\documentclass[reqno]{amsart}
\usepackage{hyperref}
\usepackage{amssymb}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 207, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/207\hfil Polytropic filtration equations]
{Solutions to polytropic filtration equations with a convection term}

\author[H. Zhan \hfil EJDE-2017/207\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 February 16, 2017. Published September 8, 2017.}
\subjclass[2010]{35L65, 35K85, 35R35}
\keywords{Polytropic filtration equation;
  convection term; stability; 
\hfill\break\indent boundary value condition}

\begin{abstract}
 We introduce a new type of the weak solution of the polytropic
 filtration equations with a convection term,
 $$
 {u_t}= \operatorname{div} (a(x)|u|^{\alpha}{| {\nabla u} |^{p-2}}\nabla u)
 +\frac{\partial b^{i}(u^m)}{\partial x_i}.
 $$
 Here, $\Omega\subset\mathbb{R}^N$ is a domain with a $C^2$
 smooth boundary $\partial \Omega$, $a(x)\in C^1(\overline{\Omega})$, $p>1$,
 $m=1+\frac{\alpha}{p-1}$, $\alpha >0$, $a(x)>0$ when $x\in \Omega$ and
 $a(x)=0$ when $x\in \partial \Omega$.  Since the equation is degenerate on
 the boundary, its weak solutions may lack the needed regularity to have a
 trace on the boundary. The main aim of the paper is to establish the
 stability of the weak solution without any  boundary value condition.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{definition}[theorem]{Definition}
\newtheorem{corollary}[theorem]{Corollary}
\allowdisplaybreaks

\section{Introduction}

Consider the polytropic filtration equation with a convection term
\begin{equation}
{u_t}= \operatorname{div} (a(x)|u|^{\alpha}{| {\nabla u} |^{p-2}}\nabla u)
 +\frac{\partial  b^{i}(u^m)}{\partial x_i},\quad
 (x,t) \in Q_T=\Omega  \times (0,T),\label{e1.1}
\end{equation}
where $p>1$,  $m=1+\frac{\alpha}{p-1}$, $\alpha >0$, $\Omega\subset\mathbb{R}^N$
is with a $C^2$ smooth boundary $\partial \Omega$, $a(x)\in C^1(\overline{\Omega})$,
$a(x)\geqslant 0$. The equations like \eqref{e1.1} arise from a variety of diffusion
phenomena, such as soil physics, fluid dynamics, combustion theory,
reaction chemistry, one can see \cite{A,WZYL} and the references therein.

 In particular, when $\alpha>0$,  $a(x)\equiv 1$, the well-posedness of
 equation \eqref{e1.1} with the usual initial-boundary value conditions
\begin{gather}
 u|_{t=0} = u_0(x),\ x\in\Omega,\label{e1.2} \\
u(x,t) = 0,\quad  (x,t)\in \Gamma_T=\partial \Omega \times (0,T),\label{e1.3}
\end{gather}
 has been studied thoroughly, one can refer to
 \cite{CW,DK,GP,Li,LPV,Lu,MV,O,YLCGX,Z1,Z2,Zh,ZY}.
  In this article, we assume that
\begin{gather*}
 a(x)>0, x\in \Omega, \\
 a(x)=0, x\in\partial \Omega.
\end{gather*}
 Consequently, equation \eqref{e1.1} is always degenerate on the boundary.
Not only the degeneracy comes from the physics quantity $u$ itself,
but also comes from the diffusion coefficient $a(x)$.

 Now, let us introduce some basic definitions and the main results.
For every fixed $t\in[0, T]$,  the Banach space
$$
V_t(\Omega) =\big\{u(x,t) : u(x,t)\in L^2(\Omega)\cap W^{1,1}_0(\Omega),
|\nabla u(x,t)|^{p}\in L^1 (\Omega)\big\},
$$
is with the norm
$$
\|u\|_{V_t(\Omega)} = \|u\|_{2,\Omega} + \|\nabla u\|_{p,\Omega} ,
$$
and we denote its dual space as  $V'_t(\Omega)$. By $ W (Q_T)$
we denote the Banach space
\begin{gather*}
 W (Q_T) = \{u : [0,T]\to V_t(\Omega)|u\in L^2(Q_T),|\nabla u|^{p}
 \in L^1(Q_T), u = 0\quad\text{on }\partial \Omega\}, \\
\|u\|_{ W (Q_T)} = \|\nabla u\|_{p,Q_T} + \|u\|_{2,Q_T} .
\end{gather*}
Here $ W '(Q_T)$ is the dual of $ W (Q_T)$
(the space of linear functionals over $ W (Q_T)$,
$ w\in  W '(Q_T)$ if
\begin{gather*}
w=w_0+\sum_{i=1}^{n}D_iw_i,\quad  w_0\in L^2(Q_T), w_i\in L^{p'}(Q_T),\\
\forall\phi\in  W (Q_T),\;
\langle w,\phi\rangle=\iint_{Q_T}\Big(w_0\phi+\sum_{i}w_iD_i\phi\Big)\,dx\,dt.
\end{gather*}
The norm in $ W '(Q_T)$ is defined by
$$
\|v\|_{ W '(Q_T)}= \sup\{\langle v,\phi\rangle :
\phi\in \mathbf{W(Q_T)},\|\phi\|_{ W (Q_T)}\leqslant 1\}.
$$

