text:
\section{Introduction}
\setcounter{equation}{0} The thermistor problem refers to the
heating of a conductor, with temperature-sensitive electric
conductivity, by electric current.
Let $\phi$ be the electric potential and let ${\bf E}=\nabla\phi$ be
the electric field. The electric current ${\bf J}$ is related to the
electric field via ${\bf J}=\sigma(u){\bf E}$, where
$\sigma(u)$ is the electric conductivity of the conductor, dependent upon the temperature $u$.
The heat produced (per unit volume) by the electric current is given
by Joule's law: ${\bf E}\cdot{\bf J}=\sigma(u)|\nabla\phi|^2$, and the
conservation of charge is described by $\nabla\cdot{\bf J}=0$.

Let $\Omega$ denote the domain possessed by the conductor. Based on
the above formulations, the temperature $u$ and the electric
potential $\phi$ are governed by the equations
\begin{align}
&\frac{\partial u}{\partial t}-\nabla\cdot(\kappa(u)\nabla
u)=\sigma(u)|\nabla\phi|^2, \label{e-heat-1}
\\[3pt]
&-\nabla\cdot(\sigma(u)\nabla\phi)=0, \label{e-heat-2}
\end{align}
for $x\in\Omega$ and $t>0$, where $\kappa(u)$ is the thermal
conductivity. In this paper, we
consider the above equations with the Dirichlet boundary/initial
conditions:
\begin{align}
\label{BC}
\begin{array}{ll}
u(x,t)=g(x,t),\quad \phi(x,t)=h(x,t)~~
&\mbox{for}~~x\in\partial\Omega~~\mbox{and}~~t>0,\\[3pt]
u(x,0)=u_0(x)~~ &\mbox{for}~~x\in\Omega .
\end{array}
\end{align}

The mathematical expressions of $\sigma(u)$ and $\kappa(u)$ depend
on the materials. For some semiconductors, the electric resistivity
$\rho(u)=1/\sigma(u)$ can be approximately expressed as
\cite{Macklen}
$$
\rho(u)=\sigma_0e^{q/u}u,
$$
and the thermal conductivity $\kappa(u)$ can be regarded as constant
(independent of $u$). For metallic conductors, the electric
conductivity and the thermal conductivity obey the Wiedemann--Franz
law \cite{Sommerfeld}:
\begin{align*}
\frac{\kappa(u)}{\sigma(u)}=Lu ,
\end{align*}
where $L=2.44\times10^{-8}{\rm W\Omega K^{-2}}$ is the Lorentz
number. In general, the electric resistivity of metals increases as
temperature grows. At high temperatures, the electric resistivity
increases approximately linearly with temperature:
$$
\rho(u)=\rho_0[1+\alpha(u-u_R)],
$$
where $u_R$ is some reference temperature and $\alpha$ is called the
temperature coefficient of resistivity. If the temperature does not vary
much, the above linear formula is often used. More precisely, the
electric resistivity is give by the Bloch--Gr\"{u}neisen formula
\cite{Ziman}:
$$
\rho(u)=\rho(0)+A\biggl(\frac{u}{\Theta}\biggl)^n\int_0^{\frac{\Theta}{u}}\frac{s^n}{(e^s-1)(1-e^{-s})}\d
s ,
$$
where $A$, $\Theta$ and $n\geq2$ are all positive physical
constants.

For both metals and semiconductors, the electric conductivity
$\sigma(u)$ tends to zero as the temperature $u$ grows to infinity.
The elliptic equation (\ref{e-heat-2}) is thus possibly degenerate,
which leads to severe difficulties for the analysis of the
coupled system.

The non-degenerate assumption
$\sigma_1\leq\sigma(u)\leq \sigma_2$ is often used to
simplify the problem.  Mathematical analysis for such
non-degenerate problem has been
studied by many authors in the last two decades.
Existence of weak solutions was studied by Antontsev and Chipot
\cite{AC}, Allegretto and Xie \cite{AX} and Cimatti \cite{Cimatti2}.
With the same non-degenerate assumption, Elliott and Larsson
\cite{EL} proved the existence of strong solutions for the 2D
problem by using the energy method (and uniqueness follows). The 3D problem
is much more difficult. To deal with the 3D problem, one has to fully
explore and make use of the coupling of the equations. The milestone
was acheived by Yuan and Liu \cite{YL,Yuan2}, who proved the existence of
$C^\alpha$ solutions for the 3D problem by using the method of Layer
potentials. Yin \cite{Yin} obtained the same result by using the techniques
of Campanato spaces.
Their results imply the existence of classical
solutions when the boundary and initial data are smooth.

Without the non-degenerate assumption, the problem becomes much more
difficult. Xu \cite{Xu} proved partial regularity of the solution,
i.e. the solution is smooth in an open subset $D\subset\Omega$ whose
complement $\Omega\backslash D$ is a set of measure zero. Later Xu
\cite{Xu1} proved existence of solutions with bounded temperature
when the boundary potential is small enough, i.e.
$\|h\|_{L^\infty(\partial\Omega\times(0,T))}$ is small enough.
Hachimi and Ammi \cite{HA} proved existence of weak solutions by the
monotonicity-compacity method. Montesinos and Gallego \cite{MG1,MG2}
proved existence of ``capacity solutions'' by considering a new
formulation with the transformation $\Phi=\sigma(u)\nabla\phi$.
Uniqueness of the weak solution and existence of global classical
solutions remain open. Overall, the main difficulty of the
degenerate problem is the lack of a $L^\infty$ bound for the
temperature $u$.

In this paper, we overcome this difficulty by establishing a
uniform-in-time ${\rm BMO}$ estimate for inhomogeneous parabolic
equations with possibly discontinuous coefficients. Applying this
estimate to the temperature equation, we obtain a uniform-in-time
${\rm BMO}$ bound of the temperature $u$, as a substitute of the
$L^\infty$ bound. Based on the BMO bound of the temperature, we
further prove that the electric conductivity $\sigma(u)$ is a $A_2$
weight uniform in time. The H\"{o}lder continuity of the electric
potential $\phi$ is then proved by applying the De
Giorgi--Nash--Moser estimate for degenerate elliptic equations with
$A_2$ coefficient. The H\"{o}lder continuity of the temperature is
proved by using the H\"{o}lder continuity of the electric potential.
Existence of a weak solution in a bounded Lipschitz domain is
proved, and uniqueness of the weak solution is proved based on the
established regularity of the solution. Our results also imply the
existence of a global classical solution when the initial and
boundary data are smooth.

For interested readers, we refer to \cite{AL, AY, EL, LS, LS2, ZW} for
numerical methods and numerical analysis of the thermistor problem.

The rest part of this paper is organized in the following way. In
Section \ref{notations} we introduce the notations to be used in
this paper and in Section \ref{dfjksl7879} we present our main
results. In Section \ref{fjkesluio99}, we establish a
uniform-in-time BMO estimate for the solutions of inhomogeneous
parabolic equations, and in Section \ref{sec5} we present H\"{o}lder
estimates of parabolic equations in terms of the Campanato spaces.
Based on the estimates obtained in Section \ref{fjkesluio99} and
Section \ref{sec5}, we prove global existence and uniqueness of a
weak solution to the degenerate thermistor problem in Section
\ref{fsdjkljl78687}. Conclusions are drawn in Section 7.

\section{Notations}\label{notations}
\setcounter{equation}{0}

Before we present our main results, we define the notations to be used in this paper.

Let $n$ be a fixed positive
integer and let $B_R(x_0)$ denote the ball of radius $R$ centered at
the point $x_0\in\R^n$. Let $\Omega$ be a bounded Lipschitz domain
in $\R^n$, i.e. $\Omega$ is a bounded domain in $\R^n$ and for any
$y\in\partial\Omega$, there exists a ball $B_R(y)$ such that through
a rotation of coordinates (if necessary),
$$
B_R(y)\cap\Omega=\{(x_1,\cdots,x_n)\in
B_R(y):x_n>\varphi(x_1,\cdots,x_{n-1})\} ,
$$
where $\varphi:\R^{n-1}\rightarrow \R$ is a Lipschitz continuous
function. For a bounded Lipschitz domain, there exists a positive
constant $R_\Omega$ and a finite number of balls
$B_{R_\Omega}(y_1)$, $B_{R_\Omega}(y_2)$, $\cdots$,
$B_{R_\Omega}(y_m)$ such that $\partial\Omega\subset
\cup_{j=1}^mB_{R_\Omega/2}(y_j)$ and through a rotation of
coordinates (if necessary),
$$
B_{2R_\Omega}(y_j)\cap\Omega=\{(x_1,\cdots,x_n)\in
B_{2R_\Omega}(y_j):x_n>\varphi_j(x_1,\cdots,x_{n-1})\}
$$
for some Lipschitz continuous function
$\varphi_j:\R^{n-1}\rightarrow \R$.

For any integer $m\geq 0$, $1\leq p\leq\infty$ and $0<\alpha<1$, let
$W^{m,p}(\Omega)$ and $C^{m+\alpha}(\overline\Omega)$ denote the
usual Sobolev space and H\"{o}lder space \cite{Adams}, respectively,
and let $C^{m+\alpha}(\Omega)$ denote the space of functions which
belong to $C^{m+\alpha}(\overline B)$ for any closed ball $\overline
B\subset\Omega$. Let $C^{m+\alpha}_0(\overline\Omega)$ be the
subspace of $C^{m+\alpha}(\overline\Omega)$ consisting of functions
vanishing on the boundary $\partial\Omega$.

Let $|D|$ denote the Lebesgue measure for any measurable subset $D$ of $\R^n$, and let $B_R(x_0)$ denote the ball of radius $R$ centered at
the point $x_0\in\R^n$. Let $\Omega$ be a bounded Lipschitz domain
in $\R^n$. We say that a positive locally integrable function $w$ defined on $\R^n$ is a $A_2$ weight if
$$
\sup_{B\subset\R^n}\biggl(\frac{1}{|B|}\int_Bw(x)\d x\biggl)\biggl(\frac{1}{|B|}\int_B\frac{1}{w(x)}\d x\biggl) \leq C
$$
for some positive constant $C$, where the supremum extends over all balls in $B$ in $\R^n$.

For any measurable
subset $D$ of $\R^n$, we let
$f_{D}=\frac{1}{|D|}\int_{D}f(x)\d x$ denote the average of $f$ over
$D$. For $1\leq p<\infty$ and $0\leq \theta\leq 1$, let
$L^{p,\theta}(\Omega)$ denote the Morrey space of measurable functions $f$ such that

\begin{align*}
\|f\|_{L^{p,\theta}(\Omega)}:=\sup_{B_R(x_0)}\biggl(\frac{1}{R^{n\theta}}\int_{B_R(x_0)\cap\Omega }|f(x)|^p\d
x\biggl)^\frac{1}{p}<\infty,
\end{align*}
where the supremum above extends over all balls $B_R(x_0)$ with
$x_0\in\overline\Omega$ and $0<R<R_\Omega$. For $1\leq p<\infty$ and
$1\leq\theta<\infty$, let ${\cal L}^{p,\theta}(\Omega)$ denote the
Campanato space of functions bounded (or vanishing for $\theta>1$)
on the boundary $\partial\Omega$, equipped with the norm
\begin{align*}
\|f\|_{{\cal L}^{p,\theta}(\Omega)}:
&= \sup_{
B_R(x_0)\cap\Omega}\biggl(\frac{1}{R^{n\theta}}\int_{
B_R(x_0)\cap\Omega}|f(x)|^p\d
x\biggl)^\frac{1}{p}\\
&~~~+\sup_{B_R(y_0)\cap\Omega}\biggl(\frac{1}{R^{n\theta}}\int_{B_R(y_0)\cap\Omega}|f(x)-f_{B_R(y_0)\cap\Omega}|^p\d
x\biggl)^\frac{1}{p},
\end{align*}
where the supremum above extends over all balls with
$x_0\in\partial\Omega$, $y_0\in\Omega$ and $0<R<R_\Omega$, and we set
$\overline{\rm BMO}={\cal L}^{1,1}(\Omega)$.

For any fixed $T>0$, we set $\Omega_T=\Omega\times(0,T]$ and $\Gamma_T=\partial\Omega\times(0,T]$. For any
point $(x_0,t_0)\in\R^{n+1}$, we set
$Q_R(x_0,t_0)=B_R(x_0)\times(t_0-R^2,t_0]$ as the parabolic cylinder centered
at $(x_0,t_0)$ of radius $R$. For integers $m,n\geq 0$,
$0<\alpha,\beta<1$ and any open subset $Q\subset \Omega_T$, let
$C^{m+\alpha,n+\beta}(\overline Q)$ denote the anistropic H\"{o}lder
space of functions, equipped with the norm
$\|f\|_{C^{m+\alpha,n+\beta}(\overline
Q)}:=\|f\|_{L^\infty(Q)}+|f|_{C^{m+\alpha,n+\beta}(\overline Q)}$,
where
\begin{align*}
|f|_{C^{m+\alpha,n+\beta}(\overline Q)}&=\sum_{|\gamma|=
m}\sup_{\begin{subarray}{l}
(x,t)\in Q\\
(y,s)\in Q
\end{subarray}}\frac{|D^\gamma_x f(x,t)-D^\gamma_x
f(y,s)|}{|x-y|^\alpha +|t-s|^\beta}\\
&~+\sum_{|\gamma|= n}\sup_{\begin{subarray}{l}
(x,t)\in Q\\
(y,s)\in Q
\end{subarray}}\frac{|D^\gamma_t f(x,t)-D^\gamma_t
f(y,s)|}{|x-y|^\alpha+|t-s|^\beta},
\end{align*}
and set
$C^\alpha(\overline\Omega_T)=C^{\alpha,\alpha}(\overline\Omega_T)$.
Let $C^{m+\alpha,n+\beta}_0(\overline Q)$ denote the subspace of
$C^{m+\alpha,n+\beta}(\overline Q)$ with functions vanishing on the
boundary $\partial\Omega$. Let $C^\infty(\overline Q)$ denote the
space of functions whose partial derivatives up to all orders are
uniformly continuous on $\overline Q$. Let
$C^{m+\alpha,n+\beta}(\Omega_T)$ and $C^\infty(\Omega_T)$ denote the
space of functions which are in $C^{m+\alpha,n+\beta}(\overline Q)$
and $C^\infty(\overline Q)$ for any closed cylinder
$\overline Q\subset\Omega_T$, respectively. For any measurable
subset $Q$ of $\R^{n+1}$ and any integrable function $f$ defined on
$Q$, we let $|Q|$ denote the Lebesgue measure of $Q$ and let
$f_{Q}=\frac{1}{|Q|}\int_{Q}f(x)\d x$ denote the average of $f$ over
$Q$.  Analogous to the Morrey space $L^{p,\theta}(\Omega)$ and the
Campanato space ${\cal L}^{p,\theta}(\Omega)$, for $1\leq p<\infty$
we can define the parabolic Morrey space $L^{p,\theta}_{\rm
para}(\Omega_T)$ equipped with the norm
\begin{align*}
&\|f\|_{L^{p,\theta}_{\rm
para}(\Omega_T)}=\sup_{Q_R}\biggl(\frac{1}{R^{(n+2)\theta}}\int_{Q_R}|f(x)|^p\d
x\biggl)^\frac{1}{p}, \quad\quad\quad\quad 0\leq \theta\leq 1,
\end{align*}
and the parabolic Campanato space ${\cal L}^{p,\theta}_{\rm
para}(\Omega_T)$ of functions vanishing on the boundary
$\Gamma_T$, equipped with the norm
\begin{align*}
\|f\|_{{\cal
L}^{p,\theta}_{\rm para}(\Omega_T)}:
&=\sup_{Q_R(x_0,t_0)\cap\Omega_T}\biggl(\frac{1}{R^{(n+2)\theta}}\int_{Q_R(x_0,t_0)\cap\Omega_T}|f(x)|^p\d
x\biggl)^\frac{1}{p},\\
&~~~ +\sup_{Q_R(y_0,s_0)\cap\Omega_T}\biggl(\frac{1}{R^{(n+2)\theta}}\int_{Q_R(y_0,s_0)\cap\Omega_T}|f(x)-f_{Q_R}|^p\d
x\biggl)^\frac{1}{p},
\end{align*}
where the supremums above extend over all cylinders with
$x_0\in\partial\Omega$, $y_0\in\Omega$, $t_0,s_0\in(0,T]$ and
$0<R<R_\Omega$.

