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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 327, pp. 1--12.\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-2016/327\hfil Parabolic equations with blowing-up coefficients]
{Nonlinear parabolic equations with blowing-up coefficients
with respect to the unknown and with soft measure data}

\author[K. Zaki, H. Redwane \hfil EJDE-2016/327\hfilneg]
{Khaled Zaki, Hicham Redwane}

 \address{Khaled Zaki \newline
 Facult\'e des Sciences et Techniques,
 Universit\'e Hassan 1, B.P. 764, Settat, Morocco}
\email{zakikhaled74@hotmail.com}

\address{Hicham Redwane \newline
 Facult\'e des Sciences Juridiques, \'Economiques et Sociales,
 Universit\'e Hassan 1, B.P. 764, Settat, Morocco}
\email{redwane\_hicham@yahoo.fr}

\thanks{Submitted September 9, 2016. Published December 22, 2016.}
\subjclass[2010]{47A15, 46A32, 47D20}
\keywords{Nonlinear parabolic equations; blowing-up coefficients;
\hfill\break\indent renormalized solutions; soft measure}

\begin{abstract}
 We establish the existence of solutions for the nonlinear parabolic problem
 with Dirichlet homogeneous boundary conditions,
 $$
 \frac{\partial u}{\partial t} - \sum_{i=1}^N\frac{\partial}{\partial x_i}
 \Big( d_i(u)\frac{\partial u}{\partial x_i} \Big) =\mu,\quad u(t=0)=u_0,
 $$
 in a bounded domain. The coefficients $d_i(s)$ are continuous on an
 interval $]-\infty,m[$,  there exists an index $p$  such that $d_p(u)$ 
 blows up at a finite value $m$ of the unknown $u$,  and $\mu$ is a diffuse  
 measure.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks

\section{Introduction}\label{s1}

In this paper we study the  existence of solutions of the  problem
\begin{gather}\label{a1}
 \frac{\partial u}{\partial t} - \sum_{i=1}^N\frac{\partial}{\partial x_i}
{\Big( d_i(u)\frac{\partial u}{\partial x_i}   \Big)}  =\mu \quad \text{in }Q, \\
\label{a2}
 u(t=0)=u_0\quad \text{in }\Omega, \\
\label{a3}
 u=0\quad\text{on }{\partial \Omega}\times (0,T),
 \end{gather}
where $\Omega$ is an open bounded subset of $\mathbb{R}^N$ ($N\geq 1$), $T$ is a
positive real number, and we have set $Q$ the cylinder $\Omega\times (0,T)$ 
and ${\partial \Omega}\times (0,T)$ its lateral surface. The coefficients 
$d_i(s)$ are continuous on an interval $]-\infty,m[$ of $\mathbb{R}$ (with $m>0$)
 with value in $\mathbb{R}^+\cup\{+\infty\}$, $d_i(s)\geq \alpha>0$, and such that there 
exists an index $p$ such that $\lim_{s\to m^-}d_p(s)=+\infty$,
and where $u_0\in L^{1}(\Omega),\ u_0\leq m$ a.e. in $\Omega$ and $\mu$ is 
a measure on $Q$ with bounded total variation.

When problem \eqref{a1}-\eqref{a3} is studied, the {\it a priori} estimates 
on the above problem do not lead in general to the existence of a weak solution 
(i.e. in the distributional sense), there are mainly two type of difficulties 
in studying problem \eqref{a1}-\eqref{a3}. One consists to define in a proper
 way the term $ d_p(u)\frac{\partial u}{\partial x_p}$ on the subset 
$\{(x,t)\in Q: u(x,t)=m\}$ of $Q$ on which $d_p(u)=+\infty$. 
As an example, one can not set in general 
$ d_p(u)\frac{\partial u}{\partial x_p}=0$ on 
$\{(x,t)\in Q: u(x,t)=m\}$ to obtain the equation in the sense of distributions.

The second difficulty is represented here by the presence of an 
$L^1$ initial datum and a measure as right-hand side term in \eqref{a1}. 
The measure $\mu$ is just assumed to be bounded total variation over $Q$ 
that do not charge the sets of zero $p$-capacity (see Section \ref{s2} 
for the definition), the so called \emph{diffuse measures} or 
\emph{soft measures}, and we will use the symbol $\mathcal{M}_0(Q)$ to denote them.

To overcome this difficulty we use the framework of
renormalized solutions. This notion was introduced by Lions and
DiPerna \cite{DL} for the study of Boltzmann equation. 
This notion was then adapted to elliptic version of
\eqref{a1}-\eqref{a3} in Boccardo,  Diaz, 
Giachetti,  Murat \cite{BGDM},  Lions and F.
Murat~\cite{M}, and  Murat \cite{M, FM}. At the same the equivalent
notion of entropy solutions was developed independently by
B\'enilan and al.~\cite{BBGGPV} for the study of nonlinear elliptic
problems.

The existence of a renormalized solution of
\eqref{a1}-\eqref{a3} was proved in \cite{BR2002} in the stationary case
where $\mu \in L^2(\Omega)$. In the stationary and evolution cases
 of $u_t-div(A(x,t,u)\nabla u)=f$ in $Q$, where the matrix $A(x,t,s)$ 
blows up (uniformly with respect to $(x,t)$) as $s \to m^-$ and where 
$f\in L^1(Q)$, the existence of renormalized solution was proved by Blanchard, 
Guib\'e and Redwane in \cite{BGR2008}.


The existence and uniqueness of renormalized solution of
\eqref{a1}-\eqref{a3} was proved in \cite{BPR2013} in the case
where $ \sum_{i=1}^N\frac{\partial}{\partial x_i}
{\big( d_i(u)\frac{\partial u}{\partial x_i}   \big)}$ is replaced by the
$p$-Laplacian operator $\Delta_p u$, $u_0 \in L^1(\Omega)$ and $u_t$ is replaced 
by $b(u)_t$ and for every measure $\mu$ which does not charge the sets 
of null parabolic $p$-capacity.