\begin{definition} \label{def1.1} \rm
 A nonnegative function $u(x,t)$ is said to be a weak solution of \eqref{e1.1}
 with the initial value \eqref{e1.2}, if $u$ satisfies
\begin{equation}
u \in{L^\infty }({Q_T}),\; \frac{\partial u}{\partial t} \in { W '}({Q_T}),\;
{a(x)}|u|^{\alpha}| {\nabla u} |^{p} \in {L^1}({Q_T}), \label{e1.4}
\end{equation}
and for any function $\varphi_1  \in  L^{1}(0,T; C_0^1(\Omega))$,
$\varphi_2\in L^{\infty}(Q_T)$ such that for any given
$t\in [0, T)$,  $\varphi_2(x,\cdot)\in W_{\rm loc}^{1,p}(\Omega)$, we have
\begin{equation}
\begin{aligned}
&\iint_{{Q_T}} \big[\frac{\partial u}{\partial t}(\varphi_1\varphi_2)
 + a(x)|u|^{\alpha}| \nabla u |^{p- 2}\nabla u \cdot \nabla
(\varphi_1\varphi_2) \\
& +b^{i}(u^m)(\varphi_1\varphi_2)_{x_i}\big]\,dx\,dt = 0.
\end{aligned} \label{e1.5}
\end{equation}
The initial value \eqref{e1.2} is satisfied in the sense that
\begin{equation}
\lim_{t\to 0}\int_{\Omega}u(x,t)\phi(x)\,dx
=\int_{\Omega}u_0(x)\phi(x)\,dx,\forall \phi(x)\in C_0^{\infty}(\Omega).\label{e1.6}
\end{equation}
If $u\in L^{\infty}(0,T; W^{1,\gamma}(\Omega))$ for some constant $\gamma>1$,
the boundary value condition \eqref{e1.3} is satisfied in the sense
of the trace, then we say $u$ is a weak solution of the initial-boundary
problem of equation \eqref{e1.1}.
\end{definition}

Clearly, if noticing $m=1+\frac{\alpha}{p-1}$, by \eqref{e1.4}, then
$$
a(x)|\nabla u^m|^p\in L^1(Q_T),
$$
and \eqref{e1.5} is equivalent to
\begin{equation}
\begin{aligned}
&\iint_{{Q_T}} \big[\frac{\partial u}{\partial t}(\varphi_1\varphi_2)
+ \frac{1}{m^{p-1}}a(x)| \nabla u^m |^{p- 2}\nabla u^m \cdot \nabla
(\varphi_1\varphi_2)\\
&+b^{i}(u^m)(\varphi_1\varphi_2)_{x_i}\big]\,dx\,dt = 0.
\end{aligned}\label{e1.7}
\end{equation}

In general, since \eqref{e1.1} is always degenerate on the boundary,
instead of $u(x,t)\in L^{\infty}(0,T; W_0^{1,p}(\Omega))$, we only
have $u(x,t)\in L^{\infty}(0,T; W^{1.p}_{\rm loc}(\Omega))$.
Thus, we can not define the trace of the weak solution $u$ on the boundary.
If $u, v$ are two weak solutions of equation \eqref{e1.1},
to prove the stability (or uniqueness) of the weak solutions,
one generally must choose a test function with the form $f(x,t, u-v)$
which involves the boundary value condition
\begin{equation}
u(x,t)=v(x,t)=0, \quad (x,t)\in \Gamma_T=\partial \Omega \times (0,T).\label{e1.8}
\end{equation}
However, the weak solution defined in this paper can not guarantee this condition.
This is the main reason that we need to choose the test function
$\varphi_1\varphi_2$ in Definition \ref{def1.1}.

If $\alpha=0$, $m=1$, $b^{i}\equiv 0$, the existence of the weak solutions
 had been proved in our previous paper \cite{Z3}. In this paper, we mainly
concern with the stability of the weak solutions of equation \eqref{e1.1}.

\begin{theorem} \label{thm1.2}
Let $u,v$ be two nonnegative solutions of  \eqref{e1.1} with the same
homogeneous boundary value condition \eqref{e1.3} and with the different
initial values $u_0$, $v_0$ respectively.
Then
\begin{equation}
\int_{\Omega}|  u(x,t)-v(x,t)|\,dx
\leqslant \int_{\Omega}| u_{0}-v_{0}|\,dx.\label{e1.9}
\end{equation}
\end{theorem}

\begin{theorem} \label{thm1.3}
Let $u,v$ be two nonnegative solutions of equation \eqref{e1.1}
with the initial values $u_0$, $v_0$ respectively. If $1<p\leqslant 2$, and
\begin{equation}
\int_{\Omega}a^{-\frac{1}{p-1}}(x)dx<\infty, \label{e1.10}
\end{equation}
then the stability of the weak solutions is true in the sense of \eqref{e1.9}.
\end{theorem}