For any Banach space $X$ and time interval $(t_1,t_2)\subset\R$, we
denote by $L^p((t_1,t_2); X)$ the Bochner space equipped with the
norm
$$
\|f\|_{L^p((t_1,t_2);X)} =\left\{
\begin{array}{ll}
\displaystyle\biggl(\int_{t_1}^{t_2}\|f(t)\|_X^pdt\biggl)^\frac{1}{p},
& 1\leq p<\infty
,\\[15pt]
\displaystyle{\rm ess\,}\!\!\sup_{\!\!\!\!\!\!t\in
(t_1,t_2)}\|f(t)\|_X, & p=\infty.
\end{array}
\right.
$$

The importance of the (parabolic) Morrey spaces is that
$L^{p,\theta}(\Omega)$ translates just like
$L^{p/(1-\theta)}(\Omega)$, i.e. through the transformation $\tilde
f(y)=f(Ry)$ we have
\begin{align*}
&\|f\|_{L^{p,\theta}(B_R)} = R^{n(1-\theta)/p}\|\tilde
f\|_{L^{p,\theta}(B_1)},\\[-25pt]
\end{align*}
just like
\begin{align*}
&\|f\|_{L^{p/(1-\theta)}(B_R)} = R^{n(1-\theta)/p}\|\tilde
f\|_{L^{p/(1-\theta)}(B_1)},
\end{align*}
for any ball $B_R\subset\Omega$. Similarly, $L^{p,\theta}_{\rm
para}(\Omega_T)$ translates just like $L^{p/(1-\theta)}(\Omega_T)$. Therefore, $L^{p,\theta}(\Omega) $ and
$L^{p,\theta}_{\rm para}(\Omega_T)$ can be used as substitute for
$L^{p/(1-\theta)}(\Omega)$ and $L^{p/(1-\theta)}(\Omega_T)$, respectively, with lower order integrability. The
importance of the (parabolic) Campanato spaces includes: \\
(1)~ ${\cal L}^{p,1}(\Omega)$ are equivalent for all $1\leq
p<\infty$, i.e. ${\cal L}^{p,1}(\Omega)\cong \overline{\rm BMO}$;\\
(2)~ If $1<\theta<(n+p)/n$, then ${\cal L}^{p,\theta}(\Omega)\cong
C^{\alpha}_0(\overline\Omega)$ for $\alpha=n(\theta-1)/p$. \\
(3)~ If $1<\theta<(n+2+p)/(n+2)$, then ${\cal L}^{p,\theta}_{\rm
para}(\Omega_T)\cong C^{\alpha,\alpha/2}_0(\overline\Omega_T)$ for
$\alpha=(n+2)(\theta-1)/p$.

These properties of the Morrey and Campanato spaces can be found in
\cite{ChenYZParabEq,Troianiello}.

In this paper, we let $C_{p_1,p_2,\cdots,p_m}$ denote a generic
positive constant which depends on the parameters
$p_1,p_2,\cdots,p_m$.

\section{Main results}\label{dfjksl7879}
First, we establish a uniform-in-time BMO estimate and
a H\"{o}lder estimate for the solution of the
parabolic equation
\begin{align}
&\left\{
\begin{array}{ll} \displaystyle\frac{\partial u}{\partial
t}-\nabla\cdot(A\nabla u)=\nabla\cdot{\vec f} + f_0,
&\mbox{in}~~\Omega\times(0,T),\\[7pt]
u=g &\mbox{on} ~~\partial\Omega\times(0,T),\\[3pt]
u(x,0)=u_0(x)  &\mbox{for} ~~x\in\Omega ,
\end{array}
\right. \label{parabEq0}
\end{align}
where $\Omega$ is a bounded Lipschitz domain in $\R^n$ and
$A(x,t)=[A_{ij}(x,t)]_{n\times n}$ is a symmetric positive definite
measurable matrix function defined on $\R^{n+1}$ such that
\begin{align}\label{djfi}
K^{-1}|\xi|^2\leq \sum_{i,j=1}^n A_{ij}(x,t)\xi_i\xi_j \leq K|\xi|^2
,\quad\mbox{for all}~\xi\in\R^n
\end{align}
holds almost everywhere for $(x,t)\in\R^{n+1}$, where $K$ is a
positive constant.

\begin{theorem}\label{Mainresult}{\bf(BMO and H\"{o}lder estimates of parabolic equations)}~\\
{\it There exist positive constants $C$ and $\alpha_0\in (0,1)$
depending only on the elliptic constant $K$, the domain $\Omega$ and
the dimension $n$ $($independent of $T)$, such that the solution of
{\rm(\ref{parabEq0})} satisfies the BMO estimate
\begin{align}
&\|u\|_{L^\infty((0,T);\overline{\rm BMO})} \label{mainest1} \leq C(
\|f_0\|_{L^{1,n/(n+2)}(\Omega_T)}+\|{\vec
f}\|_{L^{2,n/(n+2)}(\Omega_T)}+\|u_0\|_{L^\infty(\Omega)}+
\|g\|_{L^\infty(\Gamma_T)}) .
\end{align}
If the compatibility condition $u_0(x)=g(x,0)$ for
$x\in\partial\Omega$ is satisfied, then we have
\begin{align}
&\|u\|_{C^{\alpha,\alpha/2}(\overline{\Omega}_T)}
\label{mainest2}\leq
C(\|f_0\|_{L^{1,(n+\alpha)/(n+2)}(\Omega_T)}+\|{\vec
f}\|_{L^{2,(n+2\alpha)/(n+2)}(\Omega_T)}
+\|u_0\|_{C^{\alpha}(\overline{\Omega})}+
\|g\|_{C^{\alpha,\alpha/2}(\Gamma_T)}) ,
\end{align}
for $0<\alpha\leq \alpha_0$.
}
\end{theorem}
The inequality (\ref{mainest1}) is new. A similar inequality as
(\ref{mainest2}) was proved in \cite{Yin}, where
$\|f_0\|_{L^{1,(n+\alpha)/(n+2)}(\Omega_T)}$ was replaced by
$\|f_0\|_{L^{2,(n-2+2\alpha)/(n+2)}(\Omega_T)}$. Note that
$L^{2,(n-2+2\alpha)/(n+2)}(\Omega_T)$ translates in the same way as
$L^{1,(n+\alpha)/(n+2)}(\Omega_T)$ under a scale transformation but
requires higher integrability. \bigskip

Secondly, by applying Theorem \ref{Mainresult}, we prove global
existence and uniqueness of a weak solution for the degenerate
thermistor problem
under the following physical hypotheses:\\[5pt]
{\bf(H1)} The thermal conductivity is a smooth function of
temperature and satisfies that
\begin{align*}
0<\inf_{s\geq r}\kappa(s)\leq \sup_{s\geq
r}\kappa(s)<\infty,\quad\mbox{for any fixed}~\, r>0 .
\end{align*}
{\bf(H2)} The electric resistivity $\rho(u)=1/\sigma(u)$
is a smooth function of temperature such that for some $p>0$ there holds
\begin{align}\label{dfsjljiowuo}
&C_{1,r}+C_{2,r}s^{p}\leq\rho(s)\leq C_{3,r}+C_{4,r}s^{p} \quad\mbox{$\forall~s\geq r>0$},
\end{align}
where $C_{i,r}$, $i=1,\cdots,5$, are some positive c
onstants (possibly depending on $r$).

Clearly, the hypotheses (H1)-(H2) are true for metals and some
semiconductors. In particular, the electric resistivity $\rho(u)$
can be any polynomials which are positive for $u> 0$. The hypotheses
(H1)-(H2) also imply that for any given $r>0$, $\sigma(s)$ is
bounded for $s\geq r$.

\begin{theorem}\label{mainthm}{\bf(Global well-posedness of the degenerate thermistor problem)}\\
{\it Let $\Omega$ be a bounded Lipschitz domain in $\R^n$ $(n=2,3)$ and let $q_0>n$. Assume that $u_0\in W^{1,q_0}(\Omega)$, $g\in L^\infty((0,T);W^{1,q_0}(\Omega))$, $\partial_tg\in L^\infty((0,T);L^{q_0}(\Omega))$,
$h\in L^\infty((0,T);W^{1,q_0}(\Omega))$,
with
$$\min_{(x,t)\in\Gamma_T}g(x,t)>0,\qquad
\min_{x\in
\Omega}\, u_0(x)
>0,
$$
and $g(x,0)=u_0(x)$ for $x\in\partial\Omega$. Then, under the hypothesis
{\rm (H1)-(H2)}, the initial-boundary value problem
{\rm (\ref{e-heat-1})-(\ref{BC})} admits a unique weak solution $(u,\phi)$ such that
\begin{align
\begin{array}{ll}
u\in  C^{\alpha,\alpha/2}(\overline\Omega_T)\cap
L^p((0,T);W^{1,q}(\Omega)), \quad \phi\in  L^\infty((0,T);W^{1,q}(\Omega)) ,\\[3pt]
\partial_tu\in L^p((0,T);W^{-1,q}(\Omega)),
\end{array}
\label{sdfjhkl79031}
\end{align}
for some $q>n$, $0<\alpha<1$ and any $1<p<\infty$,  in the sense that the equations
\begin{align*}
&\int_0^T\int_{\Omega}\frac{\partial u}{\partial t}v\,\d x\d t
+\int_0^T\int_{\Omega}\kappa(u)\nabla
u\cdot \nabla v\,\d x\d t
=\int_0^T\int_{\Omega}\sigma(u)|\nabla\phi|^2v\,\d x\d t,\\
&\int_0^T\int_\Omega\sigma(u)\nabla\phi\cdot\nabla \varphi\,\d x\d t=0,\quad
\end{align*}
hold for any $v,\varphi\in L^2((0,T);H^1_0(\Omega))$.
}
\end{theorem}

Note that with the regularity (\ref{sdfjhkl79031}), the last equation above is equivalent to $$\int_\Omega\sigma(u)\nabla\phi\cdot\nabla \varphi\,\d x=0,\quad \forall~\varphi\in H^1_0(\Omega),\quad \mbox{a.e.}~t\in(0,T).$$

\section{BMO estimate of parabolic equations}\label{fjkesluio99}
\setcounter{equation}{0}

The solution of
(\ref{parabEq0}) can be decomposed into three parts, i.e.
the solution of  the following three problems:
\begin{align}
&\left\{
\begin{array}{ll}
\displaystyle\frac{\partial u}{\partial t}-\nabla\cdot(A\nabla u)=f_0,
&\mbox{in}~~\Omega\times(0,T),\\[7pt]
u=0 &\mbox{on} ~~\partial\Omega\times(0,T),\\[3pt]
u(x,0)=0 &\mbox{for} ~~x\in\Omega .
\end{array}
\right. \label{parabEq}\\[10pt]
&\left\{ \begin{array}{ll} \displaystyle\frac{\partial u}{\partial
t}-\nabla\cdot(A\nabla u)=\nabla\cdot{\vec f},
&\mbox{in}~~\Omega\times(0,T),\\[7pt]
u=0 &\mbox{on} ~~\partial\Omega\times(0,T),\\[3pt]
u(x,0)=0  &\mbox{for} ~~x\in\Omega ,
\end{array}
\right. \label{parabEq2}\\[10pt]
&\left\{ \begin{array}{ll} \displaystyle\frac{\partial u}{\partial
t}-\nabla\cdot(A\nabla u)=0,
&\mbox{in}~~\Omega\times(0,T),\\[7pt]
u=g &\mbox{on} ~~\partial\Omega\times(0,T),\\[3pt]
u(x,0)=u_0(x)  &\mbox{for} ~~x\in\Omega .
\end{array}
\right. \label{parabEq3}
\end{align}

From the maximum principle and the De Giorgi--Nash--Moser estimates, we know that there exist positive constants $C$ and $0<\alpha_0<1$ such that the solution of (\ref{parabEq3}) satisfies that
\begin{align*}
&\|u\|_{L^\infty(\Omega_T)} \leq \|g\|_{L^\infty(\Gamma_T)} +\|u_0\|_{L^\infty(\Omega)},\\
&\|u\|_{C^{\alpha,\alpha/2}(\overline{\Omega}_T)} \leq C
(\|g\|_{C^{\alpha,\alpha/2}(\overline\Gamma_T)}+\|u_0\|_{C^{\alpha}(\overline\Omega)}),
\end{align*}
for $0<\alpha<\alpha_0<1$ and $T>0$ (the second inequlaity above requires the compatability condition). To prove Theorem \ref{Mainresult}, it suffices to present estimates for the equations (\ref{parabEq})-(\ref{parabEq2}).

The rest part of this section is organized in the following way. In
Section \ref{secL11est}, we present local $L^1$ estimates for the
solution to (\ref{parabEq}). In Section \ref{glbest1}, we combine
the local $L^1$ estimates to derive a global BMO estimate based on
the equivalence of $\overline{\rm BMO}$ with the Campanato space
${\cal L}^{1,1}(\Omega)$. In Section \ref{sec4}, we establish the
BMO estimate for (\ref{parabEq2}) in terms of the Campanato space
${\cal L}^{2,1}(\Omega)$.

\subsection{Local $L^1$ estimates}\label{secL11est}
In this subsection, we present local $L^1$ estimates for the solution
of (\ref{parabEq}). The estimates obtained in this subsection will be used in Section \ref{glbest1} to derive a global BMO estimate uniformly with respect to time.

