta}(|\operatorname{grad}u|)}{|\operatorname{grad}u|}dx =\int_{\Omega}\frac{\partial}{\partial t}\int_0^{|\operatorname{grad}u|}S_{\eta}(\tau)d\tau dx =\frac{d}{dt}\int_{\Omega}I_{\eta}(|\operatorname{grad}u|dx, $$ where pairs of the indices of $s$ imply a summation from 1 to $N+1$, pairs of the indices of $i,j$ imply a summation from 1 to $N$, and $\{n_i\}_{i=1}^{N}$ is the inner normal vector of $\Omega$. So we have \begin{gather} \begin{aligned} &\int_{\Omega}\Delta (a(u)u_{x_s})u_{ x_s}\frac{S_{\eta}(|\operatorname{grad}u|)} {|\operatorname{grad}u|}dx\\ &=\int_{\Omega}\frac{\partial}{\partial x_i}[a'(u)u_{x_i}u_{x_s}+a(u)u_{x_ix_s}]u_{x_s} \frac{S_{\eta}(|\operatorname{grad}u|)}{|\operatorname{grad}u|}dx \\ &=\int_{\Omega}\frac{\partial}{\partial x_i}(a'(u)u_{x_i}u_{x_s})u_{ x_s} \frac{S_{\eta}(|\operatorname{grad}u|)}{|\operatorname{grad}u|}dx\\ &\quad +\int_{\Omega}\frac{\partial}{\partial x_i}(a(u)u_{x_ix_s})u_{x_s} \frac{S_{\eta}(|\operatorname{grad}u|)}{|\operatorname{grad}u|}dx\,, \end{aligned}\label{e3.12} \\ \begin{aligned} &\int_{\Omega}\frac{\partial}{\partial x_i}(a'(u)u_{x_i}u_{x_s})u_{x_s} \frac{S_{\eta}(|\operatorname{grad}u|)}{|\operatorname{grad}u|}dx \\ &=\sum_{s=1}^{N+1}\int_{\Omega}\frac{\partial}{\partial x_i}(a'(u)u_{x_i})u_{x_s}^2\frac{S_{\eta}(|\operatorname{grad}u|)} {|\operatorname{grad}u|}dx \\ &\quad +\int_{\Omega}a'(u)u_{x_i} \frac{\partial}{\partial x_i}I_{\eta} (|\operatorname{grad}u|)dx\\ &=\int_{\Omega}\frac{\partial}{\partial x_i}(a'(u)u_{x_i})|\operatorname{grad}u|S_{\eta}(|\operatorname{grad}u|)dx \\ &\quad -\int_{\Sigma}a'(u)u_{x_i}n_iI_{\eta}(|\operatorname{grad}u|)d\sigma- \int_{\Omega}I_{\eta}(|\operatorname{grad}u|)\frac{\partial}{\partial x_i}(a'(u)u_{x_i})dx \\ & =\int_{\Omega}\frac{\partial}{\partial x_i}(a'(u)u_{x_i})\left[|\operatorname{grad}u|S_{\eta}(|\operatorname{grad}u|) -I_{\eta}(|\operatorname{grad}u|)\right]dx \\ &\quad -\int_{\Sigma}a'(u)u_{x_i}n_iI_{\eta}(|\operatorname{grad}u|)d\sigma\,, \end{aligned} \label{e3.13} \\ \begin{aligned} &\int_{\Omega}\frac{\partial}{\partial x_i}(a(u)u_{x_ix_s})u_{x_s} \frac{S_{\eta}(|\operatorname{grad}u|)}{|\operatorname{grad}u|}dx \\ &=\int_{\Omega}\frac{\partial}{\partial x_i}(a(u)u_{x_ix_s}) \frac{\partial}{\partial\xi_s}I_{\eta}(|\operatorname{grad}u|)dx \\ &=-\int_{\Sigma}a(u)u_{x_ix_s}n_i\frac{\partial}{\partial\xi_s}I_{\eta} (|\operatorname{grad}u|)d\sigma \\ &\quad -\int_{\Omega}a(u)\frac{\partial^2I_{\eta} (|\operatorname{grad}u|)}{\partial\xi_s\partial\xi_{p}} u_{x_sx_i}u_{x_{p}x_i}dx\,, \end{aligned} \label{e3.14} \end{gather} where $\xi_s=u_{x_s}$. \begin{equation} \begin{aligned} &\varepsilon\int_{\Omega}\Delta u_{x_s}u_{ x_s} \frac{S_{\eta}(|\operatorname{grad}u|)}{|\operatorname{grad}u|}dx \\ &=-\varepsilon\int_{\Sigma}\frac{\partial I_{\eta}(|\operatorname{grad}u|)}{\partial x_i}n_id\sigma-\varepsilon\int_{\Omega} \frac{\partial^2I_{\eta}(|\operatorname{grad}u|)}{\partial\xi_s\partial\xi_{p}} u_{x_sx_i}u_{x_{p}x_i}dx. \end{aligned}\label{e3.15} \end{equation} From \eqref{e3.12}--\eqref{e3.15}, by the assumption $a(0)=0$, we have \begin{equation} \begin{aligned} &\frac{d}{dt}\int_{\Omega}I_{\eta}(|\operatorname{grad}u|dx \\ &=\int_{\Omega}\frac{\partial}{\partial x_i}(a'(u)u_{x_i}) \left[|\operatorname{grad}u|S_{\eta}(|\operatorname{grad}u|) -I_{\eta}(|\operatorname{grad}u|)\right]dx \\ &\quad -\int_{\Omega}a(u)\frac{\partial^2I_{\eta} (|\operatorname{grad}u|)}{\partial\xi_s\partial\xi_{p}}u_{x_sx_i}u_{x_{p}x_i}dx\\ &\quad -\varepsilon\int_{\Omega}\frac{\partial^2 I_{\eta}(|\operatorname{grad}u|)}{\partial\xi_s\partial\xi_{p}} u_{x_sx_i}u_{x_{p}x_i}dx\\ &\quad -\Big[\int_{\Sigma}a'(u)u_{x_i}n_iI_{\eta}(|\operatorname{grad}u|)d\sigma +\varepsilon\int_{\Sigma}\frac{\partial I_{\eta}(|\operatorname{grad}u|)}{\partial x_i}n_id\sigma\Big]. \end{aligned} \label{e3.16} \end{equation} Note that on $\Sigma$, we have \begin{equation} 0=\varepsilon\Delta u+\Delta A(u),\quad u=0,\label{e3.17} \end{equation} then the surface integrals in \eqref{e3.16} can be rewritten as \begin{align*} S&=-\Big[\varepsilon\int_{\Sigma}\frac{\partial I_{\eta} (|\operatorname{grad}u|)}{\partial x_i}n_id\sigma +\int_{\Sigma}a'(u)u_{x_i}n_iI_{\eta}(|\operatorname{grad}u|)d\sigma\Big] \\ &=-\varepsilon\int_{\Sigma}\big[\frac{\partial I_{\eta}(|\operatorname{grad}u|)}{\partial x_i}n_i-\Delta u \frac{I_{\eta}(|\operatorname{grad}u|)}{\frac{\partial u}{\partial n}}\big]d\sigma \\ &\quad +\int_{\Sigma}a(u)\big[\frac{\partial I_{\eta}(|\operatorname{grad}u|)}{\partial x_i}n_i-\Delta u \frac{I_{\eta}(|\operatorname{grad}u|)}{\frac{\partial u}{\partial n}}\big]d\sigma \\ &=-\varepsilon\int_{\Sigma}\big[\frac{\partial I_{\eta}(|\operatorname{grad}u|)}{\partial x_i}n_i-\Delta u \frac{I_{\eta}(|\operatorname{grad}u|)}{\frac{\partial u}{\partial n}}\big]d\sigma. \end{align*} Since $u_{x_{N+1}}|_{\Sigma}=u_{t}|_{\Sigma}=0$, we have \begin{equation} \lim_{\eta\to 0}S=-\varepsilon\int_{\Sigma}\operatorname{sign} (\frac{\partial u}{\partial n})(u_{x_ix_{j}}n_{j}n_i-\Delta u)d\sigma. \label{e3.18} \end{equation} Using the local coordinates on $V_{\tau}, \tau=1,2,\dots, n$, we have $$ y_k=F_{\tau}^k(x),\quad k=1,2,\dots, N, \quad y_{m}|_{\Sigma}=0. $$ By a direct computation (refer to \cite{WZ}), on $\Sigma\cap V_{\tau}$ we obtain \begin{gather*} u_{x_ix_{j}}=\sum_{k=1}^{N}u_{y_Ny_k}F^{N}_{x_i}F^k_{x_{j}} +\sum_{k=1}^{N-1}u_{y_Ny_k}F^{N}_{x_i}F^k_{x_{j}}+u_{y_{m}}F^{m}_{x_ix_{j}}\,,\\ u_{x_ix_{j}}n_{j}n_i =\frac{\sum_{k=1}^{N}u_{y_Ny_k}F^{N}_{x_i}F^k_{x_{j}}F^{N}_{x_{j}}F^{N}_{x_i}} {|\operatorname{grad}F^{N}|^2}+\sum_{k=1}^{N-1}u_{y_Ny_k}F^k_{x_i}F^{N}_{x_{j}} +\frac{u_{y_{m}}F^{m}_{x_ix_{j}}F^{N}_{x_{j}}F^{N}_{x_i}} {|\operatorname{grad}F^{N}|^2}, \end{gather*} in which $F^k=F^k_{\tau}$. since the inner normal vector is $$ \vec{n}=-(\frac{\partial F^{N}}{\partial x_{1}},\dots, \frac{\partial F^{N}}{\partial x_N})=-\operatorname{grad} F^{N}, $$ it follows that $$ u_{x_ix_{j}}n_{j}n_i-\Delta u=u_{y_{m}}\Big(\frac{F^{m}_{x_ix_{j}}F^{N}_{x_{j}}F^{N}_{x_i}} {|\operatorname{grad}F^{N}|^2}-F^{m}_{x_ix_i}\Big). $$ By Lemma \ref{lem3}, we see that $\lim_{\eta\to 0}S$ can be estimated by $|\operatorname{grad}u|_{L^{1}(\Omega)}$. By letting $\eta\to 0$, from $$ \lim_{\eta\to 0}[|\operatorname{grad}u|S_{\eta}(|\operatorname{grad}u|) -I_{\eta}(|\operatorname{grad}u|)]=0, $$ we have $$ \frac{d}{dt}\int_{\Omega}|\operatorname{grad}u|dx\leq c_{1}+c_2\int_{\Omega}|\operatorname{grad}u|dx. $$ Further, by Gronwall's Lemma we have \begin{equation} \int_{\Omega}|\operatorname{grad}u|dx\leq c.\label{e3.19}\end{equation} \end{proof} By \eqref{e3.5} and \eqref{e3.19}, it is easy to show that \begin{equation} \iint_{Q_T}(a(u_{\varepsilon })+\varepsilon ) | \nabla u_{\varepsilon }| ^2\,dx\,dt\leq c. \label{e3.20}\end{equation} Thus, there exists a subsequence $\{u_{\varepsilon _n}\}$ of $u_{\varepsilon}$ and a function $u\in BV(Q_T)$ $\cap L^{\infty}(Q_T)$ such that $u_{\varepsilon _n}\to u$ a.e. on $Q_T$. \begin{proof} We now prove that $u$ is a generalized solution of equation \eqref{e1.1} with the initial condition \eqref{e1.4}. For any $\varphi(x,t)\in C^{1}_0(Q_T)$, we have \begin{align*} &\iint_{Q_T}[\frac{\partial }{\partial x_i}\int_0^{u_{\varepsilon }}\sqrt{a(s)} ds-\frac{\partial }{\partial x_i}\int_0^{u}\sqrt{a(s)}ds]\varphi(x,t)\,dx\,dt \\ &=-\iint_{Q_T}\Big[\int_0^{u_{\varepsilon }}\sqrt{a(s)} ds-\int_0^{u}\sqrt{a(s)}ds\Big]\varphi_{x_i}(x,t)\,dx\,dt. \end{align*} By a limiting process, we know the above equality is also true for any $\varphi(x,t)\in L^2(Q_T)$. By \eqref{e3.20}, we have \[ \frac{\partial }{\partial x_i}\int_0^{u_{\varepsilon }}\sqrt{a(s)} ds\rightharpoonup \frac{\partial }{\partial x_i}\int_0^{u}\sqrt{a(s)}ds \] weakly in $L^2(Q_T)$ for $i=1,2,\dots, N$. This implies \begin{equation*} \frac{\partial }{\partial x_i}\int_0^{u}\sqrt{a(s)}ds\in L^2(Q_T),\quad i=1,2,\dots, N. \end{equation*} Thus $u$ satisfies \eqref{e2.3} in Definition \ref{def1}. Let $\varphi\in C_0^2(Q_T)$, $\varphi\geq 0$, and $\{n_i\}$ be the inner normal vector of $\Omega $. Multiplying \eqref{e3.5} by $\varphi S_{\eta}(u_{\varepsilon}-k)$, and integrating it over $Q_T$, we obtain \begin{equation} \begin{aligned} &\iint_{Q_T}I_{\eta}(u_{\varepsilon}-k)\varphi _{t}\,dx\,dt +\iint_{Q_T}A_{\eta}(u_{\varepsilon},k)\Delta\varphi \,dx\,dt\\ & -\varepsilon\iint_{Q_T}\nabla u_{\varepsilon}\cdot\nabla\varphi S_{\eta}(u_{\varepsilon }-k)\,dx\,dt -\varepsilon\iint_{Q_T}|\nabla u_{\varepsilon}|^2S_{\eta}'(u_{\varepsilon }-k)\varphi \,dx\,dt \\ &-\iint_{Q_T}a(u_{\varepsilon})|\nabla u_{\varepsilon}|^2S_{\eta}'(u_{\varepsilon}-k)\varphi \,dx\,dt =0. \end{aligned}\label{e3.21} \end{equation} By Lemma \ref{lem2}, \begin{equation} \begin{aligned} &\liminf_{\varepsilon \to 0} \iint_{Q_T}S_{\eta}'(u_{\varepsilon }-k)a(u_{\varepsilon})\frac{\partial u_{\varepsilon}}{\partial x_i}\frac{\partial u_{\varepsilon}}{\partial x_i}\varphi \,dx\,dt \\ &\geq \iint_{Q_T}S_{\eta}'(u-k)| \nabla \int_0^{u}\sqrt{a(s)}ds|^2\varphi \,dx\,dt. \end{aligned}\label{e3.22} \end{equation} Let $\varepsilon \to 0$ in \eqref{e3.21}. By \eqref{e3.22}, we get \eqref{e2.4}. Finally, we can prove equality \eqref{e2.5} in a similar manner as that in \cite{VH} or \cite{ZZ}, we omit the details. \end{proof} \section{Proof of Theorem \ref{thm2.3}} \begin{proof} Let $u$ and $v$ be two entropy solutions of \eqref{e1.1} in the sense of Definition \ref{def1}. Suppose the initial values are \begin{equation} u(x,0)=u_0(x),\quad v(x,0)=v_0(x)\,.