\begin{theorem} \label{thm1.4}
Let $u,v$ be two nonnegative solutions of  \eqref{e1.1}
with the initial values $u_0$, $v_0$ respectively. If $p>1$
and for small enough $\lambda>0$, $u(x)$ and
$v(x)$ satisfy
\begin{equation}
\frac{1}{\lambda}\Big(\int_{\Omega\setminus\Omega_{\lambda}}a(x)
 |\nabla u^m|^{p}dx\Big)^{\frac{p-1}{p}}
\leqslant c,\quad
 \frac{1}{\lambda}\Big(\int_{\Omega\setminus\Omega_{\lambda}}a(x)
 |\nabla v^m|^{p}dx\Big)^{\frac{p-1}{p}}\leqslant c,\label{e1.11}
\end{equation}
then  \eqref{e1.9} is true. Here $\Omega_{\lambda}=\{x\in \Omega: a(x)>\lambda\}$
\end{theorem}

\begin{theorem} \label{thm1.5}
Let $u,v$ be two weak solutions of problem \eqref{e1.1} with the initial values
$u_0(x), v_0(x)$ respectively. If $p>1$, $m>0$,
\begin{equation}
\int_{\Omega}\frac{|\nabla a|}{a}|u^m|dx\leqslant c, \quad
\int_{\Omega}\frac{|\nabla a|}{a}|v^m|dx\leqslant c, \label{e1.12}
\end{equation}
then  \eqref{e1.9} is true.
\end{theorem}

At the end,  we suggest that not any boundary value condition is required
in Theorems \ref{thm1.3}--\ref{thm1.5}. However, from my own perspective, the
condition \eqref{e1.12} in Theorem 1.5 makes a substitute of the boundary
value condition. Moreover, if $b^i\equiv 0$, i.e. equation \eqref{e1.1}
has no convection term, Theorem \ref{thm1.5} is true without the
condition \eqref{e1.12}.

\section{Proof of Theorem \ref{thm1.2}}

 Let $u,v$ are two nonnegative solutions of equation \eqref{e1.1} with
the same homogeneous boundary value and with the different initial
values $u_0$, $v_0$ respectively.
From the definition of the weak solution,  we let
$\varphi_1=\varphi  \in L^1(0,T; C_0^1(\Omega))$, $\varphi_2\equiv1$. Then
\begin{equation}
\begin{aligned}
&\int_{{\Omega}} \varphi \frac{\partial (u - v)}{\partial t}\,dx
+ \int_{{\Omega}} a(x) (u^\alpha| \nabla u |^{p - 2}\nabla u - v^\alpha
| \nabla v |^{p - 2}\nabla v) \cdot \nabla \varphi\,dx \\
&+\int_{\Omega}[b^{i}(u^m)-b^{i}(v^m)]\varphi_{x_i}\,dx
=0,
\end{aligned}\label{e2.1}
\end{equation}
or  equivalently
\begin{equation}
\begin{aligned}
&\int_{{\Omega}} \varphi \frac{\partial (u - v)}{\partial t}\,dx
+\frac{1}{m^{p-1}}\int_{{\Omega}} a(x) (| \nabla u^m |^{p - 2}\nabla u^m
 - | \nabla v^m |^{p - 2}\nabla v^m) \cdot \nabla \varphi\,dx \\
&+\int_{\Omega}[b^{i}(u^m)-b^{i}(v^m)]\varphi_{x_i}dx
=0.
\end{aligned}\label{e2.2}
\end{equation}
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}\big(1-\frac{|  s| }{\eta}\big)_{+}\,.\label{e2.3}
\end{equation}
Obviously $h_{\eta}(s)\in C(\mathbb{R})$, and
\begin{equation}
\begin{gathered}
h_{\eta}(s)\geqslant 0,\quad
|  sh_{\eta}(s)|  \leqslant 1,\quad
|  S_{\eta }(s)|  \leqslant 1,\\
\lim_{\eta \to 0} S_{\eta}(s)=\operatorname{sgn}s,\quad
\lim_{\eta \to 0} sS_{\eta}'(s)=0. 
\end{gathered}\label{e2.4}
\end{equation}