Note that, the existence result in \cite{BPR2013} is strongly based on a 
decomposition theorem given in \cite{DPP} for diffuse measure 
(i.e. $\mu \in \mathcal{M}_0(Q)$), this decomposition of $\mu$ can not be 
easily used for problem \eqref{a1}-\eqref{a3}. 
Indeed (for $p=2$), for every $\mu \in \mathcal{M}_0(Q)$ there exist 
$f\in L^1(Q),\ g \in L^2(0,T; H^1_0(\Omega))$ and $F\in L^2(0,T; H^{-1}(\Omega))$ 
such that
\begin{equation}\label{a5}
\mu=f + F+ g_t\quad \text{in } D'(Q),
\end{equation}
note that the decomposition of $\mu$ is not uniquely determined. 
Therefore, equation \eqref{a1} is equivalent to
$$    
 \frac{\partial v}{\partial t} - \sum_{i=1}^N
\frac{\partial}{\partial x_i}{\Big( d_i(v+g)\frac{\partial v}{\partial x_i}   \Big)} 
 =f + F \quad \text{in }Q,     
$$
where $v=u-g$. Since $g \not\in L^\infty(Q)$ in general and 
$ \lim_{s\to m^-}d_p(s)=+\infty$, then the term $d_p(v+g)$ can not be 
easily handled. To overcome this difficulty, we use in this paper the 
following  approximation property for the measure $\mu$ 
(see Theorem \ref{t21}). Indeed, every $\mu \in \mathcal{M}_0(Q)$ can be 
strongly approximated by measures which admit decomposition \eqref{a5} 
with $g\in L^\infty(Q)$ (see \cite[Theorem 1.1]{PPP}).

A large number of papers was then devoted to the study the existence
of renormalized solution of parabolic problems with rough data under
various assumptions and in different contexts: for a review on
classical results,  see 
\cite{BM,BMR,BP2,BR,O1,VG,PG,Peti1,Peti2,Peti3,P, R1}.


We organize this article as follows. In Section \ref{s2} we give some
preliminaries and, in particular, we provide the definition of
parabolic capacity and some basic properties of diffuse measures. 
Section \ref{s3} is devoted to specifying the assumptions on $d_i$, $u_0$ and 
$\mu$. We also give the definition of a renormalized solution of
\eqref{a1}-\eqref{a3}. In Section \ref{s4} we establish
(Theorem \ref{t3}) the existence of such a solution. 
In Section \ref{s5} (Appendix) we prove Theorem \ref{t22} that will be 
a key point in the existence result.

\section{Preliminaries on Parabolic Capacity and Diffuse Measures}\label{s2} 
We recall the notion of parabolic
$p$-capacity (with $p=2$) associated to our problem (for further details see
 \cite{Pierre,DPP}). Let $Q=\Omega\times(0,T)$ for any
fixed $T>0$, and let us recall that
$$
W=\big\{ u\in L^2(0,T;H^1_0(\Omega)): u_t \in L^2(0,T;H^{-1}(\Omega))\big\}\,,
$$ 
endowed with its natural norm 
$\|\cdot\|_{L^2(0,T;H^1_0(\Omega))}+\|\cdot\|_{L^2(0,T;H^{-1}(\Omega))}$,
remark that $W$ is continuously embedded in $C([0,T];L^2(\Omega))$
and $C^\infty_c([0,T]\times\Omega)$ is dense
in $W$. Let $U\subseteq Q$ is an open set, we define the parabolic
$2$-capacity of $U$ as
$$     
 {\rm cap}_2(U)=\inf\{ \|u\|_{W} : u\in  W,\, u\geq \chi_{U}\text{ a.e. in } Q\}\,,
$$
where as usual we set $\inf\{\emptyset\}=+\infty$.  Then for any Borel
set $B\subseteq Q$ we define
$$ 
{\rm cap}_2(B)=inf\{ {\rm cap}_2(U) : U\text{ is open subset of Q}, B\subseteq  U\} \,.  
$$

We denote by $\mathcal{M}_b(Q)$ the set of all Radon measures with
bounded variation on $Q$, while, as we already mentioned, $\mathcal{M}_0(Q)$
denotes the set of all measures with bounded variation over $Q$
that do not charge the sets of zero $2$-capacity, that is if $\mu
\in \mathcal{M}_0(Q)$, then $\mu(E)=0$, for all $E\subseteq Q$ such that
${\rm cap}_2(E)=0$.

In \cite{DPP} the authors proved the following decomposition
theorem.

\begin{theorem}\label{t2}
Let $\mu$ be a bounded measure on $Q$. If $\mu\in \mathcal{M}_0(Q)$ then
there exists $(f,F,g)$ such that $f\in L^1(Q)$,
$F\in L^2(0,T;H^{-1}(\Omega))$, $g \in L^2(0,T;H^1_0(\Omega))$ and
$$   
\int_Q \phi \,d\mu= \int_Q f\phi\,dx\,dt+\int_0^T\langle F,\phi\rangle\,dt-
\int_0^T  \langle \phi_t,g\rangle \,dt \quad\phi \in
C_c^\infty([0,T]\times\Omega).
$$
Such a triplet $(f,F,g)$ will be called a decomposition of $\mu$.
\end{theorem}

Note that the decomposition of $\mu$ is not uniquely determined.
In \cite{PPP} the authors proved the following approximation of diffuse 
measures theorem.

\begin{theorem}\label{t21}
Let $\mu\in \mathcal{M}_0(Q)$, then, for every $\varepsilon>0$ there exists $\nu\in \mathcal{M}_0(Q)$ 
such that
$$ 
\|\mu -\nu  \|_{\mathcal{M}(Q)}\leq \varepsilon\quad \text{and}\quad 
\nu=w_t -\Delta w\ \text{in } \mathcal{D}'(Q),
$$
where $w\in L^2(0,T;H^1_0(\Omega))\cap L^\infty(Q)$.
\end{theorem}

The following Theorem  will be a key point in the existence result given 
in the next section. The proof follows the arguments in 
\cite[Theorem 1.2]{ppp}.