\begin{lemma}\label{Lemma001}
{\it Let $x_0\in\Omega$ and $0<t_0<T$. There exists
$\alpha_0\in(0,1)$ and $C>0$ such that if $u$ is the solution of
{\rm(\ref{parabEq})} in $Q_R=B_R(x_0)\times I_R$ with
$I_R=(t_0-R^2,t_0]$, then
\begin{align*}
&\max_{t\in I_\rho}\|u-u_{Q_\rho}\|_{L^1(B_\rho)} \leq
C\biggl(\frac{\rho}{R}\biggl)^{n+\alpha_0}\!\!\!\!\!\!\max_{t\in
I_R}\|u-\theta\|_{L^1(B_R)} +C\|f_0\|_{L^1(Q_R)}
\end{align*}
holds for any $0<\rho\leq R\leq \min({\rm
dist}(x_0,\partial\Omega),\sqrt{t_0})$ and any $\theta\in\R$, where
the constants $C$ and $\alpha_0$ depend only on $K$ and $n$. }
\end{lemma}
\noindent{\it Proof}~~~ First, we prove the lemma for
$\theta=0$. Let $\widetilde B_r=B_r(0)$, $\widetilde
I_r=(-r^2,0]$ and $\widetilde \Gamma_r=\partial\widetilde
B_r\times\widetilde I_r$. With any function $\xi$ defined on $Q_R$, we
associate a function $\tilde \xi(y,s)=\xi(x_0+Ry,t_0+R^2s)$ defined
on
 $\widetilde Q_1:=\widetilde B_1\times\widetilde I_1$. Then $\tilde u$ is a solution to the equation
$$
\frac{\partial \tilde u}{\partial s}-\nabla_y\cdot(\tilde A \nabla_y
\tilde u)=R^2\tilde f_0
$$
in $\widetilde Q_1$. Let $w$ be the solution of
\begin{align*}
\frac{\partial w}{\partial s}-\nabla_y\cdot(\tilde A \nabla_y
w)=R^2\tilde f_0
\end{align*}
with the boundary/initial condition $w=0$ on the parabolic
boundary $\partial_{\rm p}\widetilde Q_1$ and let $\bar w$ be the
solution of
\begin{align*}
\frac{\partial \bar w}{\partial s}-\nabla_y\cdot(\tilde A \nabla_y
\bar w)=R^2|\tilde f_0| 1_{\widetilde Q_1}
\end{align*}
in $\R^{n+1}$ with the initial condition $\bar w(y,0)\equiv 0$. By
the maximum principle, we know that
\begin{align*}
&|w(y,s)|\leq |\bar
w(y,s)|\\
&\leq \int_0^s\int_{\R^n}\frac{C}{(s-s')^{n/2}}e^{-\frac{|y-y'|^2}{C(s-s')}}R^2|\tilde
f_0(y',s')| 1_{\widetilde Q_1}(y',s')\d y'\d s'.
\end{align*}
Taking the $L^1(\widetilde B_1)$ norm with respect to $y$, we derive that
\begin{align*}
\|w\|_{L^\infty(\widetilde I_1;L^1(\widetilde B_1))}\leq
CR^2\|\tilde f_0\|_{L^1(\widetilde Q_1)} .
\end{align*}
We note that $v=\tilde u-\tilde u_{\widetilde Q_1}-w$ is the
solution of
\begin{align*}
&\frac{\partial v}{\partial s}-\nabla_y\cdot(\tilde  A \nabla_y v)=0
\end{align*}
in $\widetilde Q_1$, and by the De Giorgi--Nash estimates of parabolic
equations we know that there exists $\alpha_0\in(0,1)$ such that
for $\rho\in(0,1/2]$,
\begin{align*}
&\max_{t\in \widetilde I_\rho}\frac{1}{\rho^{n+\alpha_0}}\int_{\widetilde
B_\rho}|v-v_{\widetilde Q_\rho}|\d y \nn\\
&\leq C|v|_{C^{\alpha_0,\alpha_0/2}(\widetilde Q_{1/2})} \leq
C\|v\|_{L^1(\widetilde Q_1)} \leq C\max_{t\in\widetilde
I_1}\|v\|_{L^1(\widetilde B_1)}
 .
\end{align*}
Therefore,
\begin{align*}
&\max_{t\in\widetilde I_\rho}\|\tilde u-\tilde u_{\widetilde Q_\rho}\|_{L^1(\widetilde B_\rho)}\nn\\[3pt]
&\leq \max_{t\in\widetilde I_\rho}\|v-v_{\widetilde
Q_\rho}\|_{L^1(\widetilde B_\rho)}+\max_{t\in
\widetilde I_\rho}\|w-w_{\widetilde Q_\rho}\|_{L^1(\widetilde B_\rho)} \nn\\
&\leq C\rho^{n+\alpha_0}\max_{t\in\widetilde
I_1}\|v\|_{L^1(\widetilde B_1)}+C\max_{t\in\widetilde
I_1}\|w\|_{L^1(\widetilde B_1)}   \nn\\[3pt]
&\leq C\rho^{n+\alpha_0}\max_{t\in\widetilde I_1}\|\tilde u-\tilde
u_{\widetilde Q_1}\|_{L^1(\widetilde B_1)}+C\max_{t\in\widetilde
I_1}\|w\|_{L^1(\widetilde B_1)} \nn \\
&\leq C\rho^{n+\alpha_0}\max_{t\in\widetilde I_1}\|\tilde u-\tilde
u_{\widetilde Q_1}\|_{L^1(\widetilde B_1)}+CR^2\|\tilde
f_0\|_{L^1(\widetilde Q_1)}  \nn\\
&\leq C\rho^{n+\alpha_0}\max_{t\in\widetilde I_1}\|\tilde u\|_{L^1(\widetilde B_1)}+CR^2\|\tilde
f_0\|_{L^1(\widetilde Q_1)},
\end{align*}
where we have noted that
\begin{align*}
\|\widetilde u_{\widetilde Q_1}\|_{L^1(\widetilde B_1)}=\frac{|\widetilde B_1|}{|\widetilde Q_1|}\int_{\widetilde Q_1}|\widetilde u|\d x\d t\leq \max_{t\in \widetilde I_1}\|\widetilde u\|_{L^1(\widetilde B_1)} .
\end{align*}
Transforming back to the $(x,t)$-coordinates, we complete the proof of the Lemma.
for $\theta=0$.

Then we note that $u-\theta$ is also a
solution to the equation (\ref{parabEq}) in $Q_R$ for any $\theta\in\R$.\qed

Similarly, we can prove the following local $L^1$ estimates near the boundary $\partial_p\Omega_T$.

\begin{lemma}\label{Lemma002}
{\it Let $x_0\in\Omega$ and $t_0=0$. There exists $\alpha_0\in(0,1)$
and $C>0$ such that if $u$ is the solution of {\rm (\ref{parabEq})}
in $Q_R=B_R(x_0)\times\underline{I}_R$ with
$\underline{I}_R=[0,R^2]$, then
\begin{align*}
&\max_{t\in \underline{I}_\rho}\|u\|_{L^1(B_\rho)} \leq
C\biggl(\frac{\rho}{R}\biggl)^{n+\alpha_0}\!\!\!\!\!\!\max_{t\in
\underline{I}_R}\|u\|_{L^1(B_R)} +C\|f_0\|_{L^1(Q_R)}
\end{align*}
holds for any $0<\rho\leq R\leq \min({\rm
dist}(x_0,\partial\Omega),\sqrt{T})$, where the constants $C$ and
$\alpha_0$ depend only on $K$ and $n$. }
\end{lemma}

\begin{lemma}\label{Lemma003}
{\it Let $x_0\in\partial\Omega$ and $t_0>0$. There exists
$\alpha_0\in(0,1)$ and $C>0$ such that if $u$ is the solution of
{\rm (\ref{parabEq})} in $Q_R=B_R\times I_R$, with
$B_R=B_R(x_0)\cap\Omega$ and $I_R=(t_0-R^2,t_0]$, then
\begin{align*}
&\max_{t\in I_\rho}\|u\|_{L^1(B_\rho)}\leq
C\biggl(\frac{\rho}{R}\biggl)^{n+\alpha_0}\!\!\!\!\!\!\max_{t\in
I_R}\|u\|_{L^1(B_R)} +C\|f_0\|_{L^1(Q_R)}
\end{align*}
holds for any $0<\rho\leq R\leq \min(R_\Omega,\sqrt{t_0})$, where
the constants $C$ and $\alpha_0$ depend only on $K$, $n$ and
$\Omega$. }
\end{lemma}

\begin{lemma}\label{Lemma003}
{\it Let $x_0\in\partial\Omega$ and $t_0=0$. There exists
$\alpha_0\in(0,1)$ and $C>0$ such that if $u$ is the solution of
{\rm (\ref{parabEq})} in $Q_R=B_R\times I_R$, with
$B_R=B_R(x_0)\cap\Omega$ and $I_R=[0,R^2]$, then
\begin{align*}
&\max_{t\in I_\rho}\|u\|_{L^1(B_\rho)}\leq
C\biggl(\frac{\rho}{R}\biggl)^{n+\alpha_0}\!\!\!\!\!\!\max_{t\in
I_R}\|u\|_{L^1(B_R)} +C\|f_0\|_{L^1(Q_R)} ,
\end{align*}
holds for any $0<\rho\leq R\leq \min(R_\Omega,\sqrt{T})$, where the
constants $C$ and $\alpha_0$ depend only on $K$, $n$ and $\Omega$. }
\end{lemma}

The following simple lemma can be found in
\cite{ChenYZParabEq,Lieberman}, which is widely used for estimates
in terms of the Morrey and Campanato spaces.

\begin{lemma}\label{jkddlafhlae}
{\it Let $\varphi(\cdot)$ be a nonnegative and nondecreasing
function defined on $(0,R_0]$ and suppose that for any $0<\rho\leq R\leq
R_0$,
$$
\varphi(\rho)\leq
C_1\biggl(\frac{\rho}{R}\biggl)^{\gamma_1}\varphi(R)+C_2R^{\gamma_2}
,
$$
where $C_1$, $\gamma_1$ and $\gamma_2$ are nonnegative constants
such that $0<\gamma_2<\gamma_1$. Then
$$
\frac{1}{R^{\gamma_2}}\varphi(R)\leq
C_{\gamma_1,\gamma_2,C_1}\biggl(\frac{1}{R_0^{\gamma_2}}\varphi(R_0)+C_2\biggl)
.
$$
}
\end{lemma}\medskip

From the above lemmas, we obtain the following local $L^1$
estimates. \setcounter{proposition}{\value{lemma}}
\addtocounter{lemma}{1}
\begin{proposition}\label{dnflkqhklqewad}
{\it For $x_0\in\Omega$, $t_0>0$ and $Q_R=B_R(x_0)\times I_R$ with
$I_R=(t_0-R^2,t_0]$, we have
\begin{align*}
&\frac{1}{\rho^n}\|u-u_{Q_\rho}\|_{L^\infty((t_0-\rho^2,t_0);L^1(B_\rho))}\leq
C\biggl(\frac{1}{R^{n}}\|u\|_{L^\infty((t_0-R^2,t_0);L^1(B_{R}))}
+\|f_0\|_{L^{1,n/(n+2)}(\Omega_T)}\biggl) \nn
\end{align*}
for any $0<\rho\leq R\leq\min\big({\rm
dist}(x_0,\partial\Omega),\sqrt{t_0}\big)$. }
\end{proposition}

\addtocounter{lemma}{1}
\begin{proposition}\label{dnflkqhklqewad001}
{\it For $x_0\in\Omega$, $t_0=0$ and $Q_R=B_R(x_0)\times [0,R^2]$,
we have
\begin{align*}
&\frac{1}{\rho^n}\|u\|_{L^\infty((t_0-\rho^2,t_0);L^1(B_\rho))}\leq
C\biggl(\frac{1}{R^{n}}\|u\|_{L^\infty((t_0-R^2,t_0);L^1(B_{R}))}
+\|f_0\|_{L^{1,n/(n+2)}(\Omega_T)}\biggl)  \nn
\end{align*}
for any $0<\rho\leq R\leq\min\big({\rm
dist}(x_0,\partial\Omega),\sqrt{T}\big)$. }
\end{proposition}

\begin{proposition}\label{dnflkqhklqewad092}
{\it For $x_0\in\partial \Omega$, $t_0>0$ and
$Q_R=B_R(x_0)\cap\Omega\times I_R$ with $I_R=(t_0-R^2,t_0]$, we have
\begin{align*}
&\frac{1}{\rho^n}\|u\|_{L^\infty((t_0-\rho^2,t_0);L^1(B_\rho))}\leq
C\biggl(\frac{1}{R^{n}}\|u\|_{L^\infty((t_0-R^2,t_0);L^1(B_{R}))}
+\|f_0\|_{L^{1,n/(n+2)}(\Omega_T)}\biggl)  \nn
\end{align*}
for any $0<\rho\leq R\leq \min(R_\Omega,\sqrt{t_0})$.}
\end{proposition}

\begin{proposition}\label{dnflkqhklqewad03}
{\it For $x_0\in\partial \Omega$, $t_0=0$ and
$Q_R=B_R(x_0)\cap\Omega\times [0,R^2]$, we have
\begin{align*}
&\frac{1}{\rho^n}\|u\|_{L^\infty((0,\rho^2);L^1(B_\rho))}\leq
C\biggl(\frac{1}{R^{n}}\|u\|_{L^\infty((0,R^2);L^1(B_{R}))}
+\|f_0\|_{L^{1,n/(n+2)}(\Omega_T)}\biggl) \nn
\end{align*}
for any $0<\rho\leq R\leq \min(R_\Omega,\sqrt{T})$.}
\end{proposition}

\subsection{BMO estimates via ${\cal L}^{1,1}$ }\label{glbest1}
We combine the local $L^1$ estimates obtained in the last subsection
to derive a global BMO estimate of $u$, uniform with respect to time.

\begin{proposition}\label{njdfklahkeas}
{\it The Propositions
{\rm\ref{dnflkqhklqewad}--\ref{dnflkqhklqewad03}} imply that the solution of {\rm (\ref{parabEq})} satisfies that
\begin{align}\label{BMOest2df}
\|u\|_{L^\infty((0,T);{\rm \overline{BMO}})} \leq
C\|f_0\|_{L^{1,n/(n+2)}(\Omega_T)},
\end{align}
where $C$ depends only on $K$, $n$ and $\Omega$
(independent of $T$). }
\end{proposition}
\noindent{\it Proof}~~~ Set $M=\|f_0\|_{L^{1,n/(n+2)}(\Omega_T)}$.

First, we prove the proposition for $T\geq
R_\Omega^2$.
We shall prove that for $R< R_\Omega/2$ and any set
$B_R=B_R(x_0)\cap\Omega$ with some point $x_0\in\overline\Omega$ and
$\delta={\rm dist}(x_0,\partial\Omega)$, the following estimates
hold:
\begin{align}\label{estkeaklael1}
\left\{
\begin{array}{ll}
\frac{1}{R^n}\|u\|_{L^\infty((0,T);L^1(B_R))}\leq
C\big(\|u\|_{L^\infty((0,T);L^1(\Omega))} +M\big) , &
\mbox{if}~~\delta\leq  R,\\[5pt]
\frac{1}{R^n}\|u-u_{B_R}\|_{L^\infty((0,T);L^1(B_R))}\leq
C\big(\|u\|_{L^\infty((0,T);L^1(\Omega))} +M\big) , &
\mbox{if}~~\delta> R,
\end{array}
\right.
\end{align}
{\it Case} 1: $\delta \leq R$.~~ In this case, there exists a region
$B_{2R}=B_{2R}(y_0)\cap\Omega$ with some $y_0\in\partial\Omega$ such
that $B_R\subset B_{2R}$ and so, for any given $t_0\in[0,T]$,
\begin{align}\label{fjekqhlqt}
&\|u(\cdot,t_0)\|_{L^1(B_R)}\leq \|u(\cdot,t_0)\|_{L^1(B_{2R})} .
\end{align}
Now if $t_0\leq 4R^2$, then by Proposition \ref{dnflkqhklqewad03},
\begin{align}
&\frac{1}{R^n}\|u(\cdot,t_0)\|_{L^1(B_{2R})}\leq \|u\|_{L^\infty((0,4R^2);L^1(B_{2R}))}\nn\\
&\leq
C\biggl(\frac{1}{R_\Omega^{n}}\|u\|_{L^\infty((0,R^2_\Omega);L^1(B_{R_\Omega}))}
+M\biggl) .
\end{align}
Otherwise, $t_0>4R^2$ and by Proposition \ref{dnflkqhklqewad092}, for
$R_0=\min(\sqrt{t_0},R_\Omega)$ and $R_m=\max(\sqrt{t_0},R_\Omega)$ we have
\begin{align*}
&\frac{1}{R^n}\|u\|_{L^\infty((t_0-4R^2,t_0);L^1(B_{2R}))} \\
&\leq
C\biggl(\frac{1}{R_0^{n}}\|u\|_{L^\infty((t_0-R_0^2,t_0);L^1(B_{R_0}))}
+M\biggl) \\
&= \left\{
\begin{array}{ll}
C\biggl(\frac{1}{t_0^{n/2}}\|u\|_{L^\infty((0,t_0);L^1(B_{\sqrt{t_0}}))}
+M\biggl), & \sqrt{t_0}<R_\Omega,\\[5pt]
C\biggl(\frac{1}{R_\Omega^{n}}\|u\|_{L^\infty((t_0-R_\Omega^2,t_0);L^1(B_{R_\Omega}))}
+M\biggl), & \sqrt{t_0}\geq R_\Omega .
\end{array}
\right.\\
&\leq \left\{
\begin{array}{ll}
C\biggl(\frac{1}{R_\Omega^{n}}\|u\|_{L^\infty((0,R_\Omega);L^1(B_{R_\Omega}))}
+M\biggl) , & \sqrt{t_0}<R_\Omega, \quad\mbox{(by Proposition \ref{dnflkqhklqewad03})} \\[5pt]
C\biggl(\frac{1}{R_\Omega^{n}}\|u\|_{L^\infty((t_0-R_\Omega^2,t_0);L^1(B_{R_\Omega}))}
+M\biggl), & \sqrt{t_0}\geq R_\Omega .
\end{array}
\right.
\end{align*}
To conclude, for $\delta\leq R$ and $t_0\in[0,T]$ we have
\begin{align}\label{fdjkaletehw}
\frac{1}{R^n}\|u(\cdot,t_0)\|_{L^1(B_R)}\leq
\frac{C}{R_\Omega^{n}}\|u\|_{L^\infty((0,T);L^1(\Omega))}+CM.
\end{align}