We can choose $\varphi={S_\eta}(u^m - v^m)$ as the test function, then
\begin{equation}
\begin{aligned}
&\int_{\Omega} S_{\eta}(u^m - v^m)\frac{\partial (u - v)}{\partial t}dx
+ \frac{1}{m^{p-1}}\int_{{\Omega}}a(x)(|\nabla u^m |^{p-2}
 \nabla u^m- |\nabla v^m |^{p - 2}\nabla v^m) \\
&\quad \cdot \nabla(u^m - v^m)S_\eta'(u^m-v^m)dx \\
&=-\int_{\Omega}[b^{i}(u^m)-b^{i}(v^m)](u^m - v^m)_{x_i}S_\eta'(u^m-v^m)dx.
\end{aligned} \label{e2.5}
\end{equation}
Clearly,
\begin{equation}
\begin{aligned}
\lim_{\eta\to 0}\int_\Omega  {{S_\eta}(u^m - v^m)
 \frac{{\partial (u - v)}}{{\partial t}} \,dx} 
&=\int_\Omega  {\operatorname{sgn}(u^m - v^m)\frac{{\partial (u - v)}}{{\partial t}}\,dx}\\
&=\int_\Omega  {\operatorname{sgn}(u-v)\frac{{\partial (u - v)}}{{\partial t}}\,dx}\\
&=\frac{d}{dt}\| {u - v} \|_{L^1(\Omega)},
\end{aligned}\label{e2.6}
\end{equation}
and
\begin{equation}
\int_{\Omega} a(x)|(| {\nabla u^m} |^{p - 2}
\nabla u - |{\nabla v^m} |^{p- 2}\nabla v^m) \cdot
\nabla (u^m - v^m)S_\eta'(u^m-v^m) \text{d}x \geqslant 0.\label{e2.7}
\end{equation}
At the same time,
$$
\int_{\Omega}a^{\frac{-1}{p-1}}(x)dx<\infty,
$$
using  Lebesgue dominated convergence theorem, by \eqref{e2.4}, we have
\begin{equation}
\begin{aligned}
&\lim_{\eta\to 0}
\big|\int_{\Omega}[b^{i}(u^m)-b^{i}(v^m)](u^m - v^m)_{x_i}S_\eta'(u^m-v^m)dx\big|\\
&\leqslant \lim_{\eta\to 0}\Big(\int_{\Omega}|[b^{i}(u^m)-b^{i}(v^m)]
 S_\eta'(u^m-v^m)a^{-\frac{1}{p}}|^{\frac{p}{p-1}}dx\Big)^{\frac{p-1}{p}}\\
&\quad\times
\Big(\int_{\Omega}a(x)(|\nabla u^m|^p+|\nabla v^m|^p )dx\Big)^{1/p}=0.
\end{aligned}\label{e2.8}
\end{equation}
Let $\eta\to0$  in \eqref{e2.2}. Then
\begin{equation}
\frac{d}{dt}\| {u - v} \|_{L^1(\Omega)} \leqslant 0.\label{e2.9}
\end{equation}
This implies
$$
\int_\Omega {| {u(x,t) - v(x,t)} |\,dx}
\leqslant \int_\Omega {| {{u_0} - {v_0}} |\,dx},\quad
\forall t \in [0,T). %\label{e2.10}
$$


\section{Proofs of Theorem \ref{thm1.3} and \ref{thm1.4}}


\begin{proof}[Proof of Theorem \ref{thm1.3}]
 By Definition \ref{def1.1},   for any function
$\varphi_1  \in  L^{1}(0,T; C_0^1(\Omega))$,
 $\varphi_2\in L^{\infty}(Q_T)$ such that for any given $t\in [0, T)$,
$\varphi_2(x,\cdot)\in W_{\rm loc}^{1,p}(\Omega)$, we have
\begin{equation}
\begin{aligned}
&\iint_{{Q_T}} \big[\frac{\partial (u-v)}{\partial t}(\varphi_1\varphi_2)
 + \frac{1}{m^{p-1}}a(x)(| \nabla u^m |^{p- 2}\nabla u^m \\
 & -| \nabla v^m |^{p- 2}\nabla v^m)\cdot \nabla (\varphi_1\varphi_2)
 +(b^i(u^m)-b^i(v^m))(\varphi_1\varphi_2)_{x_i}]\,dx\,dt
= 0.
\end{aligned}\label{e3.1}
\end{equation}
For a small positive constant $\lambda>0$, let
\begin{gather}
\Omega_{\lambda}=\{x\in\Omega: a(x)>\lambda\}, \nonumber \\
\phi_{\lambda}(x)=\begin{cases}
1,  & \text{if }  x\in \Omega_{\lambda},\\
\frac{1}{\lambda}a(x), &\text{if } x\in\Omega\setminus \Omega_{\lambda}.
\end{cases} \label{e3.2}
\end{gather}