\begin{theorem}\label{t22} 
Let $d_i\in \mathcal{C}^0(\mathbb{R})\cap L^\infty(\mathbb{R})$ for every 
$i\in\{1,\dots,N\}$, $\mu \in\mathcal{M}_0(Q)\cap L^2(0,T;H^{-1}(\Omega))$ and 
$u_0\in L^2(\Omega)$, let $u\in W$ be the (unique) weak solution of
\begin{equation}\label{b1bis}
    \begin{gathered}
 \frac{\partial ua}{\partial t} - \sum_{i=1}^N \frac{\partial}{\partial x_i}
(d_i(u)\frac{\partial u}{\partial x_i})=\mu \quad \text{in }Q, \\
u= 0 \quad \text{on } (0,T)\times\partial\Omega, \\
u(t=0)=u_0 \quad \text{in }\Omega.
    \end{gathered}
\end{equation}
Then
$$  
{\rm cap}_2\{{|u|>K}\}  \leq \frac{C}{\sqrt{K}}\quad\forall K\geq 1, 
$$
where $C>0$ is a constant depending on $\|\mu\|_{\mathcal{M}(Q)},\ \|u_0\|_{L^2(\Omega)}$.
\end{theorem}

The proof of the above theorem is postponed to the Appendix in Section \ref{s5}.

\begin{definition}\label{d0}  \rm
A sequence of measures $(\mu_n)$ in $Q$ is equidiffuse if for every $\eta>0$ 
there exists $\delta>0$ such that
$$ 
{\rm cap}_2(E)<\delta \Longrightarrow |\mu_n|(E)< \eta\quad \forall\ n\geq 1.
$$
\end{definition}

The following result is proved in \cite{ppp}:
\begin{lemma}\label{lem1}
Let $\rho_n$ be a sequence of mollifiers on $Q$. If $\mu \in \mathcal{M}_0(Q)$, 
then the sequence $(\rho_n\ast \mu_n)$ is equidiffuse.
\end{lemma}

Here is some notation we will use throughout the paper. For any
nonnegative real number $K$ we denote by $T_K(r)=\min(K,\max(r,-K))$ 
the truncation function at level $K$. For every $r\in\mathbb{R}$, let
$$   
\overline{T_K}(z) =\int_0^z T_K(s)\,ds              
$$
We consider the following smooth approximation of $T_K(s)$: for 
$m>0$, $\eta \in ]0,1[$ and $\sigma\in ]0,1[$, we define 
$S^m_{K,\sigma}: \mathbb{R}\to \mathbb{R}$ by
\begin{equation}\label{a4}
 S^{m,\eta}_{K,\sigma}(s)\ = \begin{cases}
 1 & \text{if } -K+\eta\leq s\leq m-2\sigma, \\
0 & \text{if $s\leq -K$ and $s\geq m-\sigma$,} \\
\text{affine}& \text{otherwise,}
    \end{cases}
\end{equation}
and let us denote $ T^{m,\eta}_{K,\sigma}(z)=\int_0^z S^{m,\eta}_{K,\sigma}(s)\,ds$ 
and
\begin{equation*}
 T^{m}_{K}(s) = \begin{cases}
 s & \text{if $ -K\leq s\leq m$,} \\
-K & \text{if $s\leq -K$,} \\
m& \text{if $s\geq m$}.
    \end{cases}
\end{equation*}


By $\langle\cdot , \cdot \rangle$  we mean the duality between
suitable spaces in which function are involved. In particular we
will consider both the duality between $H^1_0 (\Omega)$ and
$H^{-1} (\Omega)$ and  the duality between 
$H^1_0 (\Omega)\cap L^{\infty} (\Omega)$ and  
$H^{-1} (\Omega)+L^{1} (\Omega)$.

\section{Main assumptions and definition of  renormalized solution}\label{s3}

Throughout the paper, we assume that the following assumptions hold:
$\Omega$ is a bounded open set on $\mathbb{R}^N$ ($N\geq 2$), $T>0$
is given and we set $Q=\Omega\times(0,T)$.
\begin{gather}\label{b1}
d_i\in  C^0(]-\infty,m[,\mathbb{R}^+\cup\{+\infty\})\quad \text{with }
 d_i(s)<+\infty\; \forall s<m,\; \forall i\in\{1,\dots,N\};\\
\label{b2}
\exists \alpha>0 \text{ such that } d_i(s)\geq \alpha\; \forall i\in\{1,\dots,N\},\;
 \forall s\in ]-\infty,m[;\\
\label{b3}
\exists p\in \{1,\dots,N\} \text{ such that } \lim_{s\to m^-}d_p(s)=+\infty
 \text{ and } \int_0^m d_p(s)\,ds <+\infty;\\
\label{b4}
\mu \in \mathcal{M}_0(Q); \\
\label{b5}
 u_0\in L^1(\Omega) \text{ such that } u_0\leq m \text{ a.e. in } \Omega.
\end{gather}
The definition of a renormalized solution for Problem
\eqref{a1}-\eqref{a3} is as follows.

\begin{definition}\label{d1}
Let $\mu\in \mathcal{M}_0(Q)$. A function $u\in L^1(Q)$ is a renormalized solution of
Problem \eqref{a1}-\eqref{a3} if
\begin{gather}\label{b6}
u\leq m \text{ a.e. in } Q,\quad  T_K(u) \in L^2(0,T;H^{1}_0(\Omega))\quad
 \forall K>0; \\
\label{b7}
 d_i(u)\frac{\partial T_K^m(u)}{\partial x_i} \chi_{\{u<m\}}\in L^2(Q)\quad
 \forall K>0,\; \forall i\in\{1,\dots,N\},
\end{gather}
if there exists a sequence of nonnegative measures $(\Lambda_K) \in \mathcal{M}(Q)$ 
and a nonnegative measure $\Gamma \in \mathcal{M}(Q)$  such that
\begin{gather}\label{b8}
\lim_{K\to +\infty} \|\Lambda_K\|_{\mathcal{M}(Q)}=0, \\
\label{b8bis}
 \int_Q \varphi \,d\Gamma = 0 \quad \forall \varphi \in \mathcal{C}^1_0([0,T[),
\end{gather}
and if, for every $K>0$,
\begin{equation}\label{b9}
  \frac{\partial T^m_{K}(u)}{\partial t}-
\sum_{i=1}^N\frac{\partial}{\partial x_i}{\Big( d_i(u)
\frac{\partial T^m_{K}(u)}{\partial x_i} \chi_{\{u<m\}}   \Big)}
= \mu+ \Lambda_K+ \Gamma \quad \text{in } \mathcal{D}'(Q).
\end{equation}
\end{definition}

\begin{remark}\rm
(1) Note that, in view of \eqref{b6}, \eqref{b7} and \eqref{b8} all terms in 
\eqref{b9} are well defined.\\
(2) The study of \eqref{a1}-\eqref{a3} under the assumption
 $ \int_0^m d_p(s)\,ds =+\infty$ is easier (see \cite{O} for the elliptic case), 
because one can then show there exists at least a renormalized solution such 
that $u<m$ a.e. in $Q$.\\
(3) Let us point out that, in \eqref{b8bis} the function 
$\varphi\in \mathcal{C}^1_0([0,T[)$ which does not depend on the variable $x$, 
we are not able to prove \eqref{b8bis} with any function 
$\varphi \in L^2(0,T;H^1(\Omega))\cap L^\infty(Q)$ such that 
$\nabla \varphi =0$ a.e. in $\{(x,t)\ ; u(x,t)=m\}$ because of a lack of
 regularity on $u$ with respect to $t$ in the parabolic case.
\end{remark}