\noindent{\it Case} 2: $\delta>R$. ~ In this case, we set
$R_0=\min(\delta,\sqrt{t_0},R_\Omega)$. Then Proposition
\ref{dnflkqhklqewad} implies that
\begin{align*}
&\frac{1}{R^n}\|u-u_{B_R}\|_{L^\infty((t_0-R^2,t_0);L^1(B_R))} \leq \frac{1}{R^n}\|u-u_{Q_R}\|_{L^\infty((t_0-R^2,t_0);L^1(B_R))} \\
&\leq
C\biggl(\frac{1}{R_0^{n}}\|u\|_{L^\infty((t_0-R_0^2,t_0);L^1(B_{R_0}))}
+M\biggl)\\
&= \left\{
\begin{array}{ll}
C\biggl(\frac{1}{R_\Omega^{n}}\|u\|_{L^\infty((t_0-R_\Omega^2,t_0);L^1(B_{R_\Omega}))}
+M\biggl) , & \mbox{if}~~R_\Omega\leq\min(\delta,\sqrt{t_0})\\
C\biggl(\frac{1}{\delta^{n}}\|u\|_{L^\infty((t_0-\delta^2,t_0);L^1(B_\delta))}
+M\biggl) , & \mbox{else if}~~\delta\leq\min(\sqrt{t_0},R_\Omega)\\
C\biggl(\frac{1}{R_0^{n}}\|u\|_{L^\infty((0,R_0^2);L^1(B_{R_0}))}
+M\biggl) , & \mbox{else }~~t_0=R_0^2
\end{array}
\right.\\
&\leq \left\{
\begin{array}{ll}
\frac{C}{R_\Omega^{n}}\|u\|_{L^\infty((0,T);L^1(\Omega))} +CM,
~~~~~~~~~~~~~~~~~~\mbox{if}~~R_\Omega\leq\min(\delta,\sqrt{t_0})\\[8pt]
\frac{C}{R_\Omega^{n}}\|u\|_{L^\infty((0,T);L^1(\Omega))}  +CM,
~~~~~~~~~~~~~~~~~~\mbox{else if}~~\delta\leq\min(\sqrt{t_0},R_\Omega) ~~~\mbox{by}~(\ref{fdjkaletehw})\\[8pt]
\left\{
\begin{array}{ll}
C\biggl(\frac{1}{\delta^{n}}\|u\|_{L^\infty((0,\delta^2);L^1(B_\delta))}
+M\biggl), & \mbox{else if}~~\delta\leq R_\Omega\\[5pt]
C\biggl(\frac{1}{R_\Omega^{n}}\|u\|_{L^\infty((0,R_\Omega^2);L^1(B_{R_\Omega}))}
+M\biggl), & \mbox{else if}~~\delta>R_\Omega
\end{array}
\right.
\quad\mbox{(by Proposition \ref{dnflkqhklqewad03})}
\end{array}
\right.
\\[5pt]
&\leq \frac{C}{R_\Omega^{n}}\|u\|_{L^\infty((0,T);L^1(\Omega))}
+CM,\quad \mbox{again by (\ref{fdjkaletehw})} .
\end{align*}

So far we have proved (\ref{estkeaklael1}). Once we note that
$\|u\|_{L^\infty((0,T);L^1(\Omega))}\leq
C\|f_0\|_{L^1(\Omega_T)}$, we derive (\ref{BMOest2df}) from (\ref{estkeaklael1}).

Secondly, we prove the proposition for $0<T<R_\Omega$. In this case, we
consider the solution $\hat{u}$ of the equation
\begin{align}
\frac{\partial\hat u}{\partial t}-\nabla\cdot( A \nabla\hat u )=\hat
f_0
\end{align}
in the domain $\Omega_{R_\Omega}=\Omega\times(0,R_\Omega)$ with the boundary and initial conditions $\hat
u =0$ on $\partial\Omega\times(0,R_\Omega)$ and $\hat u (x,0)=0$ for
$x\in\Omega$, where
$$
\hat f_0(x,t)=\left\{
\begin{array}{ll}
f_0(x,t), &\mbox{for}~~t\in(0,T),\\[5pt]
0,      &\mbox{for}~~t\in(T,R_\Omega) .
\end{array}\right.
$$
Check that
\begin{align*}
&\|\hat f_0\|_{L^{1,n/(n+2)}(\Omega_{R_\Omega})}\leq
C\|f_0\|_{L^{1,n/(n+2)}(\Omega_T)},\\
&\|\hat f_0\|_{L^1(\Omega_{R_\Omega})}\leq C\|f_0\|_{L^1(\Omega_T)} ,\\
&\|u\|_{L^\infty((0,T);{\rm \overline{BMO}})}\leq \|\hat
u\|_{L^\infty((0,R_\Omega);{\rm \overline{BMO}})},
\end{align*}
where the constant $C$ does not depend on $T$ (as $T\rightarrow 0$). Then we
apply the inequality (\ref{BMOest2df}) to $\hat u$ with $T=R_\Omega$.\qed

\subsection{BMO estimates via ${\cal L}^{2,1}$}\label{sec4}
\setcounter{equation}{0}

In this section, we present estimates for the solution of
(\ref{parabEq2}). The idea is similar as Section \ref{glbest1}. From the proof of the following lemma we can see the main
difference between the current subsection and the last subsection.

\begin{lemma}\label{Lemma004}
{\it Let $x_0\in\Omega$ and $0<t_0<T$. There exists
$\alpha_0\in(0,1)$ and $C>0$ such that if $u$ is the solution to
{\rm (\ref{parabEq2})} in $Q_R=B_R(x_0)\times I_R$ with
$I_R=(t_0-R^2,t_0]$, then
\begin{align*}
&\max_{t\in I_\rho}\|u-u_{Q_\rho}\|_{L^2(B_\rho)}^2 \leq
C\biggl(\frac{\rho}{R}\biggl)^{n+2\alpha_0}
\!\!\!\!\!\!\max_{t\in
I_R}\|u-\theta\|_{L^2(B_R)}^2 +C\|\vec{f}\|_{L^2(Q_R)}^2
\end{align*}
holds for any $0<\rho\leq R\leq \min({\rm
dist}(x_0,\partial\Omega),\sqrt{t_0})$ and any $\theta\in\R$, where
$C$ depends only on $K$ and $n$. }
\end{lemma}
\noindent{\it Proof}~~~ Let $\widetilde B_r=B_r(0)$, $\widetilde
I_r=(-r^2,0]$ and $\widetilde \Gamma_r=\partial\widetilde
B_r\times\widetilde I_r$. With any function $w$ defined on $Q_R$, we
associate a function $\tilde \xi(y,s)=\xi(x_0+Ry,t_0+R^2s)$ defined
on
 $\widetilde Q_1:=\widetilde B_1\times\widetilde I_1$. Then $\tilde u$ is a solution of the equation
$$
\frac{\partial \tilde u}{\partial s}-\nabla_y\cdot(\tilde A \nabla_y
\tilde u)=R\nabla_y\cdot \tilde f
$$
in $\widetilde Q_1$. Let $w$ be the solution of
\begin{align*}
\frac{\partial w}{\partial s}-\nabla_y\cdot(\tilde A \nabla_y
w)=R\nabla_y\cdot \tilde f
\end{align*}
with the initial and boundary condition $w=0$ on the parabolic
boundary $\partial_{\rm p}\widetilde Q_1$. Multiplying the above
equation by $w$ and integrating the result over $\widetilde Q_1$, we
obtain that
\begin{align*}
&\|w\|_{L^\infty(\widetilde I_1;L^2(\widetilde B_1))}\leq
CR\|\tilde f\|_{L^2(\widetilde Q_1)}
\end{align*}
On the other hand, we observe that $v=\tilde u-\tilde u_{\widetilde
Q_1}-w$ is the solution of
\begin{align*}
&\frac{\partial v}{\partial s}-\nabla_y\cdot(\tilde  A \nabla_y v)=0
\end{align*}
in $\widetilde Q_1$. By the De Giorgi--Nash estimates of parabolic
equations, we know that there exists $\alpha_0\in(0,1)$ such that
for $\rho\in(0,1/2]$,
\begin{align*}
&\max_{t\in I_\rho}\frac{1}{\rho^{n+2\alpha_0}}\int_{\widetilde
B_\rho}|v-v_{\widetilde Q_\rho}|^2\d y \nn\\
&\leq C|v|_{C^{\alpha_0,\alpha_0/2}(\widetilde Q_{1/2})}^2 \leq
C\|v\|_{L^2(\widetilde Q_1)}^2 \leq C\max_{t\in\widetilde
I_1}\|v\|_{L^2(\widetilde B_1)}^2
 .
\end{align*}
Therefore,
\begin{align*}
&\max_{t\in\widetilde I_\rho}\|\tilde u-\tilde u_{\widetilde Q_\rho}\|_{L^2(\widetilde B_\rho)}^2\nn\\[3pt]
&\leq C\max_{t\in\widetilde I_\rho}\|v-v_{\widetilde
Q_\rho}\|_{L^2(\widetilde B_\rho)}^2+C\max_{t\in
\widetilde I_\rho}\|w-w_{\widetilde Q_\rho}\|_{L^2(\widetilde B_\rho)}^2 \nn\\
&\leq C\rho^{n+2\alpha_0}\max_{t\in\widetilde
I_1}\|v\|_{L^2(\widetilde B_1)}^2+C\max_{t\in\widetilde
I_1}\|w\|_{L^2(\widetilde B_1)}^2   \nn\\[3pt]
&\leq C\rho^{n+2\alpha_0}\max_{t\in\widetilde I_1}\|\tilde u-\tilde
u_{\widetilde Q_1}\|_{L^2(\widetilde B_1)}^2+C\max_{t\in\widetilde
I_1}\|w\|_{L^2(\widetilde B_1)}^2 \nn \\
&\leq C\rho^{n+2\alpha_0}\max_{t\in\widetilde I_1}\|\tilde u-\tilde
u_{\widetilde Q_1}\|_{L^2(\widetilde B_1)}^2+CR^2\|\tilde f\|_{L^2(\widetilde Q_1)}^2  \nn\\
&\leq C\rho^{n+2\alpha_0}\max_{t\in\widetilde I_1}
\|\tilde u\|_{L^2(\widetilde B_1)}^2+CR^2\|\tilde f\|_{L^2(\widetilde Q_1)}^2  .
\end{align*}
Transforming back to the $(x,t)$-coordinates, we complete the proof
of the
Lemma for $\theta=0$.

Then we note that $u-\theta$ is also a
solution to the equation (\ref{parabEq2}) in $Q_R$ for any $\theta\in\R$.\qed\medskip

In a similar way, we can prove the following lemmas and propositions.

\begin{lemma}\label{Lemma005}
{\it Let $x_0\in\Omega$ and $t_0=0$. There exists $\alpha_0\in(0,1)$
and $C>0$ such that if $u$ is the solution of {\rm (\ref{parabEq2})}
in $Q_R=B_R(x_0)\times \underline{I}_R$ with
$\underline{I}_R=[0,R^2]$, then
\begin{align*}
&\max_{t\in \underline{I}_\rho}\|u\|_{L^2(B_\rho)}^2 \leq
C\biggl(\frac{\rho}{R}\biggl)^{n+2\alpha_0}\!\!\!\!\!\!\max_{t\in
\underline{I}_R}\|u\|_{L^2(B_R)}^2  +C\|\vec{f}\|_{L^2(Q_R)}^2
\end{align*}
holds for any $0<\rho\leq R\leq \min({\rm
dist}(x_0,\partial\Omega),\sqrt{T})$,
where $C$ and $\alpha_0$ depend only on $K$ and
$n$. }
\end{lemma}

\begin{lemma}\label{Lemma006}
{\it Let $x_0\in\partial\Omega$ and $t_0>0$. There exists
$\alpha_0\in(0,1)$ and $C>0$ such that if $u$ is the solution of {\rm
(\ref{parabEq2})} in $Q_R=B_R\times I_R$ with
$B_R=B_R(x_0)\cap\Omega$ and $I_R=(t_0-R^2,t_0]$, then
\begin{align*}
&\max_{t\in I_\rho}\|u\|_{L^2(B_\rho)}^2\leq
C\biggl(\frac{\rho}{R}\biggl)^{n+2\alpha_0}\max_{t\in
I_R}\|u\|_{L^2(B_R)}^2 +C\|\vec{f}\|_{L^2(Q_R)}^2
\end{align*}
holds for any  $0<\rho\leq R\leq
\min(R_\Omega,\sqrt{t_0})$, where $C$ and $\alpha_0$ depend only on $K$, $n$ and $\Omega$. }
\end{lemma}

\begin{lemma}\label{Lemma007}
{\it Let $x_0\in\partial\Omega$ and $t_0=0$. There exists
$\alpha_0\in(0,1)$ and $C>0$ such that if $u$ is the solution to
{\rm(\ref{parabEq2})} in $Q_R=B_R\times\underline{ I}_R$, with
$B_R=B_R(x_0)\cap\Omega$ and $\underline{I}_R=[0,R^2]$, then
\begin{align*}
&\max_{t\in \underline{I}_\rho}\|u\|_{L^2(B_\rho)}^2\leq
C\biggl(\frac{\rho}{R}\biggl)^{n+2\alpha_0}\!\!\!\!\!\!\max_{t\in
\underline{I}_R}\|u\|_{L^2(B_R)}^2 +C\|\vec{f}\|_{L^2(Q_R)}^2
\end{align*}
holds for any $0<\rho\leq R\leq
\min(R_\Omega,\sqrt{T})$, where $C$ and $\alpha_0$ depend only on $K$, $n$ and $\Omega$. }
\end{lemma}

From the above lemmas, using Lemma \ref{jkddlafhlae} we can derive
the following results concerning the solution of (\ref{parabEq2}).

\setcounter{proposition}{\value{lemma}} \addtocounter{lemma}{1}
\begin{proposition}\label{dnflkqhklqewad2}
{\it For $x_0\in\Omega$, $t_0>0$, $Q_R=B_R(x_0)\times(t_0-R^2,t_0]$
and $0<\rho\leq R\leq\min\big({\rm
dist}(x_0,\partial\Omega),\sqrt{t_0}\big)$, we have
\begin{align*}
&\|u-u_{Q_\rho}\|_{L^\infty((t_0-\rho^2,t_0);L^2(B_\rho))}^2\leq
C\biggl(\frac{1}{R^{n}}\|u\|_{L^\infty((t_0-R^2,t_0);L^2(B_{R}))}^2
+\|\vec{f}\|_{L^{2,n/(n+2)}(\Omega_T)}^2\biggl)\rho^n .\nn
\end{align*}
}
\end{proposition}

\addtocounter{lemma}{1}
\begin{proposition}\label{dnflkqhklqewad12}
{\it For $x_0\in\Omega$, $t_0=0$, $Q_R=B_R(x_0)\times[0,R^2]$ and
$0<\rho\leq R\leq\min\big({\rm
dist}(x_0,\partial\Omega),\sqrt{T}\big)$, we have
\begin{align*}
&\|u\|_{L^\infty((0,\rho^2);L^2(B_\rho))}^2\leq
C\biggl(\frac{1}{R^{n}}\|u\|_{L^\infty((0,R^2);L^2(B_{R}))}^2
+\|\vec{f}\|_{L^{2,n/(n+2)}(\Omega_T)}^2\biggl)\rho^n . \nn
\end{align*}
}
\end{proposition}

\begin{proposition}\label{dnflkqhklqewad22}
{\it For $x_0\in\partial \Omega$, $t_0>0$,
$Q_R=B_R(x_0)\cap\Omega\times(t_0-R^2,t_0]$ and
$0<\rho<R\leq\min(R_\Omega,\sqrt{t_0})$, we have
\begin{align*}
&\|u\|_{L^\infty((t_0-\rho^2,t_0);L^2(B_\rho))}^2\leq
C\biggl(\frac{1}{R^{n}}\|u\|_{L^\infty((t_0-R^2,t_0);L^2(B_{R}))}^2
+\|\vec{f}\|_{L^{2,n/(n+2)}(\Omega_T)}^2\biggl)\rho^n . \nn
\end{align*}
}
\end{proposition}

\begin{proposition}\label{dnflkqhklqewad32}
{\it For $x_0\in\partial \Omega$, $t_0=0$,
$Q_R=B_R(x_0)\cap\Omega\times[0,R^2]$ and $0<\rho<R\leq
\min(R_\Omega,\sqrt{T})$, we have
\begin{align*}
&\|u\|_{L^\infty((0,\rho^2);L^2(B_\rho))}^2\leq C\biggl(\frac{1}{R^{n}}\|u\|_{L^\infty((0,R^2);L^2(B_{R}))}^2
+\|\vec{f}\|_{L^{2,n/(n+2)}(\Omega_T)}^2\biggl)\rho^n . \nn
\end{align*}
}
\end{proposition}

With the above propositions and following the outline of Section
\ref{glbest1}, we can prove the global BMO estimate below.
\begin{proposition}\label{njdfklahkeas2222}
{\it The Propositions
{\rm\ref{dnflkqhklqewad2}--\ref{dnflkqhklqewad32}} imply that the solution of {\rm (\ref{parabEq2})} satisfies that
\begin{align}\label{BMOest22}
\|u\|_{L^\infty((0,T);{\rm \overline{BMO}})} \leq C\|\vec{f}\|_{L^{2,n/(n+2)}(\Omega_T)},
\end{align}
where $C$ depends only on $K$, $n$ and $\Omega$
$($independent of $T)$. }
\end{proposition}

\section{H\"{o}lder estimate of parabolic equations}\label{sec5}
\setcounter{equation}{0}

In this section, we list the propositions to be used in deriving (\ref{mainest2}). We omit the proof of these propositions, since it is very similar as the last section,
The reason we keep these propositions in this section is that some of them are also used in the next section to prove global well-posedness of the degenerate thermistor problem.