Now, we choose $\varphi_1=\phi_{\lambda}(x)\chi_{[\tau,s]}$,
$\varphi_2=S_{\eta}(u^m-v^{m})$, and then integrate it over $\Omega$, to have
\begin{equation}
\begin{aligned}
&\int_{\tau}^{s} \int_{\Omega} \phi_{\lambda}(x)S_{\eta}(u^{m}-v^{m})
\frac{\partial (u - v)}{\partial t}\,dx\,dt 
  + \frac{1}{m^{p-1}}\int_{\tau}^{s}\int_{\Omega} \phi_{\lambda}(x)a(x) \\
&\times \big(|\nabla u^{m}|^{p-2}\nabla u^{m}
-|\nabla v^{m}|^2\nabla v^{m}\big)
 \cdot \nabla (u^{m}-v^{m})S'_{\eta}(u^{m}-v^{m})\,dx\,dt\\
&+ \frac{1}{m^{p-1}}\int_{\tau}^{s}\int_{\Omega} a(x)
 (|\nabla u^{m}|^{p-2}\nabla u^{m}
-|\nabla v^{m}|^{p-2}\nabla v^{m})  \\
&\cdot \nabla\phi_{\lambda}(x)
  S_{\eta}(u^{m}-v^{m}) \,dx\,dt\\
&+\int_{\tau}^{s}\int_{\Omega}[b^i(u^m)-b^i(v^m)]
 [\phi_{\lambda}(x)S'_{\eta}(u^{m}-v^{m})(u^{m}-v^{m})_{x_i} \\
&+S_{\eta}(u^{m}-v^{m})\phi_{\lambda x_i}(x)]\,dx\,dt
=  0.
\end{aligned}\label{e3.3}
\end{equation}
Clearly,
\begin{equation}
\begin{aligned}
&\int_{\Omega} \phi_{\lambda}(x)a(x)(|\nabla u^{m}|^{p-2}\nabla u^{m}
-|\nabla v^{m}|^2\nabla v^{m}) \\
&\cdot \nabla (u^{m}-v^{m})S'_{\eta}(u^{m}-v^{m})\,dx \geqslant 0.
\end{aligned}\label{e3.4}
\end{equation}
\begin{equation}
\begin{aligned}
&\big|\int_{\Omega} a(x)(|\nabla u^{m}|^{p-2}\nabla u^{m}
-|\nabla v^{m}|^2\nabla v^{m})  \cdot \nabla\phi_{\lambda}(x) S_{\eta}(u^{m}-v^{m})
\,dx\big| \\
&\leqslant \int_{\Omega\setminus\Omega_{\lambda}} a(x) |(|\nabla u^{m}|^{p-2}\nabla u^{m}
-|\nabla v^{m}|^2\nabla v^{m})
\cdot \nabla\phi_{\lambda}(x) S_{\eta}(u^{m}-v^{m})|dx
\\
&\leqslant \int_{\Omega\setminus\Omega_{\lambda}} a(x) |(|\nabla u^{m}|^{p-2}\nabla u^{m}
-|\nabla v^{m}|^2\nabla v^{m}) | |\nabla\phi_{\lambda}(x)|dx
\\
&\leqslant \frac{c}{\lambda}\Big[\int_{\Omega\setminus\Omega_{\lambda}} a(x) |\nabla u^{m}|^{p-1}|\nabla a|
\,dx
+\int_{\tau}^{s}\int_{\Omega\setminus\Omega_{\lambda}} a(x)
  |\nabla v^{m}|^{p-1}|\nabla a|dx\Big].
\end{aligned}\label{e3.5}
\end{equation}
Since $1<p\leqslant 2$, $|\nabla a|\leqslant c$ and
$$
\int_{\Omega\setminus\Omega_{\lambda}}|\nabla a|^{p}dx
\leqslant c\lambda\leqslant c\lambda^{p-1},
$$
it follows that
\begin{equation}
\frac{c}{\lambda}\Big(\int_{\Omega\setminus\Omega_{\lambda}}a(x)|\nabla a|^{p}dx
 \Big)^{1/p}
\leqslant \frac{c}{\lambda}\Big(\lambda\int_{\Omega\setminus\Omega_{\lambda}}
|\nabla a|^{p}dx\Big)^{1/p}\leqslant c.\label{e3.6}
\end{equation}
By \eqref{e3.5}-\eqref{e3.6}, using the H\"older inequality,
\begin{equation}
\begin{aligned}
&\big|\int_{\Omega} a(x)(|\nabla u^{m}|^{p-2}\nabla u^{m}
-|\nabla v^{m}|^2\nabla v^{m})  \cdot
\nabla\phi_{\lambda}(x) S_{\eta}(u^{m}-v^{m})\,dx\big| \\
&\leqslant \frac{c}{\lambda}\Big[\int_{\Omega\setminus\Omega_{\lambda}} a(x)
 |\nabla u^{m}|^{p-1}|\nabla a|\,dx
+\int_{\tau}^{s}\int_{\Omega\setminus\Omega_{\lambda}} a(x)
 |\nabla v^{m}|^{p-1}|\nabla a|dx\Big]\\
&\leqslant \frac{c}{\lambda}\Big(\int_{\Omega\setminus\Omega_{\lambda}}a
 |\nabla a|^{p}dx\Big)^{1/p}
\Big(\int_{\Omega\setminus\Omega_{\lambda}}a(x)|\nabla u^m|^{p}dx
 \Big)^{\frac{p-1}{p}} \\
&\quad +\frac{c}{\lambda}\Big(\int_{\Omega\setminus\Omega_{\lambda}}a(x)
 |\nabla a|^{p}dx\Big)^{1/p}
\Big(\int_{\Omega\setminus\Omega_{\lambda}}a(x)|\nabla v^m|^{p}dx
\Big)^{\frac{p-1}{p}}\\
&\leqslant c\Big(\int_{\Omega\setminus\Omega_{\lambda}}a(x)|\nabla u^m|^{p}dx
\Big)^{\frac{p-1}{p}}
+c\Big(\int_{\Omega\setminus\Omega_{\lambda}}a(x)|\nabla v^m|^{p}dx