\section{Existence of solutions}\label{s4}

This section is devoted to establish the following existence
theorem.

\begin{theorem}\label{t3}
Under assumptions \eqref{b1}-\eqref{b7} there exists at least a
renormalized solution $u$ of Problem \eqref{a1}-\eqref{a3}.
\end{theorem}

\begin{proof} 
The proof is divided into 4 steps. In Step 1, we introduce an
approximate problem. Step 2 is devoted to establish a few {\it a
priori} estimates. At last, Step 3 and Step 4 are devoted to
prove that $u$ satisfies \eqref{b7}, \eqref{b8}, \eqref{b8bis} and \eqref{b9} of
Definition \ref{d1}.
\smallskip

\noindent\textbf{Step 1.} Let us introduce the following
regularization of the data: for $n\geq 1$ fixed
\begin{gather}\label{c1}
d_i^n(s)=d_i(T_{m-\frac{1}{n}}(s^+)-T_{n}(s^-))\quad
 \forall s\in \mathbb{R},\; \forall i\in \{1,\dots,N\}, \\
\label{c2}
u_{0n}\in C^\infty_c(\Omega):
u_{0n}\to u_0 \text{ strongly in $L^1(\Omega)$ as $n$ tends to $+\infty$},
\end{gather}
we consider a sequence of mollifiers $(\rho_n)$, and we define the convolution 
$\rho_n\ast \mu$ for every $(t,x)\in Q$ by
\begin{equation}\label{c3}
 \mu^n (t,x)= \rho_n\ast \mu (t,x)=\int_Q \rho_n(t-s,x-y)d\mu(s,y).
\end{equation}
Let us now consider the  regularized problem
\begin{gather}\label{c4}
  \frac{\partial u_n}{\partial t}
 - \sum_{i=1}^N\frac{\partial}{\partial x_i}{\Big( d_i^n(u_n)
\frac{\partial u_n}{\partial x_i}  \Big)}  =\mu^n \quad
\text{in }Q, \\
\label{c5}
 u_n(t=0)=u_{0n}\quad \text{in }\Omega, \\
\label{c6}
 u_n=0\quad\text{on }{\partial \Omega}\times (0,T).
 \end{gather}
As a consequence, proving existence of a weak solution
$u_n \in L^2(0,T;H^{1}_0(\Omega))$ of
\eqref{c4}-\eqref{c6} is an easy task (see e.g. \cite{JL}).
\smallskip

\noindent\textbf{Step 2.} 
Using $T_K(u_n)$ as a test function in \eqref{c4} leads to
\begin{equation}\label{c7}
\int_\Omega {\overline{{T_K}}}(u^n)\,dx
+ \sum_{i=1}^N \int_Q d_i^n(u^n)\Big|\frac{\partial T_K(u^n)}{\partial x_i}\Big|^2 
\,dx\,dt 
\leq K (\|\mu_n\|_{L^1(Q)}+\|u_0\|_{L^1(\Omega)})
\end{equation}
for almost every $t$ in $(0,T)$, and where 
${\overline{ T_K}}(r)= \int_0^r T_K(s)\,ds$. The properties 
${\overline{{T_K}}} 
\big(\overline{T_K} \geq 0,\overline{T_K}(s) \geq |s|-1\ \forall
 s\in \mathbb{R}\big)$, and
since $\| \mu^n \|_{L^1(Q)}$ and $\|u_{0n} \|_{L^1(\Omega)}$ are bounded, we
deduce from \eqref{c7} that
\begin{gather}\label{c8}
u^n \text{ is bounded in }L^\infty(0,T;L^1(\Omega)), \\
\label{c9}
T_K(u^n) \text{ is bounded in }L^2(0,T;H^1_0(\Omega)), \\
\label{c10}
d_i^n(u^n)^{1/2}\frac{\partial T_K(u^n)}{\partial x_i} 
\text{ is bounded in } L^2(Q)
\end{gather}
independently of $n$ for any $K\geq 0$ and any $i\in\{1,2,\dots,N\}$.