There exist positive constants $\alpha_0$ and $C$ such that the following propositions hold.

\setcounter{proposition}{\value{lemma}} \addtocounter{lemma}{1}
\begin{proposition}\label{dnflkqhklqewad3}
{\it For $x_0\in\Omega$, $t_0>0$, $Q_R=B_R(x_0)\times(t_0-R^2,t_0]$,
$0<2\rho\leq R\leq\min\big({\rm
dist}(x_0,\partial\Omega),\sqrt{t_0}\big)$ and $0<\alpha<\alpha_0$,
the solution of {\rm (\ref{parabEq})} satisfies that
\begin{align*}
&\frac{1}{\rho^{n+2+\alpha}}\|u-u_{Q_\rho}\|_{L^1(Q_\rho)}\leq
C\biggl(\frac{1}{R^{n+2+\alpha}}\|u-\theta\|_{L^1(Q_{R})}
+\frac{1}{R^{n+\alpha}}\|f_0\|_{L^1(Q_R)}\biggl), \nn
\end{align*}
where $\theta$ is an arbitrary constant. }
\end{proposition}

\addtocounter{lemma}{1}
\begin{proposition}\label{dnflkqhklqewad13}
{\it For $x_0\in\Omega$, $t_0=0$, $Q_R=B_R(x_0)\times[0,R^2]$,
$0<\rho\leq R\leq\min\big({\rm
dist}(x_0,\partial\Omega),\sqrt{T}\big)$ and $0<\alpha<\alpha_0$,
the solution of {\rm (\ref{parabEq})} satisfies that
\begin{align*}
&\frac{1}{\rho^{n+2+\alpha}}\|u\|_{L^1(Q_\rho)}\leq C\biggl(\frac{1}{R^{n+2+\alpha}}\|u\|_{L^1(Q_{R})}
+\frac{1}{R^{n+\alpha}}\|f_0\|_{L^1(Q_R)}\biggl)
. \nn
\end{align*}
}
\end{proposition}

\begin{proposition}\label{dnflkqhklqewad23}
{\it For $x_0\in\partial \Omega$, $t_0>0$,
$Q_R=B_R(x_0)\cap\Omega\times(t_0-R^2,t_0]$, $0<\rho\leq
R\leq\min(R_\Omega,\sqrt{t_0})$ and $0<\alpha<\alpha_0$, the
solution of {\rm (\ref{parabEq})} satisfies that
\begin{align*}
&\frac{1}{\rho^{n+2+\alpha}}\|u\|_{L^1(Q_\rho)}\leq
C\biggl(\frac{1}{R^{n+2+\alpha}}\|u\|_{L^1(Q_{R})}
+\frac{1}{R^{n+\alpha}}\|f_0\|_{L^1(Q_R)}\biggl) . \nn
\end{align*}
}
\end{proposition}

\begin{proposition}\label{dnflkqhklqewad33}
{\it For $x_0\in\partial \Omega$, $t_0=0$,
$Q_R=B_R(x_0)\cap\Omega\times[0,R^2]$, $0<\rho\leq R\leq
\min(R_\Omega,\sqrt{T})$  and $0<\alpha<\alpha_0$, the solution of
{\rm (\ref{parabEq})} satisfies that
\begin{align*}
&\frac{1}{\rho^{n+2+\alpha}}\|u\|_{L^1(Q_\rho)}\leq
C\biggl(\frac{1}{R^{n+2+\alpha}}\|u\|_{L^1(Q_{R})}
+\frac{1}{R^{n+\alpha}}\|f_0\|_{L^1(Q_R)}\biggl)
. \nn
\end{align*}
}
\end{proposition}

With the above propositions and following the outline of Section
\ref{glbest1}, we can derive the following estimate in terms of the Campanato space.
\begin{proposition}\label{njdfklahkeas3333}
{\it The solution of {\rm(\ref{parabEq})} satisfies that
\begin{align}\label{BMOest333}
\|u\|_{{\cal L}^{1,1+\alpha/(n+2)}_{\rm para}(\Omega_T)} \leq
 C\|f_0\|_{L^{1,(n+\alpha)/(n+2)}(\Omega_T)},
\end{align}
where $C$ depends only on $K$, $n$ and $\Omega$
$($independent of $T)$. }
\end{proposition}

The local and global estimates in ${\cal L}^{2,\theta}_{\rm para}(\Omega_T)$
follow in a similar way. To conclude, we have

\begin{proposition}\label{dnflkqaghu}
{\it For $x_0\in\Omega$, $t_0>0$, $Q_R=B_R(x_0)\times(t_0-R^2,t_0]$,
$0<2\rho\leq R\leq\min\big({\rm
dist}(x_0,\partial\Omega),\sqrt{t_0}\big)$ and $0<\alpha<\alpha_0$,
the solution of {\rm (\ref{parabEq2})} satisfies that
\begin{align*}
&\frac{1}{\rho^{n+2+2\alpha}}
\|u-u_{Q_\rho}\|_{L^2(Q_\rho)}^2\leq
C\biggl(\frac{1}{R^{n+2+2\alpha}}
\|u-\theta\|_{L^2(Q_{R})}^2
+\frac{1}{R^{n+2\alpha}}\|\vec{f}\|_{L^2(Q_R)}^2
\biggl), \nn
\end{align*}
where $\theta$ is an arbitrary constant. }
\end{proposition}

\addtocounter{lemma}{1}
\begin{proposition}\label{dnflkqas13}
{\it For $x_0\in\Omega$, $t_0=0$,
$Q_R=B_R(x_0)\cap\Omega\times[0,R^2]$, $0<\rho\leq
R\leq\min\big({\rm dist}(x_0,\partial\Omega),\sqrt{T}\big)$ and
$0<\alpha<\alpha_0$, the solution of {\rm (\ref{parabEq2})}
satisfies that
\begin{align*}
&\frac{1}{\rho^{n+2+2\alpha}}
\|u\|_{L^2(Q_\rho)}^2\leq C\biggl(\frac{1}{R^{n+2+2\alpha}}
\|u\|_{L^2(Q_{R})}^2
+\frac{1}{R^{n+2\alpha}}\|\vec{f}\|_{L^2(Q_R)}^2
\biggl)
. \nn
\end{align*}
}
\end{proposition}

\begin{proposition}\label{dnflkqhksgd23}
{\it For $x_0\in\partial \Omega$, $t_0>0$,
$Q_R=B_R(x_0)\cap\Omega\times(t_0-R^2,t_0]$,
$0<\rho<R\leq\min(R_\Omega,\sqrt{t_0})$ and $0<\alpha<\alpha_0$, the
solution of {\rm (\ref{parabEq2})} satisfies that
\begin{align*}
&\frac{1}{\rho^{n+2+2\alpha}}\|u\|_{L^1(Q_\rho)}\leq
C\biggl(\frac{1}{R^{n+2+2\alpha}}\|u\|_{L^2(Q_{R})}^2
+\frac{1}{R^{n+2\alpha}}\|\vec{f}\|_{L^2(Q_R)}^2
\biggl) . \nn
\end{align*}
}
\end{proposition}

\begin{proposition}\label{dnflkqsady63}
{\it For $x_0\in\partial \Omega$, $t_0=0$,
$Q_R=B_R(x_0)\cap\Omega\times[0,R^2]$, $0<\rho\leq R\leq
\min(R_\Omega,\sqrt{T})$ and $0<\alpha<\alpha_0$, the solution of
{\rm (\ref{parabEq2})} satisfies that
\begin{align*}
&\frac{1}{\rho^{n+2+2\alpha}}\|u\|_{L^2(Q_\rho)}^2\leq
C\biggl(\frac{1}{R^{n+2+2\alpha}}\|u\|_{L^2(Q_{R})}^2
+\frac{1}{R^{n+2\alpha}}\|\vec{f}\|_{L^2(Q_R)}^2
\biggl)
. \nn
\end{align*}
}
\end{proposition}

\begin{proposition}\label{njdfklah5yj}
{\it The solution of {\rm(\ref{parabEq2})} satisfies that
\begin{align}\label{BMOest333}
\|u\|_{{\cal L}^{2,1+2\alpha/(n+2)}_{\rm para}(\Omega_T)} \leq
C\|\vec{f}\|_{L^{2,(n+2\alpha)/(n+2)}(\Omega_T)}
,
\end{align}
where $C$ depends only on $K$, $n$ and $\Omega$
$($independent of $T)$. }
\end{proposition}

Proposition \ref{njdfklahkeas3333} and Proposition
\ref{njdfklah5yj} imply the global H\"{o}lder estimate
(\ref{mainest2}).

\bigskip

\section{The degenerate thermistor problem}\label{fsdjkljl78687}
\setcounter{equation}{0}

In this section, we prove Theorem \ref{mainthm} concerning
global well-posedness of the degenerate thermistor problem.
Before we prove the theorem, we introduce some lemmas to be used.

\subsection{Preliminaries}

\begin{lemma}\label{BMOandrecipr}
{\it Let $p>0$. If $u\in$ BMO$(\R^n)$, $u\geq 0$, and $C_1+C_2|s|^p\leq\rho(s)\leq C_3+C_4|s|^p$ for $s\geq 0$, then $\rho(u)$ is a $A_2$ weight in the sense that
\begin{align*}
\biggl(\frac{1}{|B|}\int_{B}\rho(u)\d
x\biggl)\biggl(\frac{1}{|B|}\int_{B}\frac{1}{\rho(u)}\d x\biggl)\leq
C
\end{align*}
for any ball $B\subset\R^n$, where the constant $C$ depends on $C_1,C_2,C_3,C_4, p$ and $\|u\|_{\rm BMO}$.
}
\end{lemma}
\noindent{\it Proof}~~
For any ball $B\subset\R^n$, we set $B_1=\{x\in B|\, |u(x)-u_B|<\frac{1}{2}u_B\}$ and $B_2=B\backslash B_1$. By the Nirenberg inequality \cite{Grafakos} we have
$|B_2|/|B|\leq e^{-Cu_B/\|u\|_{\rm BMO}}$. Clearly, $\rho(u)\geq C\rho(u_B)$ on $B_1$. Therefore,
\begin{align*}
\frac{1}{|B|}\int_{B}\rho(u)\d x
&\leq \frac{C}{|B|}\int_{B}(1+|u-u_B|^p)\d x + \frac{C}{|B|}\int_{B}|u_B|^p\d x\\
&\leq C + C|u_B|^p\leq C\rho(u_B) ,\\[8pt]
\frac{1}{|B|}\int_{B}\frac{1}{\rho(u)}\d x
&\leq \frac{1}{|B|}\int_{B_1}\frac{1}{\rho(u)}\d x + \frac{C|B_2|}{|B|}\\
&\leq \frac{C}{\rho(u_B)}+e^{-Cu_B/\|u\|_{\rm BMO}}
\leq \frac{C}{\rho(u_B)} .
\end{align*}
The last two inequalities imply that $\rho(u)$ is a $A_2$ weight.
\qed

The following lemma concerns maximal regularity of parabolic
equations, which is an application of the maximal regularity of \cite{Wood} and  \cite{JK} (with the perturbation method for the
treatment of operators with merely continuous coefficients).
\begin{lemma}\label{maxregpara}
{\it Let $u$ be the solution of the
parabolic problem {\rm (\ref{parabEq0})} in $\R^n$ $(n=2,3)$
with the Dirichlet boundary/initial conditions $u\equiv g\equiv f_0\equiv 0$, and assume that the coefficient matrix $A$ is continuous. Then we have
$$
\|u\|_{L^p(I;W^{1,q}(\Omega))}\leq
C_{p,q}\|\vec{f}\|_{L^p(I;L^q(\Omega))}
$$
for some $q>n$ and any $1<p<\infty$. The constant $C_{p,q}$ depends only on $p,q$, $K$, the domain $\Omega$ and the modulo of continuity of $A$.
}
\end{lemma}

The analogus result for elliptic equations is given below, which can be proved by applying the $W^{1,q}$ estimate of \cite{JK} with a perturbation argument.
\begin{lemma}\label{maxregellp}
{\it
Let $A_{ij}$, $i,j=1,\cdots,n$, be continuous functions defined on
$\Omega$, satisfying
\begin{align*}
K^{-1}|\xi|^2\leq \sum_{i,j=1}^n A_{ij}(x)\xi_i\xi_j \leq K|\xi|^2
,\quad\mbox{for all}~\xi\in\R^n,~~\mbox{a.e. $x\in\R^{n}$ $(n=2,3)$},
\end{align*}
where $K$ is a
positive constant.
Let $u$ be the solution of the
elliptic equation
$$
-\nabla\cdot(A\nabla
u)=\nabla\cdot\vec{f}\quad\mbox{in}~~\Omega,
$$
with the Dirichlet boundary/initial conditions $u=0$
on $\partial\Omega$. Then we have
$$
\|u\|_{W^{1,q}(\Omega)}\leq
C_{q}\|\vec{f}\|_{L^q(\Omega)}
$$
for some $q>n$. The constant $C_{q}$ depends only on $q$, $\Lambda$, the domain $\Omega$ and the modulo of continuity of $A$.
}
\end{lemma}

The following lemma is concerned with H\"{o}lder estimates for inhomogeneous parabolic equations \cite{AS}, which is also a consequence of Theorem \ref{Mainresult}.
\begin{lemma}\label{Mainresult4}
{\it The solution of {\rm(\ref{parabEq0})} with $u_0\equiv g\equiv 0$
satisfies that
\begin{align*}
&\|u\|_{C^{\alpha,\alpha/2}(\overline{\Omega}_T)}
\leq C(
\|f_0\|_{L^{p}((0,T);L^{q}(\Omega))}+\|{\vec
f}\|_{L^{2p}((0,T);L^{2q}(\Omega))}),
\end{align*}
for some $0<\alpha<1$, provided $1\leq p,q\leq\infty$ and $\,
2/p+n/q<2$. }
\end{lemma}

The following lemma concerns an estimate of $\nabla u$ in the Morrey space for the parabolic equation (\ref{parabEq0}), which was proved in \cite{Yin} for $u_0\equiv g\equiv f_0\equiv 0$.
\begin{lemma}\label{Mainrasdh8}
{\it The solution of {\rm(\ref{parabEq0})} with $f_0\equiv 0$
satisfies that
\begin{align*}
&\|\nabla u\|_{L^{2,n/(n+2)}_{\rm para}(\Omega_T)} \leq C(\|{\vec
f}\|_{L^{2,n/(n+2)}_{\rm para}(\Omega_T)} +\|\nabla
g\|_{L^{2,n/(n+2)}_{\rm
para}(\Omega_T)}+\|\partial_tg\|_{L^{2,n/(n+2)}_{\rm
para}(\Omega_T)}+\|u_0\|_{L^\infty(\Omega)}).
\end{align*}
}
\end{lemma}