 \Big)^{\frac{p-1}{p}}.
\end{aligned}\label{e3.7}
\end{equation}
Then, we have
\begin{equation}
\begin{aligned}
&\lim_{\lambda\to 0}\big| \int_{\Omega} a(x)(|\nabla u^{m}|^{p-2}\nabla u^m
-|\nabla v^{m}|^{p-2}\nabla v^m) \\
& \cdot \nabla\phi_{\lambda}(x) S_{\eta}(u^{m}-v^{m})\,dx|=0.
\end{aligned}\label{e3.8}
\end{equation}
At the same time,
by that $\int_{\Omega}a^{-\frac{1}{p-1}}(x)dx<c$, using \eqref{e2.4}
and the Lebesgue dominated convergence theorem, we also have
\begin{equation}
\lim_{\eta\to 0}\int_{\Omega}\phi_{\lambda}[b^i(u^m)-b^i(v^m)]
S_{\eta}'(u^m-v^m)(u^m-v^m)_{x_i}dx=0,\label{e3.9}
\end{equation}
and
\begin{equation}
\begin{aligned}
&\lim_{\lambda\to 0}|\int_{\Omega}\phi_{\lambda x_i}[b_i(u^m)-b_i(v^m)]
S_{\eta}(u^m-v^m)dx|
\\
&\leqslant \lim_{\lambda\to 0}\frac{c}{\lambda}
 \int_{\Omega\setminus\Omega_{\lambda}}|\nabla a|dx
\\
&\leqslant\lim_{\lambda\to 0}\frac{c}{\lambda}
\Big(\int_{\Omega\setminus\Omega_{\lambda}}a(x)|\nabla a|^pdx\Big)^{1/p}
\Big(\int_{\Omega\setminus\Omega_{\lambda}}a^{-\frac{1}{p-1}}(x)dx
\Big)^{\frac{p-1}{p}}
=0,
\end{aligned}\label{e3.10}
\end{equation}
by \eqref{e3.6} and $\int_{\Omega}a^{-\frac{1}{p-1}}(x)dx<c$.
At last,
\begin{equation}
\begin{aligned}
&\lim_{\eta\to 0} \lim_{\lambda\to 0}
\int_{\tau}^{s}\int_{\Omega} \phi_{\lambda}(x)S_{\eta}(u^{m}
- v^{m})\frac{\partial (u - v)}{\partial t}\,dx\,dt
\\
&=\lim_{\eta\to 0}\int_{\tau}^{s} \int_{\Omega} S_{\eta}(u^m
 - v^m)\frac{\partial (u - v)}{\partial t}\,dx\,dt
\\
&=\int_{\tau}^{s} \int_{\Omega} \operatorname{sgn}(u^m- v^m)
 \frac{\partial (u - v)}{\partial t}\,dx\,dt \\
& =\int_{\tau}^{s} \int_{\Omega} \operatorname{sgn}(u- v)
 \frac{\partial (u - v)}{\partial t}\,dx\,dt
\\
&=\int_{\tau}^{s}\frac{d}{dt}\|u-v\|_{L^1(\Omega)}dt.
\end{aligned}\label{e3.11}
\end{equation}
 Now, after letting $\lambda\to 0$,  let $\eta\to 0$  in \eqref{e3.2}.
Then by \eqref{e3.4}, \eqref{e3.8}-\eqref{e3.11},
$$
\int_\Omega  | {u(x,t) - v(x,t)} |dx \leqslant \int_\Omega  | {{u_0} - {v_0}} |dx.
$$
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1.4}]
 As in the proof of Theorem \ref{thm1.3}, we have \eqref{e3.3}- \eqref{e3.5}.
Since  $u(x)$ and $v(x)$ satisfy \eqref{e1.11}
by \eqref{e3.6}-\eqref{e3.7}, using the H\"older inequality, we have
\begin{equation}
\begin{aligned}
&\big|\int_{\Omega} a(x)(|\nabla u^{m}|^{p-2}\nabla u^{m}
-|\nabla v^{m}|^2\nabla v^{m})  \cdot \nabla\phi_{\lambda}(x) S_{\eta}(u^{m}-v^{m})
\,dx\big| \\
&\leqslant \frac{c}{\lambda}\Big(\int_{\Omega\setminus\Omega_{\lambda}}
a|\nabla a|^{p}dx\Big)^{1/p}
\Big(\int_{\Omega\setminus\Omega_{\lambda}}a(x)|\nabla u^m|^{p}dx\Big)^{\frac{p-1}{p}}
\\
&\quad +\frac{c}{\lambda}\Big(\int_{\Omega\setminus\Omega_{\lambda}}
a(x)|\nabla a|^{p}dx\Big)^{1/p}
\Big(\int_{\Omega\setminus\Omega_{\lambda}}a(x)|\nabla v^m|^{p}dx\Big)^{\frac{p-1}{p}}
\\
&\leqslant c\Big(\int_{\Omega\setminus\Omega_{\lambda}}a|\nabla a|^{p}dx\Big)^{1/p}
+c\Big(\int_{\Omega\setminus\Omega_{\lambda}}a(x)|\nabla a|^{p}dx\Big)^{1/p},
\end{aligned}\label{e3.12}
\end{equation}
which approaches zero as $\lambda\to 0$ since that 
$a(x)\in C^{1}(\overline{\Omega})$, we have \eqref{e3.8}.
 At last, since $\int_{\Omega}a^{-\frac{1}{p-1}}(x)dx<\infty$,
similar as the proof of Theorem \ref{thm1.3}, we have \eqref{e3.9}-\eqref{e3.10}.
 So, as the proof of Theorem \ref{thm1.3}, we know that the stability
\eqref{e1.9} is true.
\end{proof}