In view of \eqref{b1}-\eqref{b3}, we have that for any $K\geq 0$,
 $$ 
\Big|\int_0^{u^n} d_i^n(s)\chi_{\{-K\leq s\leq m\}}\,dx\Big|\leq \int_{-K}^m d_i(s)\,ds \equiv C_K <+\infty,$$ then we can use $ \int_0^{u^n} d_i^n(s)\chi_{\{-K\leq s\leq m\}}\,ds$ in $L^2(0,T;H^1_0(\Omega))\cap L^\infty(Q_T)$ as a test function
in \eqref{c4} obtaining
\begin{equation}\label{c11}
\begin{aligned}
&\int_\Omega \int_0^{u^n} \int_0^z d_i^n(s)\chi_{\{-K\leq s\leq m\}}\,ds\,dz\,dx
+  \int_Q (d_i^n(u^n))^2\Big|
\frac{\partial T^m_K(u^n)}{\partial x_i}\Big|^2 \,dx\,dt
\\
&\leq  (\|\mu_n\|_{L^1(Q_T)}+\|u_0\|_{L^1(\Omega)})\max_{i} \int_{-K}^m d_i(s)\,ds
\end{aligned}
\end{equation}
for all $i\in\{1,2,\dots,N\}$. Since  
$ \int_\Omega \int_0^{u^n} \int_0^z d_i^n(s)\,ds\,dz\,dx$ 
is positive and $\| \mu^n \|_{L^1(Q)}$ and $\|u_{0n}\|_{L^1(\Omega)}$ 
are bounded,  from \eqref{c11} we deduce that
\begin{equation}\label{c12}
d^n(u^n)\nabla T^m_K(u^n) \ \text{is bounded in}\
(L^2(Q))^N.
\end{equation}
For any $S \in W^{2,\infty}(\mathbb{R})$ such that $S'$ has a compact support
($\operatorname{supp}(S')\subset [-K,m]$), we have
\begin{gather}\label{c13}
 S(u^n) \text{ is bounded in  $L^2(0,T;H^1_0(\Omega))$}, \\
\label{c14}
    \frac{\partial S(u^n)}{\partial t} \text{ is bounded in
    $L^1(Q)+L^2(0,T;H^{-1}(\Omega))$},
\end{gather}
independently of $n$. In fact, as a consequence of
\eqref{c9}, by Stampacchia's Theorem, we obtain \eqref{c13}. To
show that \eqref{c14} holds true, we multiply the equation
\eqref{c4} by $S'(u^n)$ to obtain
\begin{equation}\label{c15}
\begin{aligned}
  \frac{\partial S(u^n)}{\partial t}
&= \sum_{i=1}^N\frac{\partial}{\partial x_i}{\Big( d_i^n(u_n)
\frac{\partial S(u_n)}{\partial x_i} \Big)} \\
&\quad - \sum_{i=1}^N  d_i^n(u_n) \Big|\frac{\partial u_n}{\partial x_i}
\Big|^2 S''(u^n) + \mu^n S'(u^n)\quad \text{in }\mathcal{D}'(Q),
\end{aligned}
\end{equation}
as a consequence of \eqref{c3}, \eqref{c10}, \eqref{c12}, we obtain \eqref{c14}.

Arguing again as in \cite{BM, BMR, BMR1, BR} estimates \eqref{c13} and \eqref{c14}
imply that, for a subsequence still indexed by $n$,
\begin{gather}\label{c16}
u^n \to u\quad \text{almost every where in $Q$}, \\
\label{c17}
T_K(u^n)\rightharpoonup T_K(u)\quad \text{weakly in $L^2(0,T;H_0^{1}(\Omega))$},\\
\label{c18}
 (d^n(u^n))^{1/2}\nabla T_K(u^n)\rightharpoonup X_{K}
\quad \text{weakly in } (L^2(Q))^N, \\
\label{c19}
 d^n(u^n)\nabla T^m_K(u^n)  \rightharpoonup Y_{K}
\quad \text{weakly in } (L^2(Q))^N,
\end{gather}
as $n$ tends to $+\infty$, for any $K>0$.

Using the admissible test function $T_{2m}^+(u^n)-T_{m}^+(u^n)$ 
in \eqref{c4} and the Poincar\'e inequality, leads to
\begin{equation}\label{c20}
 d_p(m-\frac{1}{n}) \int_Q
\Big|T_{2m}^+(u^n)-T_{m}^+(u^n)\Big|^2 \,dx\,dt
\leq m( \|\mu_n\|_{L^1(Q)}  + \|u_{0n}\|_{L^1(\Omega)}).
\end{equation}
In view of \eqref{b3}, \eqref{c2} and \eqref{c16} 
(since $d_p(m-\frac{1}{n})\to +\infty$ as $n$ tends $+\infty$) passing 
to the limit in \eqref{c20} as $n$ tends to $+\infty$, we deduce that
$T_{2m}^+(u)-T_{m}^+(u)=0$ a.e. in $Q$, hence
\begin{equation}\label{c21}
u\leq m\quad \text{a.e. in } Q.
\end{equation}
Now, in view of \eqref{c18}, \eqref{c19} and \eqref{c21} we deduce
\begin{equation}\label{c22}
X_{K}=d(u)^{1/2}\nabla T_K(u)\ \text{and}\ Y_K = d(u)\nabla T^m_K(u)\quad 
\text{a.e. in } \{(x,t)\in Q:u(x,t)< m\},
\end{equation}
for any $K\geq 0$.