\label{e4.1} \end{equation} By Definition \ref{def1}, for any $\varphi\in C_0^2(Q_T)$, $\varphi\geq 0$, and $\eta>0$, $k,l\in \mathbb{R}$, we have \begin{gather} \iint_{Q_T}\Big[I_{\eta}(u-k)\varphi_{t}+A_{\eta}(u,k)\Delta \varphi-S_{\eta }^{\prime }(u-k)| \nabla \int_0^{u}\sqrt{a(s)}ds| ^2\varphi\Big]\,dx\,dt\geq 0, \label{e4.2}\\ \iint_{Q_T}\Big[I_{\eta}(v-l)\varphi_{\tau}+A_{\eta}(v,l)\Delta \varphi-S_{\eta }'(v-l)| \nabla \int_0^v\sqrt{a(s)}ds|^2\varphi \Big]\,dy\,d\tau\geq 0.\label{e4.3} \end{gather} Let $\psi (x,t,y,\tau)=\phi (x,t)j_{h}(x-y,t-\tau )$, where $\phi (x,t)\geq 0,\;\phi (x,t)\in C_0^{\infty}(Q_T)$, and \begin{gather} j_{h}(x-y,t-\tau )=\omega_{h}(t-\tau)\Pi_{i=1}^{N}\omega _{h}(x_i-y_i),\label{e4.4} \\ \begin{gathered} \omega _{h}(s)=\frac{1}{h}\omega(\frac{s}{h}),\quad \omega(s)\in C_0^{\infty}(R),\quad \omega(s)\geq 0,\quad \omega (s)=0\\ \text{if }| s| >1,\quad \int_{-\infty}^{\infty}\omega (s)ds=1. \end{gathered}\label{e4.5} \end{gather} We choose $k=v(y,\tau)$, $l=u(x,t)$, $\varphi_{1}=\psi (x,t,y,\tau)$ in \eqref{e4.2}-\eqref{e4.3}, integrating over $Q_T$ we obtain \begin{equation} \begin{aligned} &\iint_{Q_T}\iint_{Q_T}[I_{\eta}(u-v)(\psi _{t}+\psi_{\tau}) +A_{\eta}(u,v)\Delta_{x}\psi+A_{\eta }(v,u)\Delta_y\psi]\\& -S_{\eta}'(u-v)\Big(| \nabla \int_0^{u}\sqrt{a(s)}ds| ^2+| \nabla \int_0^v\sqrt{a(s)}ds| ^2\Big)\psi \,dx\,dt\,dy\,d\tau=0. \end{aligned}\label{e4.6} \end{equation} Clearly, \begin{gather*} \frac{\partial j_{h}}{\partial t}+\frac{\partial j_{h}}{\partial \tau }=0,\quad \frac{\partial j_{h}}{\partial x_i}+\frac{\partial j_{h}}{\partial y_i}=0,\quad i=1,\dots ,N; \\ \frac{\partial \psi}{\partial t}+\frac{\partial \psi}{\partial \tau }=\frac{\partial \phi}{\partial t}j_{h},\quad \frac{\partial \psi}{\partial x_i}+\frac{\partial \psi}{\partial y_i}=\frac{\partial \phi}{\partial x_i}j_{h}. \end{gather*} For the third and the fourth terms in \eqref{e4.6}, we have \begin{align*} &\iint_{Q_T}[A_{\eta}(u,v)\Delta_{x}\psi +A_{\eta}(v,u)\Delta _y\psi]\,dx\,dt\,dy\,d\tau \\ &=\iint_{Q_T}\iint_{Q_T}\{A_{\eta}(u,v)(\Delta_{x}\phi j_{h}+2\phi_{x_i}j_{hx_i}+\phi \Delta j_{h})+A_{\eta}(v,u)\phi \Delta_yj_{h}\}\,dx\,dt\,dy\,d\tau \\ &=\iint_{Q_T}\iint_{Q_T}\{A_{\eta}(u,v)\Delta_{x}\phi j_{h}+A_{\eta}(u,v)\phi_{x_i}j_{hx_i}+A_{\eta}(v,u)\phi _{x_i}j_{hy_i}\}\,dx\,dt\,dy\,d\tau \\ &\quad -\iint_{Q_T}\iint_{Q_T}\{\widehat{a(u)S_{\eta}(u-v)} \frac{\partial u}{ \partial x_i} -\widehat{\int_u^va(s)S_{\eta}'(s-v)ds} \frac{\partial u}{\partial x_i})\phi j_{hx_i}\}\,dx\,dt\,dy\,d\tau, \end{align*} %\label{e4.7} where \begin{gather*} \widehat{a(u)S_{\eta}(u-v)}=\int_0^{1}a(su^{+}+(1-s)u^{-})S_{\eta }(su^{+}+(1-s)u^{-}-v)ds, \\ \int_u^v\widehat{a(s)S_{\eta}'(s-v)}ds=\int_0^{1} \int_{su^{+}+(1-s)u^{-}}^va(\sigma)S_{\eta}(\sigma -su^{+}-(1-s)u^{-})d\sigma ds. \end{gather*} Since \begin{align*} &\iint_{Q_T}\iint_{Q_T}S_{\eta}'(u-v) \Big(| \nabla_{x}\int_0^{u}\sqrt{a(s)}ds|^2 +|\nabla_y\int_0^v\sqrt{a(s)}ds|^2\Big)\psi \,dx\,dt\,dy\,d\tau\\ &=\iint_{Q_T}\iint_{Q_T}S_{\eta}'(u-v) \Big(| \nabla _{x}\int_0^{u}\sqrt{a(s)}ds| -| \nabla_y\int_0^v\sqrt{a(s)}ds|\Big)^2\psi \,dx\,dt\,dy\,d\tau \\ &\quad +2\iint_{Q_T}\iint_{Q_T}S_{\eta}'(u-v)\nabla_{x}\int_0^{u} \sqrt{a(s)}ds\cdot \nabla_y\int_0^v\sqrt{a(s)}ds\psi \,dx\,dt\,dy\,d\tau\,. \end{align*} %\label{e4.8} By Lemma \ref{lem1}, we have \begin{align*} &\iint_{Q_T}\iint_{Q_T}\nabla_{x}\nabla _y\int_{v}^{u}\sqrt{a(\delta)}\int_{\delta}^v\sqrt{a(\sigma )}S_{\eta}'(\sigma -\delta)\,d\sigma\,d\delta \psi \,dx\,dt\,dy\,d\tau \\ &=\iint_{Q_T}\iint_{Q_T}\int_0^{1}\int_0^{1}\sqrt{ a(su^{+}+(1-s)u^{-})}\sqrt{a(\sigma v^{+}+(1-\sigma)v^{-}} \\ &\quad \times \times S_{\eta}'[\sigma v^{+}+(1-\sigma )v^{-}-su^{+}-(1-s)u^{-}]\,d\,d\sigma \nabla _{x}u\nabla _yv\,dx\,dt\,dy\,d\tau \\ &=\iint_{Q_T}\iint_{Q_T}\int_0^{1}\int_0^{1}S_{\eta}' [\sigma v^{+}+(1-\sigma)v^{-}-su^{+}-(1-s)u^{-}]\,d\,d\sigma \\ &\quad\times \widehat{\sqrt{a(u)}}\nabla_{x}u\widehat{\sqrt{a(v)}}\nabla _yv\,dx\,dt\,dy\,d\tau \\ &=\iint_{Q_T}\iint_{Q_T}\int_0^{1}\int_0^{1}S_{\eta }^{\prime }(v-u)\nabla _{x}\int_0^{u}\sqrt{a(s)}ds\nabla _y\int_0^v\sqrt{a(s)}ds\,dx\,dt\,dy\,d\tau. \end{align*} %\label{e4.9} and \begin{align*} &\iint_{Q_T}\iint_{Q_T}\nabla _{x}\nabla_y\int_{v}^{u} \sqrt{a(\delta)}\int_{\delta }^v\sqrt{a(\sigma)}S_{\eta}'(\sigma-\delta) \,d\sigma\,d\delta \psi \,dx\,dt\,dy\,d\tau \\ &=\iint_{Q_T}\iint_{Q_T}\int_0^{1}\sqrt{a(su^{+}+(1-s)u^{-})} \\ &\quad \times\int_{su^{+}+(1-s)u^{-}}^v\sqrt{a(\sigma)}S_{\eta}' (\sigma -su^{+}-(1-s)u^{-})d\sigma ds\frac{\partial u}{\partial x_i}j_{hx_i}\phi \,dx\,dt\,dy\,d\tau. \end{align*} %\label{e4.10} We further have \begin{align*} &\iint_{Q_T}\iint_{Q_T}(\widehat{a(u)S_{\eta}(u-v)}\frac{\partial u}{\partial x_i}-\widehat{\int_u^va(s)S_{\eta}' (s-u)ds}\frac{\partial u}{\partial x_i})j_{hx_i}\phi \,dx\,dt\,dy\,d\tau \\ &\quad +2\iint_{Q_T}\iint_{Q_T}S_{\eta}'(u-v)\nabla_{x}\int_0^{u} \sqrt{a(s)}ds\cdot \nabla_y\int_0^v\sqrt{a(s)}ds\psi \,dx\,dt\,dy\,d\tau \\ &=\iint_{Q_T}\iint_{Q_T}\Big[\int_0^{1}a(su^{+}+(1-s)u^{-}) S_{\eta}(su^{+}+(1-s)u^{-}-v)ds \\ &\quad -\int_0^{1}\int_{su^{+}+(1-s)u^{-}}^va(\sigma)S_{\eta }'(\sigma-su^{+}-(1-s)u^{-})d\sigma ds \\ &\quad +2\int_0^{1}\sqrt{a(su^{+}+(1-s)u^{-})}\int_{su^{+}+(1-s)u^{-}}^v \sqrt{a(\sigma)}S_{\eta }'(\sigma-su^{+}\\ &\quad -(1-s)u^{-})d\sigma ds\Big] \frac{\partial u}{\partial x_i}j_{hx_i}\phi \,dx\,dt\,dy\,d\tau \\ &=-\iint_{Q_T}\iint_{Q_T}\int_0^{1}\int_{su^{+}+(1-s)u^{-}}^v[ \sqrt{a(\sigma)}-\sqrt{a(su^{+}+(1-s)u^{-})}] \\ &\quad \times S'_{\eta} (\sigma-su^{+}-(1-s)u^{-})d\sigma ds\frac{\partial u}{\partial x_i}j_{hx_i}\phi \,dx\,dt\,dy\,d\tau \to 0, \end{align*} %\label{e4.11} as $\eta \to 0$. Since $$ \lim_{\eta \to 0}A_{\eta}(u,v)=\underset{\eta \to 0}{\lim}A_{\eta }(v,u)=\operatorname{sign}(u-v)[A(u)-A(v)], $$ we have \begin{equation} \lim_{\eta \to 0}[A_{\eta}(u,v)\phi _{x_i}j_{hx_i}+A_{\eta }(u,v)\phi _{y_i}j_{hy_i}]=0.\label{e4.12} \end{equation} By \eqref{e4.6}--\eqref{e4.12} and letting $\eta \to 0,h\to 0$ in \eqref{e4.6}, we obtain \begin{equation} \iint_{Q_T}\left[| u(x,t)-v(x,t)|\phi_{t}+ |A(u)-A(v)|\Delta \phi\right]\,dx\,dt\geq 0.\label{e4.13} \end{equation} Let $\delta_{\varepsilon}$ be the mollifier. For any given $\varepsilon>0$, $y=(y_{1},\dots, y_N)$, $\delta_{\varepsilon}(y)$ is defined by $$ \delta_{\varepsilon}(y)=\frac{1}{\varepsilon^{N}}\delta(\frac{y}{\varepsilon}), $$ where $$ \delta(y)=\begin{cases} \frac{1}{A}e^{\frac{1}{|y|^2-1}},&\text{if } |y|<1,\\ 0,&\text{if } |y|\geq 1, \end{cases} $$ with $$ A=\int_{B_{1}(0)}e^{\frac{1}{|y|^2-1}}dx. $$ Especially, we can choose $\phi$ in \eqref{e4.13} by $$ \phi (x,t)=\omega_{\lambda \varepsilon}(x)\eta (t), $$ where $\eta (t)\in C_0^{\infty}(0,T)$, and $\omega_{\lambda \varepsilon}(x)$ is the mollified function of $\omega_{\lambda}$. Let $\omega_{\lambda}(x)\in C_0^2(\Omega)$ be defined as follows: for any given small enough $0<\lambda$, $0\leq\omega_{\lambda}\leq 1$, $\omega|_{\partial \Omega}=0$ and $$ \omega _{\lambda}(x)=1, if\ d(x)=\operatorname{dist}(x,\partial \Omega)\geq \lambda, $$ where $0\leq d(x)\leq \lambda$ and $$ \omega_{\lambda}(d(x))=1-\frac{(d(x)-\lambda)^2}{\lambda^2}. $$ Then $\omega_{\lambda \varepsilon}=\omega_{\lambda}\ast \delta_{\varepsilon}(d)$, \begin{align*} \omega'_{\lambda\varepsilon}(d) &=-\int_{\{|s|<\varepsilon\}\cap\{0