\subsection{Construction of approximating solutions}

For the non-degenerate problem, the existence of a $C^\alpha$ solution was proved by Yuan and Lin \cite{YL,Yuan2}. Based on their result, for any given $\varepsilon>0$, there exists a weak solution $(u^\varepsilon,\phi^\varepsilon)$ such that $\phi^\varepsilon\in L^\infty((0,T);H^1(\Omega))$
and $u^\varepsilon\in C^{\alpha,\alpha/2}(\overline\Omega_T)\cap L^2((0,T);H^1(\Omega))$, to the following equations
\begin{align}\label{sdbgkfo99033}
\left\{
\begin{array}{ll}
\displaystyle
\frac{\partial u^\varepsilon}{\partial t}-\nabla\cdot(\kappa(u^\varepsilon)
\nabla
u^\varepsilon)=\nabla\cdot[
(\sigma(u^\varepsilon)+\varepsilon)\phi^\varepsilon\nabla\phi^\varepsilon]
&\mbox{in}~~\Omega,
\\[5pt]
u^\varepsilon=g  & \mbox{on}~~\partial\Omega ,\\
u^\varepsilon(x,0)=u_0(x)  & \mbox{for}~~x\in \Omega ,
\end{array}\right.
\end{align}
\begin{align}\label{sfdhyy733}
\left\{
\begin{array}{ll}
\displaystyle -\nabla\cdot\big(
(\sigma(u^\varepsilon)+\varepsilon)
\nabla\phi^\varepsilon\big)=0 &\mbox{in}~~\Omega,
\\[3pt]
\phi^\varepsilon=h & \mbox{on}~~\partial\Omega .
\end{array}\right.
\end{align}
We also note that, by the maximum principle, the solution $u^\varepsilon$ of  (\ref{sdbgkfo99033}) satisfies that
\begin{align}
u^\varepsilon\geq c:=
\min\big(\min_{x\in\Omega}u_0(x),\min_{x\in\partial\Omega}g(x)\big)>0
,
\end{align}
and the solution $\phi$ of (\ref{sfdhyy733}) satisfies that
\begin{align}\label{sdfjkljoo}
\|\phi^\varepsilon\|_{L^\infty(\Omega_T)}\leq \|h\|_{L^\infty(\Gamma_T)} .
\end{align}
By the hypotheses (H1)-(H2), we have
\begin{align}
\kappa_0\leq\kappa(u^\varepsilon)\leq \kappa_1,\qquad \varepsilon\leq\sigma(u^\varepsilon)+\varepsilon\leq 2\sigma_0:=\sup_{s\geq c}\,\sigma(s),
\end{align}
for some positive constants $\kappa_0,\kappa_1$ and $\sigma_0$, where we choose $\varepsilon<\sigma_0$.

\begin{proposition}\label{fdsjklhiy87979}
{\it
The solution $(u^\varepsilon,\phi^\varepsilon)$ of {\rm (\ref{sdbgkfo99033})-(\ref{sfdhyy733})} satisfies that
\begin{align*}
&\|u^\varepsilon\|_{C^{\alpha,\alpha/2}(\overline\Omega_T)}
+\|u^\varepsilon\|_{L^p((0,T);W^{1,q}(\Omega))}
+\|\partial_tu^\varepsilon\|_{L^p((0,T);W^{-1,q}(\Omega))}
+\|\phi^\varepsilon\|_{L^\infty((0,T);W^{1,q}(\Omega))}\leq C ,
\end{align*}
and
$$
\|\phi^\varepsilon\|_{C^{\alpha,\alpha/2}(\overline B_R\times[0,T])}\leq C_{{\rm dist}(\overline B_R,\partial\Omega)}
$$
for any closed ball $\overline B_R\subset\Omega$, where the constants $C$ and $C_{{\rm dist}(\overline B_R,\partial\Omega)} $ are independent of $\varepsilon$.
}
\end{proposition}

\noindent{\it Proof}~~~
First, we show that $\sigma(u^\varepsilon)+\varepsilon$ is a $A_2$ weight, uniformly with respect to time and $\varepsilon$.

Let $x_0\in\Omega$, $t_0>0$ and let
$R_0=\frac{1}{2}\min(\sqrt{t_0},{\rm dist}(x_0,\partial\Omega))$.
For any ball $B_R$ of radius $R$ centered at $x_0$, we let $\zeta$
be a smooth function defined on $\R^n$ which satisfies $0\leq
\zeta\leq 1$, $\zeta=1$ in $B_R$ and $\zeta=0$ outside $B_{2R}$. For
any interval $I_R=(t_0-R^2,t_0]$, we let $\chi$ be a smooth function
defined on $\R$ which satisfies $0\leq \chi\leq 1$, $\chi=1$ on
$I_R$ and $\chi=0$ on
$(-\infty,t_0-4R^2]$.
Let $Q_R=B_R\times I_R$ so that $(u^\varepsilon,\phi^\varepsilon)$ is a
solution of (\ref{sdbgkfo99033})-(\ref{sfdhyy733}) in $Q_{2R_0}$.
Multiplying (\ref{sfdhyy733}) by $\varphi=\phi^\varepsilon\zeta^2$, we obtain
\begin{align*}
\int_{B_R}(\sigma(u^\varepsilon)+\varepsilon)|\nabla\phi^\varepsilon|^2\d
x\leq\int_{B_{2R}}
(\sigma(u^\varepsilon)+\varepsilon)
|\phi^\varepsilon|^2|\nabla\zeta|^2\d x\leq
C\|\phi^\varepsilon\|_{L^\infty(\Omega)}^2R^{n-2} .
\end{align*}
Integrating the above inequality with respect to time and using (\ref{sdfjkljoo}), we get
\begin{align}\label{jkdfabe}
\iint_{Q_R}(\sigma(u^\varepsilon)+\varepsilon)|\nabla\phi^\varepsilon|^2\d x\d t\leq
C\|h\|_{L^\infty(\Gamma_T)}^2R^n.
\end{align}
Similarly, for $x_0\in\partial\Omega$, $t_0>0$, $R<\frac{1}{2}\min(\sqrt{t_0},R_\Omega)$, $B_R:=B_R(x_0)\cap\Omega$ and $Q_R=Q_R(x_0,t_0)\cap\Omega_T$, we also have (\ref{jkdfabe}).
From the last inequality we see that
\begin{align}\label{jkdfabe1}
\big\|\sqrt{\sigma(u^\varepsilon)+\varepsilon}
\nabla\phi^\varepsilon\big \|_{L^{2,n/(n+2)}_{\rm
para}(\Omega_T)}\leq C .
\end{align}

By Theorem \ref{Mainresult}, the solution of (\ref{sdbgkfo99033}) satisfies that
\begin{align}
\|u^\varepsilon\|_{L^\infty((0,T);\overline{{\rm BMO}})}\leq
C\big\|\sqrt{\sigma(u^\varepsilon)+\varepsilon}
\nabla\phi^\varepsilon\big \|_{L^{2,n/(n+2)}_{\rm
para}(\Omega_T)}+C\|u_0\|_{L^\infty(\Omega)}+C\|g\|_{L^\infty(\Gamma_T)}\leq
C .
\end{align}
Applying Lemma \ref{Mainrasdh8} to the equation (\ref{sdbgkfo99033}) and using (\ref{jkdfabe1}), we derive that
\begin{align}\label{sfdjkl79009}
&\|\nabla u^\varepsilon\|_{L^{2,n/(n+2)}_{\rm para}(\Omega_T)} \leq
C\big\|\sqrt{\sigma(u^\varepsilon)+\varepsilon}
\nabla\phi^\varepsilon\big \|_{L^{2,n/(n+2)}_{\rm
para}(\Omega_T)}+C\leq C .
\end{align}

We extend the function $u^\varepsilon$ defined on $\Omega$ to $\R^n$ by setting
$u^\varepsilon(x)=c$ for $x\in\R^n\backslash\Omega$ so that
$$\|u^\varepsilon\|_{L^\infty((0,T); {\rm BMO}(\R^n))}\leq C.$$

Since (\ref{dfsjljiowuo}) holds, from Lemma \ref{BMOandrecipr} we see that $\rho(u^\varepsilon)$ (and also $\sigma(u^\varepsilon)=1/\rho(u^\varepsilon)$) is a $A_2$ weight uniform with respect to time and $\varepsilon$.
It follows that, for any ball $B\subset \R^n$,
\begin{align*}
&\biggl(\frac{1}{|B|}\int_B(\sigma(u^\varepsilon)+\varepsilon) \d x\biggl)\biggl(\frac{1}{|B|}\int_B\frac{1}{\sigma(u^\varepsilon)+\varepsilon}\d x\biggl)  \\\
&=\biggl(\frac{1}{|B|}\int_B\sigma(u^\varepsilon) \d x\biggl)\biggl(\frac{1}{|B|}\int_B\frac{1}{\sigma(u^\varepsilon)+\varepsilon}\d x\biggl)+\frac{1}{|B|}\int_B\frac{\varepsilon}{\sigma(u^\varepsilon)+\varepsilon}\d x\\
&\leq \biggl(\frac{1}{|B|}\int_B\sigma(u^\varepsilon) \d x\biggl)\biggl(\frac{1}{|B|}\int_B\frac{1}{\sigma(u^\varepsilon)}\d x\biggl)+1\\
&\leq C ,
\end{align*}
which says that $\sigma(u^\varepsilon)+\varepsilon$ is also a $A_2$ weight, uniform with respect to time and $\varepsilon$.

Secondly, we estimate the H\"{o}lder norms of $\phi^\varepsilon$ and $u^\varepsilon$, respectively. In fact,
from \cite{FKS} we know that any solution of
the elliptic equation (\ref{sfdhyy733}) with the $A_2$ coefficient
$\sigma(u^\varepsilon)+\varepsilon$ satisfies the H\"{o}lder estimates:
\begin{align}\label{holderest}
\|\phi^\varepsilon(\cdot,t)\|_{C^\alpha(\overline\Omega)}\leq
C\|h^\varepsilon(\cdot,t)\|_{C^\alpha(\partial\Omega)}
\leq C,
\quad\mbox{for}~~ t\in(0,T),\quad\forall~\alpha\in(0,\alpha_0),
\end{align}
for some fixed constant $\alpha_0\in (0,1)$.

We proceed to the H\"{o}lder estimate of $u^\varepsilon$. For any fixed $x_0\in\Omega$,
we decompose the function $u^\varepsilon$ as
$u^\varepsilon=u_1^\varepsilon+u_2^\varepsilon$, where $u_1^\varepsilon$ and $u_2^\varepsilon$ are weak solutions of the equations
\begin{align*}
\left\{
\begin{array}{ll}
\displaystyle
\frac{\partial u_1^\varepsilon}{\partial t}-\nabla\cdot(\kappa(u^\varepsilon)
\nabla
u_1^\varepsilon)=0
&\mbox{in}~~\Omega,
\\[5pt]
u_1^\varepsilon=g & \mbox{on}~~\partial\Omega ,\\
u_1^\varepsilon(x,0)=u_0(x) & \mbox{for}~~x\in \Omega ,
\end{array}\right.
\end{align*}
and
\begin{align*}
\left\{
\begin{array}{ll}
\displaystyle
\frac{\partial u_2^\varepsilon}{\partial t}-\nabla\cdot(\kappa(u^\varepsilon)
\nabla
u_2^\varepsilon)=\nabla\cdot\big[
(\phi^\varepsilon-\phi^\varepsilon(x_0,t))
(\sigma(u^\varepsilon)+\varepsilon)
\nabla\phi^\varepsilon\big]
&\mbox{in}~~\Omega,
\\[5pt]
u_2^\varepsilon=0 & \mbox{on}~~\partial\Omega ,\\
u_2^\varepsilon(x,0)=0 & \mbox{for}~~x\in \Omega ,
\end{array}\right.
\end{align*}
respectively.
By the De Giorgi--Nash--Moser estimates, we have
$$\|u_1^\varepsilon\|_{C^{\alpha,\alpha/2}(\overline\Omega_T)}\leq C(\|g\|_{C^{\alpha,\alpha/2}(\overline\Gamma_T)}+\|u_0\|_{C^\alpha(\overline\Omega)}).$$
and in order to estimate $\|u_2^\varepsilon\|_{C^{\alpha,\alpha/2}(\overline\Omega_T)}$, we set $\vec{f}=(\phi^\varepsilon-\phi^\varepsilon(x_0,t))
(\sigma(u^\varepsilon)+\varepsilon)
\nabla\phi^\varepsilon$ and apply Proposition \ref{dnflkqaghu}--Proposition \ref{dnflkqsady63}. We see that
for $x_0\in\Omega$, $t_0>0$, $0<2\rho\leq R\leq\min\big({\rm
dist}(x_0,\partial\Omega),\sqrt{t_0}\big)$,
we have
\begin{align*}
&\frac{1}{\rho^{n+2+2\alpha}}
\|u_2^\varepsilon-(u_2^\varepsilon)_{Q_\rho}\|_{L^2(Q_\rho)}^2\\
&\leq
C\biggl(\frac{1}{R^{n+2+2\alpha}}
\|u_2^\varepsilon-\theta\|_{L^2(Q_{R})}^2
+\frac{1}{R^{n+2\alpha}}\|\vec{f}\|_{L^2(Q_R)}^2
\biggl), \\
&\leq
C\biggl(\frac{1}{R^{n+2+2\alpha}}
\|u_2^\varepsilon-\theta\|_{L^2(Q_{R})}^2
+\frac{1}{R^{n}}\|\phi^\varepsilon
\|_{L^\infty(I;C^\alpha(\overline\Omega))}^2\big\|
\sqrt{\sigma(u^\varepsilon)+\varepsilon}
\nabla\phi^\varepsilon\big\|_{L^2(Q_R)}^2
\biggl) \\
&\leq
C\biggl(\frac{1}{R^{n+2+2\alpha}}
\|u_2^\varepsilon-\theta\|_{L^2(Q_{R})}^2
+\frac{1}{R^{n}}\big\|
\sqrt{\sigma(u^\varepsilon)+\varepsilon}
\nabla\phi^\varepsilon\big\|_{L^2(Q_R)}^2
\biggl) .
\end{align*}
Similarly, for $x_0\in\Omega$, $t_0=0$, $Q_R=B_R(x_0)\times[0,R^2]$ and $0<\rho<R\leq\min\big({\rm
dist}(x_0,\partial\Omega),\sqrt{T}\big)$, we
have
\begin{align*}
&\frac{1}{\rho^{n+2+2\alpha}}
\|u_2^\varepsilon\|_{L^2(Q_\rho)}^2\leq C\biggl(\frac{1}{R^{n+2+2\alpha}}
\|u_2^\varepsilon\|_{L^2(Q_{R})}^2
+\frac{1}{R^{n}}\big\|
\sqrt{\sigma(u^\varepsilon)+\varepsilon}
\nabla\phi^\varepsilon\big\|_{L^2(Q_R)}^2
\biggl)
. \nn
\end{align*}
For $x_0\in\partial \Omega$, $t_0>0$, $Q_R=B_R(x_0)\cap\Omega\times(t_0-R^2,t_0]$ and $0<\rho<R\leq{\rm
dist}(R_\Omega,\sqrt{t_0})$, we have
\begin{align*}
&\frac{1}{\rho^{n+2+2\alpha}}
\|u_2^\varepsilon\|_{L^2(Q_\rho)}^2\leq
C\biggl(\frac{1}{R^{n+2+2\alpha}}\|u_2^\varepsilon\|_{L^2(Q_{R})}^2
+\frac{1}{R^{n}}\big\|
\sqrt{\sigma(u^\varepsilon)+\varepsilon}
\nabla\phi^\varepsilon\big\|_{L^2(Q_R)}^2
\biggl) . \nn
\end{align*}
For $x_0\in\partial \Omega$, $t_0=0$, $Q_R=B_R(x_0)\cap\Omega\times[0,R^2]$ and $0<\rho<R\leq
\min(R_\Omega,\sqrt{T})$, we have
\begin{align*}
&\frac{1}{\rho^{n+2+2\alpha}}
\|u_2^\varepsilon\|_{L^2(Q_\rho)}^2\leq
C\biggl(\frac{1}{R^{n+2+2\alpha}}
\|u_2^\varepsilon\|_{L^2(Q_{R})}^2
+\frac{1}{R^{n}}
\big\|
\sqrt{\sigma(u^\varepsilon)+\varepsilon}
\nabla\phi^\varepsilon\big\|_{L^2(Q_R)}^2
\biggl)
. \nn
\end{align*}

Combining the last four inequalities and following the outline of Section \ref{glbest1}, we can derive that
\begin{align*}
\|u_2^\varepsilon\|_{{\cal L}^{2,1+2\alpha/(n+2)}_{\rm
para}(\Omega_T)} \leq C\big\|
\sqrt{\sigma(u^\varepsilon)+\varepsilon} \nabla\phi^\varepsilon\big
\|_{L^{2,n/(n+2)}_{\rm para}(Q_R)}.
\end{align*}
With (\ref{jkdfabe1}) and the equivalence relation ${\cal L}^{2,1+2\alpha/(n+2)}_{\rm para}(\Omega_T)\cong C^{\alpha,\alpha/2}(\overline\Omega_T)$, we see that
\begin{align*}
\|u_2^\varepsilon\|_{C^{\alpha,\alpha/2}
(\overline\Omega_T)} \leq
C .
\end{align*}
Therefore,
\begin{align}
\|u^\varepsilon\|_{C^{\alpha,\alpha/2}
(\overline\Omega_T)}\leq \|u_1^\varepsilon\|_{C^{\alpha,\alpha/2}
(\overline\Omega_T)}+\|u_2^\varepsilon\|_{C^{\alpha,\alpha/2}
(\overline\Omega_T)} \leq
C .
\end{align}