For fixed $K\geq 1, \eta\in ]0,1[$ and $\sigma\in ]0,1[$, we define the functions,  
$h_{K,\eta}$ and $Z_\sigma$ by
\begin{equation}\label{c23}
 h_{K,\eta}(s) =  \begin{cases}
 0 & \text{ if $-K\leq s$} \\
-1 & \text{ if $s\leq -K-\eta$} \\
\text{affine}& \text{ otherwise},
    \end{cases}
\quad  Z_{\sigma}(s) =  \begin{cases}
 0 & \text{ if $s\leq m-2\sigma$} \\
1 & \text{ if $s\geq m-\sigma$} \\
\text{affine}& \text{ otherwise}.
    \end{cases}
\end{equation}
We remark that $\max(\|h_{K,\eta}\|_{L^\infty(\mathbb{R})}, 
\|Z_{\sigma}\|_{L^\infty(\mathbb{R})})=1$ for any $K\geq 1$ any $0<\eta<1$ 
and any $0<\sigma<1$. Using the admissible test functions
$ h_{K,\eta}(u^n)$ and $Z_\sigma(u^n)$ in \eqref{c4}
leads to
\begin{equation}\label{c24}
\begin{aligned}
&\int_\Omega \overline{h_{K,\eta}}(u^n(T))\,dx+ \sum_{i=1}^N \int_Q
d_i^n(u_n)\frac{\partial u_n}{\partial x_i} 
\frac{\partial h_{K,\eta}(u_n)}{\partial x_i} \,dx\,dt \\
& = \int_Q h_{K,\eta}(u_n) \mu_n\,dx\,dt +
\int_\Omega \overline{h_{K,\eta}}(u_{0n})\,dx,
\end{aligned}
\end{equation}
and
\begin{equation}\label{c25}
\begin{aligned}
&\int_\Omega \overline{Z_{\sigma}}(u^n(T))\,dx+ \sum_{i=1}^N \int_Q
d_i^n(u_n)\frac{\partial u_n}{\partial x_i}
\frac{\partial Z_{K,\sigma}(u_n)}{\partial x_i} \,dx\,dt \\
& = \int_Q Z_{K,\sigma}(u_n) \mu_n\,dx\,dt +
\int_\Omega \overline{Z_{K,\sigma}}(u_{0n})\,dx,
\end{aligned}
\end{equation}
where 
\[
 \overline{h_{K,\eta}}(r)=\int_0^r h_{K,\eta}(s)\,ds \geq 0,\quad
\overline{Z_{\sigma}}(r)=\int_0^r Z_{\sigma}(s)\,ds \geq 0.
\]
 Hence, using \eqref{c2}, \eqref{c3} and dropping a nonnegative term,
\begin{equation}\label{c26}
\begin{aligned}
&\sum_{i=1}^N \frac{1}{\eta}\int_{\{-K-\eta\leq
u^n\leq -K\}} d_i^n(u^n)\Big|\frac{\partial u^n}{\partial x_i}\Big|^2\,dx\,dt\\
& \leq \int_{\{u^n\leq -K\}}|\mu^n| \,dx\,dt
  + \int_{\{u_{n0}\leq -K\}}|u_{0n}|\,dx\leq C_1,
\end{aligned}
\end{equation}
and
\begin{equation}\label{c27}
\begin{aligned}
&\sum_{i=1}^N \frac{1}{\sigma}\int_{\{m-2\sigma\leq
u^n\leq m-\sigma\}} d_i^n(u^n)\Big|\frac{\partial u^n}{\partial x_i}
\Big|^2\,dx\,dt \\
&\leq  \int_{\{u^n\geq m-2\sigma\}}  Z_{\sigma}(u_n)     \mu^n \,dx\,dt
    + \int_{\{u_{n0}\geq m-2\sigma\}}|u_{0n}|\,dx\leq C_2.
\end{aligned}
\end{equation}
Thus, there exists a bounded Radon measures $\lambda^n_K$ and $\nu_\sigma$ 
such that, as $\eta$ tends to zero and $n$ tends to infinity
  \begin{equation}\label{c28}
   \lambda^{n,\eta}_K\equiv\sum_{i=1}^N \frac{1}{\eta} d_i^n(u^n)
\Big|\frac{\partial u^n}{\partial x_i}\Big|^2\chi_{\{-K-\eta\leq u^n\leq -K\}}
 \rightharpoonup  \lambda^n_K \quad \ast\text{-weakly in}\ \mathcal{M}(Q),
  \end{equation}
  and
   \begin{equation}\label{c29}
 \nu_\sigma^n\equiv \sum_{i=1}^N \frac{1}{\sigma}d_i^n(u^n)
\Big|\frac{\partial u^n}{\partial x_i}\Big|^2\chi_{\{m-2\sigma\leq u^n
\leq m-\sigma\}} \rightharpoonup \nu_\sigma \quad\ast\text{-weakly in}\ \mathcal{M}(Q).
  \end{equation}
\smallskip

\noindent\textbf{Step 3.}
 In this step, $u$ is shown to satisfy \eqref{b9}.
For all real numbers $\eta>0,\ \sigma>0$ and $K>0$, let $S^{m,\eta}_{K,\sigma}$ 
be the function defined by \eqref{a4}, and let us denote 
$ T^{m,\eta}_{K,\sigma}(z)=\int_0^z S^{m,\eta}_{K,\sigma}(s)\,ds$. 
Since $\operatorname{supp}(S^{m,\eta}_{K,\sigma})'\subset [-K-\eta,-K]
\cup [m-2\sigma,m-\sigma]$, the equation \eqref{c15} with 
$S=T^{m,\eta}_{K,\sigma}$ gives
\begin{equation}\label{c30}
\begin{aligned}
&\frac{\partial T^{m,\eta}_{K,\sigma}(u^n)}{\partial t}
-\sum_{i=1}^N \frac{\partial}{\partial x_i} 
\Big( d_i^n(u_n)\frac{\partial T^{m,\eta}_{K,\sigma}(u_n)}{\partial x_i} \Big)\\
&= \mu^n + (S^{m,\eta}_{K,\sigma}(u^n)-1)\mu^n  
+ \frac{1}{\eta}\sum_{i=1}^N  d_i^n(u_n) 
 \Big|\frac{\partial u_n}{\partial x_i}\Big|^2
\chi_{\{-K-\eta<u_n<-K\}} \\
&\quad  + \frac{1}{\sigma} \sum_{i=1}^N  d_i^n(u_n)
\Big|\frac{\partial u_n}{\partial x_i}\Big|^2
\chi_{\{m-2\sigma<u_n<m-\sigma\}}
\end{aligned}
\end{equation}
in $\mathcal{D}'(Q)$. Passing to the limit in \eqref{c30} as $\eta$ tends to zero, 
and using \eqref{c17}, \eqref{c19}, \eqref{c21}, \eqref{c22}, \eqref{c28} 
and \eqref{c29}, we deduce
\begin{equation}\label{c31}
\begin{aligned}
&\frac{\partial T^{m}_{K,\sigma}(u^n)}{ \partial t}-
\sum_{i=1}^N\frac{\partial}{\partial x_i}{\Big( d_i^n(u_n)
\frac{\partial T^m_{K,\sigma}(u_n)}{ \partial x_i}   \Big)} \\
&= \mu^n -\mu^n \chi_{\{u^n<-K\}}-\mu^n Z_\sigma(u^n) +\lambda^n_K+\nu^n_\sigma
\end{aligned}
\end{equation}
in $\mathcal{D}'(Q)$. Now, using the properties of convolution 
$\mu_n=\rho_n\ast \mu$ and in view of \eqref{c26}, \eqref{c27}, \eqref{c28} 
and \eqref{c29}, we deduce that
$\Lambda_K^n\equiv -\mu^n \chi_{\{u^n<-K\}} +\lambda^n_K$ and 
$\Gamma_\sigma^n\equiv-\mu^n Z_\sigma(u^n) +\nu^n_\sigma$ are bounded in 
$L^1(Q)$. Then there exists a bounded measures $\Lambda_K$ and $\Gamma_\sigma$ 
such that $(-\mu^n \chi_{\{u^n<-K\}} +\lambda^n_K)_n$ converges to
 $\Lambda_K$ and  $(-\mu^n Z_\sigma(u^n) +\nu^n)_n$ converges to 
$\Gamma_\sigma$ in $\ast-$weakly in $\mathcal{M}(Q)$. 
From \eqref{c16}, \eqref{c17}, \eqref{c19}, \eqref{c21}, \eqref{c22} and 
\eqref{c31} We deduce that $u$ satisfies
\begin{equation}\label{c32}
 \frac{\partial T^m_{K,\sigma}(u)}{\partial t}
 -\sum_{i=1}^N\frac{\partial}{\partial x_i}
{\Big( d_i^n(u)\frac{\partial T^m_{K,\sigma}(u)}{\partial x_i} \chi_{\{u<m\}} \Big)}
= \mu+ \Lambda_K+ \Gamma_\sigma \ \text{in}\ \mathcal{D}'(Q).  
 \end{equation}
To complete this step, we use
\begin{align*}
\int_Q|\Gamma_\sigma|\,dx\,dt 
&\leq \liminf_{n\to +\infty} \int_Q |\Gamma^n_\sigma|\,dx\,dt \\
&=\liminf_{n\to +\infty} \int_Q |-\mu^n Z_\sigma(u^n) +\nu^n_\sigma|\,dx\,dt \\
&\leq 2\|\mu\|_{\mathcal{M}(Q)} +\|u_0\|_{L^1(\Omega)} 
\end{align*}
then there exists a bounded measure $\Gamma$ such that $\Gamma_\sigma$ 
converges to $\Gamma$ in $\ast-$weakly in $\mathcal{M}(Q)$. 
Therefore, as $\sigma$ tends to zero in \eqref{c32}, it is easy to see that 
$u$ satisfies \eqref{b9}.
\smallskip