\section{Proof Theorem \ref{thm1.5}}

It is not difficult to show that the following definition is equivalent
to Definition \ref{def1.1}.

 \begin{definition} \label{def4.1} \rm
 A function $u(x,t)$ is said to be a weak solution of  \eqref{e1.1} with
 initial value \eqref{e1.2}, if
\begin{equation}
u \in L^{\infty}(Q_T),\quad u_{t}\in L^2(Q_T),\quad
 a(x){| {\nabla u} |^p} \in {L^1}({Q_T}),\label{e4.1}
\end{equation}
and for any function $g(s)\in C^1(\mathbb{R})$, $g(0)=0$,
$\varphi_1  \in  C^1_0(\Omega)$,
$\varphi_2\in L^\infty(0,T; W_{\rm loc}^{1,p}(\Omega))$,
\begin{equation}
\begin{aligned}
&\iint_{{Q_T}} [u_tg(\varphi_1\varphi_2)
 + a(x)| \nabla u |^{p - 2}\nabla u \cdot \nabla g(\varphi_1\varphi_2)
 +u \big(b_{ix_i}(x)g(\varphi_1\varphi_2) \\
&\quad +b_i(x)g_{x_i}(\varphi_1\varphi_2)\big)
-c(x,t)ug(\varphi_1\varphi_2)+f(x,t)g(\varphi_1\varphi_2)]\,dx\,dt = 0.
\end{aligned}\label{e4.2}
\end{equation}
The initial value is satisfied in the sense that
\begin{equation}
\lim_{t\to 0}\int_{\Omega}u(x,t)\phi(x)\,dx
=\int_{\Omega}u_0(x)\phi(x)\,dx,\forall \phi(x)\in C_0^{\infty}(\Omega).\label{e4.3}
\end{equation}
\end{definition}

\begin{proof}[Proof of Theorem \ref{thm1.5}]
Let  $u$, $v$ be two solutions of equation \eqref{e1.1} with
the initial values $u_0(x), v_0(x)$.
We can choose $S_{\eta}(a^{\beta}(u^m-v^m))$ as the test function.
 Then
\begin{equation}
\begin{aligned}
&\int_{\Omega} S_{\eta}(a^\beta(u^m - v^m))\frac{\partial (u - v)}{\partial t}dx
 + \frac{1}{m^{p-1}}\int_{\Omega} a^{\beta+1}(x)
\big(| \nabla u^m |^{p- 2}\nabla u^m \\
&- | \nabla v^m |^{p- 2}\nabla v^m\big)
\cdot  \nabla (u^m - v^m)S'_{\eta} (a^\beta(u^m-v^m))\,dx\\
&+ \int_{\Omega} a(x)(| \nabla u^m |^{p - 2}\nabla u^m - | \nabla v^m |^{p- 2}
 \nabla v^m) \\
&\cdot \nabla a^\beta  (u^m - v^m)S'_{\eta}(a^\beta(u^m-v^m))\,dx\\
&+ \int_{\Omega}[b_i(u^m)-b_i(v^m)][S'_\eta(a^\beta(u^m-v^m))\\
&(a^{\beta}_{x_i}(u^m-v^m)+a^{\beta}(u^m-v^m)_{x_i}dx=0.
\end{aligned}\label{e4.4}
\end{equation}
Thus
\begin{gather}
\lim_{\eta\to 0}\int_{\Omega} S_{\eta}(a^\beta(u^m - v^m))
\frac{\partial (u - v)}{\partial t}dx  = \frac{d}{dt}\| u - v \|_1,\label{e4.5}
\\
\begin{aligned}
&\int_{\Omega} a^{\beta+1}(x)(| \nabla u^m |^{p- 2}\nabla u^m -
| \nabla v^m |^{p - 2}\nabla v^m) \\
&\cdot \nabla (u^m - v^m)S'_{\eta}(a^\beta(u^m - v^m))
\,dx \geqslant 0.
\end{aligned}\label{e4.6}
\end{gather}