Thirdly, we present $W^{1,q}$ estimates of $\phi^\varepsilon$ and $u^\varepsilon$.
Note that the last inequality implies that
\begin{align}
C^{-1}\leq \sigma(u^\varepsilon)+\varepsilon\leq C,\quad \|\sigma(u^\varepsilon)+\varepsilon
\|_{C^{\alpha,\alpha/2}(\overline\Omega_T)}
\leq C,\quad  \|\kappa(u^\varepsilon)
\|_{C^{\alpha,\alpha/2}(\overline\Omega_T)}
\leq C.
\end{align}
With the H\"{o}lder estimates of  $\sigma(u^\varepsilon)+\varepsilon$ and $\kappa(u^\varepsilon)$, we
apply Lemma \ref{maxregpara} -- \ref{maxregellp} and derive that
\begin{align}
&\|\phi^\varepsilon\|_{L^\infty((0,T);W^{1,q}(\Omega))} \leq
C\|h\|_{L^\infty((0,T);W^{1,q}
(\Omega))}\leq C
,\\
&\|u^\varepsilon\|_{L^p((0,T);W^{1,q}(\Omega))} \leq
C_p\|\phi^\varepsilon\|_{L^p((0,T);W^{1,q}(\Omega))}+C_p
\leq C_p ,
\end{align}
for some $q>n$ and any $1< p<\infty$. From the equation (\ref{sdbgkfo99033}) we also see that
\begin{align}
\|\partial_tu^\varepsilon\|_{L^p(I;W^{-1,q}(\Omega))} \leq
C(\|u^\varepsilon\|_{L^p((0,T);W^{1,q}(\Omega))}+\|\nabla\phi^\varepsilon\|_{L^p((0,T);W^{1,q}(\Omega))})\leq C.
\end{align}

Finally, we estimate the interior space-time H\"{o}lder norm of $\phi^\varepsilon$, which is used to obtain pointwise convergence of the approximating solutions in the next subsection.
For the simplicity of notations, we set $A^\varepsilon=\sigma(u^\varepsilon)+\varepsilon$. From (\ref{sfdhyy733}) we see that
\begin{align*}
\displaystyle -\nabla\cdot\biggl(
A^\varepsilon(x,t_1)
\nabla[\phi^\varepsilon(x,t_1)
-\phi^\varepsilon(x,t_2)]\biggl)
=\nabla\cdot\biggl(
(A^\varepsilon(x,t_1)-A^\varepsilon(x,t_2))
\nabla\phi^\varepsilon(x,t_2)\biggl) .
\end{align*}
By applying the interior $W^{1,q}$ estimate to the above equation, we find that for any closed ball $\overline B_R$ contained in $\Omega$ there holds
\begin{align*}
\|\phi^\varepsilon(x,t_1)
-\phi^\varepsilon(x,t_2)\|_{
L^\infty((0,T);W^{1,q}(B_R))} &\leq
C_{{\rm dist}(\overline B_R,\partial\Omega)}
\|A^\varepsilon(x,t_1)-A^\varepsilon(x,t_2)
\|_{L^\infty(\Omega_T)}\\
&\leq
C_{{\rm dist}(\overline B_R,\partial\Omega)}
\|A^\varepsilon\|_{C^{\alpha,\alpha/2}
(\overline\Omega_T)}|t_1-t_2|^{\alpha/2} ,
\end{align*}
which reduces to
\begin{align*}
\|\phi^\varepsilon\|_{
C^{\alpha/2}([0,T]
;W^{1,q}(B_R))} \leq
C_{{\rm dist}(\overline B_R,\partial\Omega)} .
\end{align*}
Since $W^{1,q}(B_R)\hookrightarrow C^\alpha(\overline\Omega)$, the last inequality implie that
\begin{align}
\|\phi^\varepsilon\|_{
C^{\alpha,\alpha/2}(\overline B_R\times[0,T])} \leq
C_{{\rm dist}(\overline B_R,\partial\Omega)} .
\end{align}

The proof of Proposition \ref{fdsjklhiy87979} is complete.\qed

\subsection{Existence of solution}
Since $C^{\alpha,\alpha/2}
(\overline\Omega_T)$ is compactly embedded into $C(\overline\Omega_T)$ and $C^{\alpha,\alpha/2}(\overline B_R\times[0,T])$ is compactly embedded into $C(\overline B_R\times[0,T])$
there exist functions $u\in C^{\alpha,\alpha/2}
(\overline\Omega_T)$, $\phi\in L^\infty(I;W^{1,q}(\Omega))$ with $\phi\in C^{\alpha,\alpha/2}(\overline B_R\times[0,T])$ for any closed ball $\overline B_R$ contained in $\Omega$, and a sequence $\varepsilon_k\rightarrow 0$, such that
$u^{\varepsilon_k}$ converges to $u$ in the norm of $C(\overline\Omega_T)$, $u^{\varepsilon_k}$ converges weakly to $u$ in $L^p(I;W^{1,q}(\Omega))$, $\partial_tu^{\varepsilon_k}$ converges weakly to $\partial_tu$ in $L^p(I;W^{-1,q}(\Omega))$, $\phi^{\varepsilon_k}$ converges weakly$^*$ to $\phi$ in $L^\infty(I;W^{1,q}(\Omega))$, and $\phi^{\varepsilon_k}$ converges to $\phi$ pointwise uniformly in each compact subset of $\Omega\times[0,T]$.

From (\ref{sfdhyy733}) we see that
\begin{align*}
\int_\Omega(\sigma(u^{\varepsilon_k})+\varepsilon_k)
\nabla\phi^{\varepsilon_k}\cdot\nabla\varphi \,\d x=0
\quad\mbox{for any $\varphi\in H^1_0(\Omega)$} .
\end{align*}
By taking the limit $k\rightarrow\infty$, we obtain
\begin{align}\label{sdfhjkyo79}
\int_\Omega\sigma(u)\nabla\phi\cdot\nabla\varphi \,\d x=0, \quad\mbox{for any $\varphi\in H^1_0(\Omega)$ and a.e. $t\in (0,T)$.}
\end{align}
Therefore, for any function $v\in C^\infty_0(\Omega)$,
\begin{align*}
\lim_{k\rightarrow\infty}
\int_\Omega\nabla\cdot\biggl(
\phi^{\varepsilon_k}(\sigma(u^{\varepsilon_k})+\varepsilon_k)
\nabla\phi^{\varepsilon_k}\biggl)v \d x
&=
-\lim_{k\rightarrow\infty}\int_\Omega
\phi^{\varepsilon_k}(\sigma(u^{\varepsilon_k})+\varepsilon_k)
\nabla\phi^{\varepsilon_k}\cdot\nabla v \d x\\
&=
-\int_\Omega
\phi\sigma(u)
\nabla\phi\cdot\nabla v\d x\\
&=
-\int_\Omega
\sigma(u)
\nabla\phi\cdot[\nabla(\phi v)-v\nabla\phi] \d x\\
&=\int_\Omega
\sigma(u)
|\nabla\phi|^2v\d x .
\end{align*}

From (\ref{sdbgkfo99033}) we know that for any $v\in L^\infty((0,T);C^\infty_0(\Omega))$,
\begin{align*}
\int_0^T\int_\Omega\frac{\partial u^{\varepsilon_k}}{\partial t}v \d x\d t+\int_0^T\int_\Omega
\kappa(u^{\varepsilon_k})_{\varepsilon_k}
\nabla
u^\varepsilon\cdot\nabla v\d x\d t\\
=\int_0^T\int_\Omega\nabla\cdot\biggl(
\phi^{\varepsilon_k}\frac{1}{
\rho(u^{\varepsilon_k})_{\varepsilon_k}}
\nabla\phi^{\varepsilon_k}\biggl)v\,\d x\d t .
\end{align*}
By taking the limit $k\rightarrow\infty$, we get
\begin{align}\label{sdfhjkyoasdf}
\int_0^T\int_\Omega\frac{\partial u}{\partial t}v\, \d x\d t+\int_0^T\int_\Omega
\kappa(u)
\nabla
u\cdot\nabla v\, \d x\d t=\int_0^T\int_\Omega\sigma(u)
|\nabla\phi|^2v\,\d x\d t .
\end{align}

From the regularity of $u$ and $\phi$, we know that the equations (\ref{sdfhjkyo79})-(\ref{sdfhjkyoasdf}) actually hold for any $\varphi\in H^1_0(\Omega)$ and $v\in L^2((0,T);H^1_0(\Omega))$.

To conclude, we have proved the existence of a weak solution $(u,\phi)$ to the equations (\ref{e-heat-1})-(\ref{BC}) with the regularity (\ref{sdfjhkl79031}).

\subsection{Uniqueness of solution}

Suppose that $(u_1,\phi_1)$ and $(u_2,\phi_2)$ are two pairs of
solutions to the initial-boundary value problem
(\ref{e-heat-1})-(\ref{BC}), both satisfying  (\ref{sdfjhkl79031}). Let $\bar u=u_1-u_2$ and $\bar
\phi=\phi_1-\phi_2$. Then $\bar u$ and $\bar \phi$ are weak solutions to the equations
\begin{align}
&\frac{\partial \bar u}{\partial
t}-\nabla\cdot(\kappa(u_1)\nabla\bar
u)=\nabla\cdot((\kappa(u_1)-\kappa(u_2))\nabla
u_2) \nonumber \\
&~~~+(\sigma(u_1)-\sigma(u_2))|\nabla\phi_1|^2+\sigma(u_2)\nabla(\phi_1+\phi_2)\cdot\nabla\bar\phi
\label{diffEqu}\\[5pt]
&-\nabla\cdot(\sigma(u_1)\nabla\bar\phi)
=\nabla\cdot\Big((\sigma(u_1)-\sigma(u_2))\nabla\phi_2\Big),
\label{diffEqphi}
\end{align}
with the following boundary and initial conditions:
\begin{align}
\begin{array}{ll}
\bar u(x,t)=0,\quad \bar\phi(x,t)=0~~
&\mbox{for}~~x\in\partial\Omega,~~t\in[0,T],\\[3pt]
\bar u(x,0)=0~~ &\mbox{for}~~x\in\Omega .
\end{array}
\end{align}

For any $\tau\in(0,T)$, we denote $I_\tau=(0,\tau)$ and
$\Omega_\tau=\Omega\times I_\tau$. By applying Lemma \ref{Mainresult4}
to the parabolic equation (\ref{diffEqu}), we see that for $q>n$ there
exists $1<p<\infty$ such that
\begin{align*
\|\bar u\|_{L^\infty(\Omega_\tau)}&\leq
C\|(\kappa(u_1)-\kappa(u_2))\nabla u_2\|_{L^p(I_\tau;L^q(\Omega))} \nn\\
&~~~+C\|(\sigma(u_1)-\sigma(u_2))|\nabla\phi_1|^2\|_{L^p(I_\tau;L^{q/2}(\Omega))} \nn\\
&~~~+C\|\sigma(u_2)\nabla(\phi_1+\phi_2)\cdot\nabla\bar\phi\|_{L^p(I_\tau;L^{q/2}(\Omega))} \nn\\
&\leq C\|\bar u\|_{L^\infty(\Omega_\tau)}\big(\tau^{1/2p}\|\nabla
u_2\|_{L^{2p}(I_\tau;L^q(\Omega))}+\|\nabla\phi_1\|_{L^{2p}(I_\tau;L^q(\Omega))}^2\big) \nn\\
&~~~+C\tau^{1/2p}\|\nabla(\phi_1+\phi_2)\|_{L^\infty(I_\tau;L^q(\Omega))}
\|\nabla\bar\phi\|_{L^{2p}(I_\tau;L^q(\Omega))} \nn\\
&\leq C\tau^{1/2p}\|\bar
u\|_{L^\infty(\Omega_\tau)}+C\tau^{1/2p}\|\nabla\bar\phi\|_{L^\infty(I_\tau;L^q(\Omega))}
,
\end{align*}
where the constant $C$ is independent of $\tau$.
With the H\"{o}lder regularity of $u_1$, by applying the $W^{1,q}$
estimates to (\ref{diffEqphi}), we obtain
\begin{align*
\|\nabla\bar\phi\|_{L^\infty(I_\tau;L^q(\Omega))}&\leq
C\|(\sigma(u_1)-\sigma(u_2))\nabla\phi_2\|_{L^\infty(I_\tau;L^q(\Omega))}
\leq C\|\bar u\|_{L^\infty(\Omega_\tau)} .
\end{align*}
There exists $T_0$ such that for $\tau<T_0$, the last two
inequalities imply that
\begin{align*
\|\bar
u\|_{L^\infty(\Omega_\tau)}+\|\nabla\bar\phi\|_{L^\infty(I_\tau;L^q(\Omega))}
=0 .
\end{align*}
By dividing the interval $(0,T)$ into small parts $(T_k,T_{k+1}]$,
$k=0,1,\cdots$, each part satisfying $T_{k+1}-T_k<T_0$, we find that
$\bar u(\cdot,T_k)\equiv\bar \phi(\cdot,T_k)\equiv0$ implies that
$\bar u(\cdot,t)\equiv\bar \phi(\cdot,t)\equiv0$ for
$t\in[T_k,T_{k+1}]$. This proves the uniqueness of solution.

\medskip

\section{Conclusions}
\setcounter{equation}{0}

In this paper, we proved global existence and uniqueness of a weak
solution to the degenerate thermistor problem by establishing a
uniform-in-time BMO estimate for parabolic equations with possibly
discontinuous coefficients. The physical hypothesis (H1)-(H2) are
satisfied by metals and some semiconductors. The BMO estimate of
parabolic equations established in this paper may be applied to many
other equations of mathematical physics.

redpajama_set_name: RedPajamaArXiv


text:
Acer Therapeutics Inc (ACER.OQ)
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UPDATE 2-FDA declines to approve Acer Therapeutics' rare genetic disorder treatment
June 25Acer Therapeutics Inc said on Tuesday the U.S. Food and Drug Administration declined to approve its treatment for a severe, rare genetic disorder that can cause blood vessels to fatally rupture, sending its shares plunging as much as 78%.
FDA declines to approve Acer Therapeutics' treatment for rare genetic disorder
Acer Therapeutics Inc said on Tuesday its treatment for a severe, rare genetic disorder that can cause blood vessels to fatally rupture was declined approval from the U.S. Food and Drug Administration.
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redpajama_set_name: RedPajamaCommonCrawl


Buchegger F., Trowbridge I. S., Liu L. F., White S., Collawn J. F.
Chimeric human/chicken transferrin receptors have been constructed using the polymerase chain reaction. Different regions of the 671-residue external domain of the human transferrin receptor were replaced by the corresponding sequences from the chicken transferrin receptor. As chicken transferrin receptors do not bind human transferrin, functional analysis of such chimeric receptors provides an approach to define the ligand-binding site of the human transferrin receptor. Four of 16 chimeric human/chicken transferrin receptors expressed in chick embryo fibroblasts were efficiently transported to the plasma membrane and displayed on the cell surface. Studies of the four chimeric receptors indicated that binding of human transferrin was abolished if the carboxy terminal 192 amino acids of the human transferrin receptor (residues 569-760) were replaced with the corresponding region from the chicken transferrin receptor. Further, a chimeric receptor in which the carboxy-terminal 72 residues were derived from the chicken transferrin receptor exhibited a 16-fold decrease in binding affinity for human transferrin. In contrast, analysis of the other two chimeric receptors showed that 340 amino acids of the human transferrin receptor external domain more proximal to the transmembrane region (residues 151-490) could be replaced with the corresponding region from the chicken transferrin receptor without loss of high-affinity ligand binding. In contrast, two mAbs against the human transferrin receptor external domain, B3/25 and D65.3, that do not compete with transferrin binding, do not bind the chimeric transferrin receptors in which the membrane proximal part is replaced by chicken sequences, while they do bind the two other chimeric transferrin receptors with high affinity. These data indicate that sequence differences in the carboxy-terminal region of human and chicken transferrin receptor external domains are important for the species specificity of transferrin binding and imply that this portion of the human transferrin receptor is critical for ligand binding.