\noindent\textbf{Step 4.}
 In this step, $\Lambda_K$ and $\Gamma$ are shown to satisfy \eqref{b8} 
and \eqref{b8bis}.
From \eqref{c26} and \eqref{c28} we deduce
\begin{equation}\label{c33}
\begin{aligned}
\|\Lambda_K^n\|_{L^1(Q)}
&=\|\ -\mu^n \chi_{\{u^n<-K\}} +\lambda^n_K\|_{L^1(Q)} \\
&\leq  2\int_{\{u^n<-K\}} |\mu^n|\,dx\,dt + \int_{\{u_{0n}<-K\}} |u_{0n}|\,dx.
\end{aligned}
\end{equation}
Since 
\[
\| \lambda_K \|_{\mathcal{M}(Q)}
\leq \liminf_{n\to +\infty}\|\-\mu^n \chi_{\{u^n<-K\}} 
+\lambda^n_K\|_{\mathcal{M}(Q)},
\]
 the sequence $(\mu_n)$ is equidiffuse, and the function $u_{0n}$ converges 
to $u_0$ strongly in $L^1(\Omega)$, we deduce from Theorem \ref{t22} and
 \eqref{c33} that $\| \Lambda_K \|_{\mathcal{M}(Q)}$ tends to zero as $K$
 tends to infinity, then we obtain \eqref{b8}.

On the other hand, for all $\varphi \in \mathcal{C}^1_0([0,T[)$, we can write
\begin{equation}\label{c34}
 \int_Q \varphi \,d\Gamma = \lim_{\sigma\to 0} \int_Q \varphi \,d\Gamma_\sigma 
 = \lim_{\sigma\to 0} \lim_{n\to +\infty} \int_Q \varphi \Gamma^n_\sigma \,dx\,dt
\end{equation}
where 
\[
 \Gamma^n_\sigma \equiv \frac{1}{\sigma} \sum_{i=1}^N  d_i^n(u_n) 
\Big|\frac{\partial u_n}{\partial x_i}\Big|^2
\chi_{\{m-2\sigma<u_n<m-\sigma\}}- Z_\sigma(u^n) \mu^n.
\]
Using the admissible function $Z_{\sigma}(u^n)\varphi$ in \eqref{c4}, 
since $\varphi \in \mathcal{C}^1_0([0,T[)$,  it is easy to see that
\begin{equation}\label{c35}
\begin{aligned}
&\int_\Omega \overline{Z_\sigma}(u^n_0)\varphi(0)\,dx 
 + \int_Q  \overline{Z_\sigma}(u^n)\varphi_t \,dx\,dt \\
&= \frac{1}{\sigma}\sum_{i=1}^N  
 \int_{\{m-2\sigma<u_n<m-\sigma\}} d_i^n(u_n) 
\Big|\frac{\partial u_n}{\partial x_i}\Big|^2 \varphi \,dx\,dt
 - \int_Q Z_{\sigma}(u^n)\mu^n \varphi \,dx\,dt \\
&\equiv\int_Q \varphi \Gamma^n_\sigma \,dx\,dt,
\end{aligned}
\end{equation}
 where $ \overline{Z_\sigma}(r)=\int_0^r Z_\sigma(s)\,ds$. 
Next we pass to the limit in \eqref{c35} as $n$ tends to infinity, 
and then $\sigma$ tends to zero. 
Since $\overline{Z_\sigma}(u^n)$ converges to $\overline{Z_\sigma}(u)$ 
strongly in $L^1(Q)$ and $\overline{Z_\sigma}(u^n_0)$ converges to 
$\overline{Z_\sigma}(u_0)$ strongly in $L^1(\Omega)$ as $n$ tends to infinity, 
we deduce
\begin{equation}\label{c36}
\begin{gathered}
 \lim_{n\to +\infty}\int_Q \overline{Z_\sigma}(u^n)\varphi_t\,dx 
=\int_Q \overline{Z_\sigma}(u)\varphi_t\,dx\\
\lim_{n\to +\infty}\int_\Omega \overline{Z_\sigma}(u^n_0)\varphi\,dx 
=\int_\Omega \overline{Z_\sigma}(u_0)\varphi\,dx
\end{gathered}
\end{equation}
Moreover, since $\overline{Z_\sigma}(r)$ converges to $ (r-m)^+$ for all 
$r\in \mathbb{R}$ and $u\leq m, u_0\leq m$ almost everywhere, then it is 
easy to see that
\begin{gather}\label{c37}
 \lim_{\sigma\to 0} \lim_{n\to +\infty}\int_Q \overline{Z_\sigma}(u^n)\varphi_t\,dx
 =\int_Q (u-m)^+\varphi_t\,dx=0, \\
\label{c38}
 \lim_{\sigma\to 0}\lim_{n\to +\infty}\int_\Omega \overline{Z_\sigma}(u^n_0)
\varphi\,dx =\int_\Omega (u_0-m)^+\varphi\,dx=0.
\end{gather}
Then, from \eqref{c34}, \eqref{c35}, \eqref{c37} and \eqref{c38} we deduce 
\eqref{b8bis}.