From $|\nabla a(x)|\leqslant c$  in $\Omega$, we have
\begin{equation}
\begin{aligned}
&\big|\int_{\Omega} a(x)  (u^m - v^m)S'_{\eta}(a^\beta(u^m - v^m))
(| \nabla u^m |^{p- 2}\nabla u^m - | \nabla v^m |^{p - 2}\nabla v^m) \\
&\cdot \nabla a^\beta\,dx\big| \\
&\leqslant c\big|\int_{\Omega} a^\beta (u^m - v^m)S'_{\eta}(a^\beta(u^m - v^m))\\
&\quad \times(| \nabla u^m|^{p- 2}\nabla u^m - | \nabla v^m |^{p - 2}\nabla v^m)
\,dx|,
\end{aligned}\label{e4.7}
\end{equation}
\begin{equation}
\begin{aligned}
&\big|\int_{\Omega} a^\beta (u^m - v^m)S'_{\eta}(a^\beta(u^m - v^m)) \\
&\times (| \nabla u^m|^{p- 2}\nabla u^m - | \nabla v^m |^{p - 2}\nabla v^m)\,dx\big|
\\
&=\big|\int_{\Omega: a^{\beta}|u^m-v^m|<\eta} a^{-\frac{p-1}{p}}
a^\beta (u^m - v^m)S'_{\eta}\\
&\quad \cdot(a^\beta(u^m - v^m))a^{\frac{p-1}{p}}(| \nabla u^m|^{p- 2}\nabla u^m
- | \nabla v^m |^{p - 2}\nabla v^m)\,dx\big|
\\
&\leqslant\Big(\int_{\Omega: a^{\beta}|u-v|<\eta} |a^{-\frac{p-1}{p}}
a^\beta (u^m - v^m)S'_{\eta}(a^\beta(u^m - v^m))|^{p}dx\Big)^{1/p} \\
&\quad \times \Big(\int_{\Omega: a^{\beta}|u-v|<\eta}a(x)(|\nabla u^m|^{p}
+|\nabla v^m|^{p})dx\Big)^{\frac{p-1}{p}}.
\end{aligned} \label{e4.8}
\end{equation}
If $\{x\in \Omega: u^m-v^m=0\}$ has $0$ measure, since
$\int_{\Omega}a^{p-1}(x)dx<\infty$,
we have
$$
\big|\int_{\{\Omega: a^{\beta}|u^m-v^m|<\eta\}} |a^{-\frac{p-1}{p}}a^\beta
(u^m - v^m)S'_{\eta}(a^\beta(u^m - v^m))|^{p}dx\big|
$$
and
\begin{equation}
\begin{aligned}
&\lim_{\eta\to 0}\Big(\int_{\{\Omega: a^{\beta}|u^m-v^m|<\beta\}}
a(x)(|\nabla u^m|^{p}+|\nabla v^m|^{p})dx\Big)^{\frac{p-1}{p}} \\
&=\Big(\int_{\{\Omega: |u^m-v^m|=0\}}a(x)
 (|\nabla u^m|^{p}+|\nabla v^m|^{p})dx\Big)^{\frac{p-1}{p}}=0.
\end{aligned}\label{e4.9}
\end{equation}
If $\{x\in \Omega: u^m-v^m=0\}$ has a positive measure, obviously
\begin{equation}
\begin{aligned}
&\lim_{\eta\to 0}\Big(\int_{\{\Omega: a^{\beta}|u^m-v^m|<\eta\}}
|a^{-\frac{p-1}{p}}a^\beta (u^m - v^m)S'_{\eta}(a^\beta(u^m - v^m))|^{p}dx\Big)^{1/p}\\
&=\Big(\int_{\{\Omega: |u^m-v^m|=0\}} |a^{-\frac{p-1}{p}}
 a^\beta (u^m - v^m)S'_{\eta}(a^\beta(u^m - v^m))|^{p}dx\Big)^{1/p}=0.
\end{aligned}\label{e4.10}
\end{equation}
By \eqref{e2.2} and \eqref{e2.4},  using the Lebesgue controlled
convergence theorem, in both cases, we have
$$
\lim_{\eta\to 0}|\int_{\Omega} a^\beta (u^m - v^m)S'_{\eta}(a^\beta(u^m - v^m))
(| \nabla u^m|^{p- 2}\nabla u^m - | \nabla v^m |^{p - 2}\nabla v^m)dx|=0.
%\label{e4.11}
$$
In addition,
\begin{equation}
\begin{aligned}
&\big|\int_{\Omega}[b_i(u^m)-b_i(v^m)]a_{x_i}^{\beta}(u^m-v^m)S'_{\eta}
(a^{\beta}(u^m-v^m))dx\big| \\
&\leqslant c\int_{\Omega}(|u^m|+|v^m|)\frac{|\nabla a|}{a}
a^{\beta}(u^m-v^m)S'_{\eta}(a^{\beta}(u^m-v^m))dx
\to 0,
\end{aligned} \label{e4.12}
\end{equation}
as $\eta\to 0$ by \eqref{e1.12},
 \begin{equation}
\begin{aligned}
&\big|\int_{\Omega}[b_i(u^m)-b_i(v^m)]a^{\beta}(u^m-v^m)_{x_i}S'_{\eta}
(a^{\beta}(u^m-v^m))dx\big| \\
&=|\int_{\Omega}a^{\beta-\frac{1}{p}}[b_i(u^m)-b_i(v^m)]
S'_{\eta}(a^{\beta}(u^m-v^m))a^{-\frac{1}{p}}(u^m-v^m)_{x_i}dx|
\\
&\leqslant c\Big(|a^{-\frac{1}{p}}a^{\beta}(u^m-v^m)
S'_{\eta}(a^{\beta}(u^m-v^m))|^{\frac{p}{p-1}}\Big)^{\frac{p-1}{p}} \\
&\quad\times \Big(\int_{\Omega}a(x)(|\nabla u^m|^p+|\nabla v^m|)\Big)^{1/p}
\to 0,
\end{aligned} \label{e4.13}
\end{equation}
as $\eta\to 0$ by \eqref{e2.4}.

Now, let $\eta\to 0$  in \eqref{e4.4}.  Then
\begin{equation}
\int_\Omega  {| {u(x,t) - v(x,t)} |\,dx}
\leqslant \int_\Omega  {| {{u_0} - {v_0}} |\,dx},\quad \forall t \in [0,T).
\label{e4.14}
\end{equation}
Theorem \ref{thm1.5} is proved.
\end{proof}

\begin{corollary} \label{coro4.1}
Let $u,v$ be two weak solutions of equation \eqref{e1.1} with the
initial values $u_0(x), v_0(x)$ respectively. If $b_i\equiv0$,
then  \eqref{e4.14} is true without any boundary value condition.
\end{corollary}

\begin{proof}
 We notice that,  in the proof of Theorem \ref{thm1.5},  condition \eqref{e1.12}
is only used to deal with the convection term to obtain \eqref{e4.12}
and \eqref{e4.13}. Consequently, when $b_i\equiv0$, the stability
is \eqref{e4.14} is true.
\end{proof}

\subsection*{Acknowledgments}
This  work was supported by the NSF of Fujian Province (no: 2015J01592),
 by the Science Foundation of
Xiamen University of Technology (no: XYK201448), China.


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\end{document}