As a conclusion from Step 1, Step 2, Step 3 and Step 4, the proof  is complete.
\end{proof}

\section{Appendix: Proof of Theorem \ref{t22}}\label{s5}


\begin{proof}[Sketch of the Proof] 
For simplicity we assume that $\mu\geq 0$ and $u_0\geq 0$. Using the admissible 
test function $T_K(u)$ in \eqref{b1bis} leads to
\begin{equation}\label{e1}
\begin{aligned}
&\int_\Omega {\overline{T_K}}(u)\,dx
 +\sum_{i=1}^n \int_Q  \Big|d_i(u)^{1/2}\frac{\partial T_K(u)}{\partial x_i}
\Big|^2\,dx\,dt \\
&\leq K\big[ \|\mu\|_{\mathcal{M}(Q)} +  \|u_0\|_{L^1(\Omega)} \big]
\equiv K M,
\end{aligned}
\end{equation}
for almost any $t$ in $]0,T[$ and where $ {\overline{T_K}}(r) = \int_0^r T_K(s)\,ds$.
Since $ \frac{1}{2}T_K^2(r)\leq {\overline{T_K}}(r) \leq K r$,  from \eqref{e1}
we deduce that
\begin{equation}\label{e2}
\max\big\{\| T_K(u) \|^2_{L^\infty(L^2(\Omega))}: 
\|\nabla T_K(u) \|^2_{L^2(Q)} \big\}   \leq K M,\quad
\| T_K(u) \|^2_{L^2(H^1_0(\Omega))} \leq K \frac{M}{\alpha}.
\end{equation}
Moreover, for $i\in \{1,\dots, N\}$ let us choose 
$ \int_0^{T_K(u)} d_i(r)\,dr\in L^2(0,T;H^1_0(\Omega))\cap L^\infty(Q)$ 
as test function in \ref{b1bis}. Then 
\begin{equation}\label{e3}
 \sum_{i=1}^n \int_Q  \Big|d_i(u)
\frac{\partial T_K(u)}{\partial x_i}\Big|^2\,dx\,dt
\leq K\big[ \|\mu\|_{\mathcal{M}(Q)} +  \|u_0\|_{L^1(\Omega)} \big]
\|d_i\|_{L^\infty(\mathbb{R})}.
\end{equation}
Let $v\in W$ be the solution of
\begin{equation}\label{e4}
\begin{gathered}
 -\frac{\partial v}{\partial t} 
- \sum_{i=1}^N \frac{\partial}{\partial x_i}(d_i(u)\frac{\partial v}{\partial x_i})
=- 2\sum_{i=1}^N \frac{\partial}{\partial x_i}(d_i(u)
\frac{\partial T_K(u)}{\partial x_i})\quad \text{in $Q$,} \\
v= 0 \quad \text{on $(0,T)\times\partial\Omega$,} \\
v(t=T)=T_K(u(t=T)) \quad \text{in $\Omega$.}
    \end{gathered}
\end{equation}
Using the admissible test function $v$ in \eqref{e4} and integrate between 
$\tau$ and $T$, and by Young's inequality we obtain
\begin{equation}\label{e5}
\begin{aligned}
&\int_\Omega \frac{|v(\tau)|^2}{2}\,dx+ \frac{\alpha}{2} \int_Q  |\nabla v |^2
\,dx\,dt \\
&\leq  C\sum_{i=1}^n \int_Q  \Big|d_i(u)
\frac{\partial T_K(u)}{\partial x_i}\Big|^2\,dx\,dt 
+\int_\Omega {\overline{T_K}}(u(t=T))\,dx
\end{aligned}
\end{equation}
In view of \eqref{e2}, \eqref{e3} and \eqref{e5}, we deduce that
\begin{equation}\label{e6}
\max\big\{\| v \|^2_{L^\infty(0,T;L^2(\Omega))}:
  \|\nabla v \|^2_{L^2(Q)} \big\}  \leq C K M.
\end{equation}
Moreover, by \eqref{e4} we obtain
\begin{equation}\label{e7}
\| v_t \|_{L^2(0,T;H^{-1}(\Omega))} 
\leq C \Big(   \| v \|_{L^2(0,T;H^1_0(\Omega))}  
+  \| T_K(u) \|_{L^2(0,T;H^1_0(\Omega))}   \Big).
\end{equation}
Hence, by \eqref{e6} and \eqref{e7} we conclude that
\begin{equation}\label{e8}
 \| v\|_W \leq C\sqrt{K}.
\end{equation}
Since $\mu\geq 0$ and $u_0\geq 0$, it follows that
\[
\frac{\partial ua}{\partial t} - \sum_{i=1}^N 
\frac{\partial}{\partial x_i}(d_i(u)\frac{\partial u}{\partial x_i})\geq 0
\]
 and $u\geq 0$ in $Q$, and by a nonlinear version of Kato's inequality for 
parabolic equations (see \cite{ppp}), we deduce that
\[
 \frac{\partial T_K(u)}{\partial t} - \sum_{i=1}^N 
\frac{\partial}{\partial x_i}(d_i(u)\frac{\partial T_K(u)}{\partial x_i})\geq 0,
\]
 hence by \eqref{e4}, we obtain 
\[ 
-\frac{\partial v}{\partial t} - \sum_{i=1}^N 
\frac{\partial}{\partial x_i}(d_i(u) \frac{\partial v}{\partial x_i})
\geq  -\frac{\partial T_K(u)}{\partial t} 
 - \sum_{i=1}^N \frac{\partial}{\partial x_i}(d_i(u)
\frac{\partial T_K(u)}{\partial x_i})\quad \text{in } \mathcal{D}'(Q). 
\]
Now using the standard comparison argument, we easily see that 
$v\geq T_K(u)$ a.e. in $Q$, hence $v\geq K$ a.e. on $\{u>K\}$, and by 
\eqref{e8} we conclude that 
\[
 cap_2\{ u>K \}\leq \big\| \frac{v}{K} \big\|_W 
\leq \frac{C}{\sqrt{K}},
\]
 the proof is complete.
\end{proof}

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