This rémoulade sauce from the Bountiful cookbook is the perfect companion for crab cakes.
Blend all ingredients in a food processor until vegetables are finely chopped. Season with salt and pepper. Chill until ready to use.


Dr Ulla Rajala was granted a British Academy Small Grant for the study of the Roman pottery of the Nepi Survey Project in 2007. Dr Phil Mills (University of Leicester) carried out the detailed analysis of the Roman surface material. The main fabric series was published in 2011 in the Papers of the British School at Rome, but Rajala and Mills continue to publish more in depth papers that give new insights into the development of this minor town and its hinterland between the third century BC and seventh century AD. They also continue developing the theoretical and methodological concept of ceramiscene and its use in the characterisation of rural landscapes in the Mediterranean and beyond.
Mills, P., and Rajala, U., 2011. 'Interpreting a ceramiscene landscape – the Roman pottery from the Nepi Survey Project', in D. Mladenović and B. Russell (eds.). TRAC 2010. Proceedings of the Twentieth Annual Theoretical Roman Archaeology Conference, Oxford 2010, 1-17. Oxford: Oxbow.
Mills, P., and Rajala, U., 2011. 'The Roman ceramic material from the fieldwalking in the environs of Nepi', Papers of the British School at Rome 79, 147-240.
Mills, P., and Rajala, U., in press. 'Supply and Distribution of Late Roman Coarsewares from the Nepi Survey Project', in LRCW4, Late Roman Coarse Wares, Cooking Wares and Amphorae in the Mediterranean, Thessaloniki, 7-11 April 2011 (BAR International Series). Oxford: Archaeopress.
Rajala, U., and Mills, P., in press. 'Defining Edges and Districts – ceramiscene in the territory of Nepi (VT, Lazio, Italy)', in J. Ikäheimo, A.-K. Salmi and T. Äikäs (eds.). NTAG 2012 (Monographs of the Archaeological Society of Finland 2).


We love wedding photography, since we do it for a living that should be obvious! However, there is so much more to us than photography!
We recently became the proud parents of a baby boy. His name is Zayden and we are basically obsessed. We struggled for 6 years to have him, so now that he is here, we couldn't be happier. He loves to growl like a dinosaur, play with things he is not allowed to have, and laughs at everything. He is the cutest and sweetest baby you'll ever see and we aren't even being biased wen we say that!
We also have 3 fur children who basically run our lives. Charlie, our welsh terrier, has more "go" than the energizer bunny. Della, our Great Dane/German Shepherd mix, looks eerily similar to Yoda and loves to cuddle with anyone and everyone. And last, but definitely not least, our Great Dane Ollie, who weighs more than the average size adult and has more stuffed animals than the average child.
When we are not with our dogs, you will most likely find us spending time with our family or friends. We enjoy anything from wake surfing on the lake, exploring caves out on the Buffalo, to hanging out on a patio! We spend most of our free time outside and with the people we love! We love to travel and are trying to take as many trips as possible with our son before he turns 2 and we have to start paying for his flight tickets. We are also active members of Fellowship Bible Church in Fayetteville and always look forward to the time we get to spend with our church community group.
Aside from being the crazy dog people who love our friends and family, we simply love people. We enjoy getting to know couples and their stories! That's why we enjoy shooting weddings so much! You may think all weddings look the same: bride and groom get ready, they say "I Do," followed by a reception with cake cutting and dancing, yet, weddings are so much more than these traditions! How did that moment make you and your guests feel? What was your most memorable part? Your wedding day is so much more important than just capturing the pretty images. We want you to look back on your photos in 30 years and be reminded how it felt when you said "I do!" We do this by capturing your wedding day from a unique and in the moment viewpoint.
In short, YES, we love taking amazing and beautiful images, but to us, it's more important that your photos tell a story, bring you back to that moment, and make you feel.


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Carbondale Community High School » Administration » Employment Opportunities
Administrative Position: Assistant Principal: Carbondale Community High School District 165 is accepting applications for an Assistant Principal for the 2019-20 school year. This is a 10.5-month full-time administrative position with a salary of $75,000.00. Applicants must have an Illinois Professional Educator License with a General Administrative Endorsement for the position. Teacher evaluation experience preferred. To request an Application for Employment please contact Ms. Lewis at (618) 457-4722, extension 21 or you may download an application below. Submit Letter of Application, Application for Employment, Resume, Copy of Illinois Professional Educator License Credentials and Letters of Reference to: Mr. Stephen Murphy, Superintendent, Carbondale Community High School District 165, Administrative Center, 330 S. Giant City Road, Carbondale, IL 62902. Applications will be accepted until the position is filled. AN EQUAL OPPORTUNITY EMPLOYER. (Posted July 17, 2019)
2019-20 Administrative Position: Dean of Students: Carbondale Community High School District 165 is accepting applications for a Dean of Students for the 2018-19 school year. This is a 10.5-month full-time administrative position with a salary of $69,500.00. Applicants must have an Illinois Professional Educator License or Illinois School Service Personnel License for the position. Three years of experience in secondary education preferred. To request an Application for Employment please contact Ms. Lewis at (618) 457-4722, extension 21 or you may download an application below. Submit Letter of Application, Application for Employment, Resume, Copy of Illinois Professional Educator License Credentials and Letters of Reference to: Mr. Stephen Murphy, Superintendent, Carbondale Community High School District 165, Administrative Center, 330 S. Giant City Road, Carbondale, IL 62902. Applications will be accepted until the position is filled. AN EQUAL OPPORTUNITY EMPLOYER. (Posted July 17, 2019)
Physical Education Teacher (Temporary Full-time As Needed Position): Carbondale Community High School District 165 is accepting applications for a temporary full-time as needed physical education teacher for the 2019-20 school year. Applicants must have Illinois Professional Educators License in the respective area. To request an Application for Employment please contact Ms. Lewis at (618) 457-4722, extension 21 or you may download an application below. Submit Letter of Application, Application for Employment, Resume, Copy of Illinois Professional Educator License Credentials and Letters of Reference to: Mr. Stephen Murphy, Superintendent, Carbondale Community High School District 165 Administrative Center, 330 South Giant City Road, Carbondale, IL 62902. Applications will be accepted until the position is filled. (Posted July 10, 2019)
Teacher Aides and Hall Monitors: Carbondale Community High School District 165 is accepting applications for teacher aides and hall monitors for the 2019-20 school year.
These are non-certified IMRF positions. Applicants must have a valid Illinois Professional Educator License, or an Illinois Paraprofessional Educator License registered in Jackson County.
The rate of pay is $12.50 per hour for individuals with an Illinois Professional Educator License, and $10.82 per hour for individuals with an Illinois Paraprofessional Educator License.
To request an Application for Employment please contact Ms. Lewis at (618) 457-4722, extension 21 or you may download an application below. Applications may be picked up at the Principal's Office, Carbondale Community High School, 1301 East Walnut Street or at the District 165 Administrative Center, 330 South Giant City Road, Carbondale.
To apply for a position, submit the following documents:
1. Application for Employment
2. Copy of University Transcript (an official transcript will be required upon employment)
3. Copy of Illinois Professional/Paraprofessional Educator License Credentials
Ryan Thomas, Principal
1301 East Walnut Street
Applications will be accepted until the positions are filled. AN EQUAL OPPORTUNITY EMPLOYER. (Posted May 17, 2019)
Substitute Employment Opportunities
Substitute Teachers: Carbondale Community High School District 165 is accepting applications for Substitute Teachers for the 2019-20 school year. The Substitute Teacher daily rate of pay is $85.00 (including TRS). Substitute Teachers are employed on an as needed basis. Substitute Teachers are paid by direct deposit on the 25th of each month for the days worked the preceding month. The following documentation is required to be placed on the district roster of Substitute Teachers:
1. Completed Substitute Teacher Employment Packet.
2. Regional Office of Education #30 Substitute Teacher Authorization Form with verification that it is registered for the 2019-20 school year. Contact the ROE #30 Annex Office, 819 Walnut Street, Murphysboro, IL 62966, Telephone: 618-687-7290 to obtain the Authorization Form.
Contact Ms. Lewis at (618) 457-4722, extension 21 to request a CCHS District 165 Substitute Teacher Employment Packet or you may pick one up at the CCHS District 165 Administrative Center, 330 South Giant City Road, Carbondale. AN EQUAL OPPORTUNITY EMPLOYER.
Substitute Clerical Workers/Substitute Cafeteria Workers/Substitute Maintenance Workers: Carbondale Community High School District 165 is accepting applications for CCHS District 165 Substitute Support Staff (substitute clerical workers, cafeteria workers and maintenance workers). Substitute personnel are employed on an as needed basis and are paid by direct deposit. Substitute Clerical and Cafeteria Workers are paid minimum wage per hour. Substitute Maintenance Workers are paid $10.13 per hour. The following documentation is required to be placed on the district roster of substitute support staff:
1. Completed Substitute Support Staff Employment Packet
2. Copy of Fingerprint-Based Criminal History Background Investigation Results (ISP and FBI)
3. Copy of Physical Health Examination.
Contact Ms. Kelso at (618) 457-4722, extension 28 to request a CCHS District 165 Substitute Support Staff Employment Packet or you may pick one up at the CCHS District 165 Administrative Center, 330 South Giant City Road, Carbondale. AN EQUAL OPPORTUNITY EMPLOYER.
Completed Employment Applications for licensed positions should include the following information with the application:
Letter of Application
Copy of Illinois Professional Educator License Credentials
Copy of Transcripts
Applications may be picked up at the
Administrative Center
330 South Giant City Road
Or you may download an application below:
Download an Application for Employment Here
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General Information
School: Carbondale Community High School District 165 opened a new high school in June 2003. Grades: 9-12, Alternative Education Program (Rebound), Drug and Rehab facility (Gateway); Health Clinic (Terrier Care), Students: 1,050; two Assistant Principals, Dean of Students, Special Education Director, Curriculum Director, Athletic Director; Career and Industrial Technology Director, School Resource Officer, Student Activities Director, Vocational Program Coordinator, Wellness Coordinator, Nine Department Chairs; Teachers: 89; Support Personnel: 54.
Curriculum: The high school provides a comprehensive educational program that provides vocational, special education, remedial and college preparatory programs. Comprehensive co-curricular activities are offered in a broad range of programs in athletics, the arts and student clubs. The high school enjoys a well-established tradition of excellence and has the support of the community for its school, faculty and administration.
Recognitions:
•Class of 2018 earned over $3,000,000 in scholarships and will matriculate to over 50 colleges and universities throughout the US including Bradley University, DePaul University, John A. Logan College, Oberlin College, Saint Louis University, Southern Illinois University, Vanderbilt University, Washington University and Yale University. Graduates have committed to service the US Army and US Marines.
•37 Illinois State Scholars, 5 National Merit Commended Honorees, and 4 National Merit Finalists.
•106 Students took 185 AP exams in 15 subject areas in the spring of 2018, with an average score of 3.8.
•Nationally Ranked Scholar Bowl Team.
•State Champion and Nationally Recognized Physics Team.
•Music Program Top 3 State Finalist eight years in a row (State Champion in 2014).
•National Scholastic Art Winners four years in a row (One of only eight Gold Portfolios Awarded Nationally in 2017).
•State Finalists in Speech and History Fair, Science Fair, Science/Engineering Team, and Skills USA.
•Garwin Family Foundation sponsorship of students to pursue rigorous summer academic and vocational programs at Brown University, UT-Dallas, U Missouri-St. Louis, Standford University, University of Michigan and Tom's Place in DeSoto.
•IHSA All State Academic Team three of the five years.
•South Seven Gold Cup Champions.
Information About the Area:
Location: Carbondale Community High School is in Carbondale, Illinois, home of Southern Illinois University. A city of approximately 26,000 residents, Carbondale is proud to offer all the luxuries of a larger city while maintaining the comforts of a small town.
Services Available: Every service imaginable is available within or near Carbondale, including a major hospital, two airports, rail and bus service.
Recreation: Carbondale offers numerous recreation opportunities to its residents. These include scenic lakes, a national forest, collegiate spectator sports, abundance of cultural opportunities, numerous parks and recreation fields, plus shopping, dining and entertainment.
Colleges: Carbondale is home to Southern Illinois University, a major state university with approximately 15,000 students. John A. Logan Community College is located ten miles east of Carbondale.
Additional Information about the Carbondale area may be obtained at the city of Carbondale website:
www.ci.carbondale.il.us, or at the Carbondale Chamber of Commerce website: www.carbondalechamber.com
AN EQUAL OPPORTUNITY EMPLOYER FOR ALL: CARBONDALE COMMUNITY HIGH SCHOOL DISTRICT 165 DOES NOT DISCRIMINATE ON THE GROUNDS OF RACE, COLOR, RELIGION, SEX, AGE OR DISABILITY IN ADMISSION TO, OR TREATMENT OF EMPLOYMENT IN ITS PROGRAMS OR ACTIVITIES.


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Infrastructure law may help Colorado's aim to tackle poverty and climate change
CPR News | By Sam Brasch
Colorado is trying to fight both poverty and climate change. They're doing so by retrofitting low-income homes after the state got a big boost from the infrastructure law. In this encore presentation, Sam Brasch of Colorado Public Radio reports.
SAM BRASCH: Hi.
LAURA RASCON: Hi.
BRASCH: I'm Sam.
RASCON: Laura.
BRASCH: "It was always really cold," she says. "Cold air came in through the door, and I had two broken windows."
Her family's propane furnace also had a habit of breaking down. To stay comfortable, they would carry space heaters from room to room. Rascon says her daughter brought one too close to her bed one night three years ago. A blanket caught fire. Rascon woke up just in time to hurl it out the front door before anyone got hurt.
BRASCH: "I was so scared," she says, "because we didn't know if it was going to happen again at night or when she was home alone after school."
Today, she's looking forward to a far less anxious winter. That's because the state updated her home with new windows, insulation and something that's not usually included in government-backed efficiency projects - all new electric appliances.
Doug Jones is with the Northwest Colorado Council of Governments, which coordinated the work.
BRASCH: If heat pump prices come down, Harry says the state could expand to far more common natural gas-heated homes and start moving thousands of low-income residents away from fossil fuels for good.
For NPR News, I'm Sam Brasch in Dotsero, Colo.
Morning Edition on WCAI
Surgeon General Murthy details the administration's next move against COVID
The Supreme Court delivered 2 decisions on vaccine rules for U.S. workers
A man in Florida may regret not leaving a tip at Key West bar
State laws that add restrictions on voting are a serious problem, Waldman says


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UPDATE: Handcuffed fugitive nabbed after wiggling under fence
Jim Holt and Austin Dave
A handcuffed fugitive eluding local sheriff's deputies since his arrest Sunday night was tracked down and arrested Monday morning following a foot pursuit through the dusty back lots of businesses in the Centre Pointe area and across the Santa Clara River wash.
The suspect, who was arrested by deputies of the Santa Clarita Valley Sheriff's Station Sunday night about 9 p.m. near Golden Valley Road and McKeon Way, escaped custody and eluded capture until about 10:15 a.m. Monday.
Monday's arrest of John Jacob Kealey, 23, of Canyon Country, on a misdemeanor drug offense and for escaping custody, also a misdemeanor, marks the 5th time in a year he's been arrested.
Arrest documents maintained by the Los Angeles County Sheriff's Station show Kealey was arrested in the SCV five times since October 2016. His most recent prior arrest happened less than a month ago, when he was arrested on suspicion of possessing a controlled substance on Sept. 23.
On Monday morning, deputies arrested him in an industrial area near the intersection of Oak Avenue and Soledad Canyon Road, and near the Santa Clara River wash.
The suspect, found by deputies wearing a plastic bag as a shirt, was arrested shortly after he wiggled under a fence to elude capture.
The fugitive was nabbed about 1,000 feet from the place of his earlier arrest.
The manhunt began shortly after deputies were called to the Rattler's Restaurant on Carl Boyer Drive Sunday.
"They were called for a man passed out in the bathroom of Rattlers," Lt. Doug Mohroff told The Signal Monday.
Deputies and paramedics with the Los Angeles County Fire Department responded to the medical call and assisted in the man's "rescue," he said.
"As he was being treated deputies became aware that he might be in possession of narcotics," Mohroff said. "He was going to be taken to the hospital.
"As he was leaving the restaurant, he (suspect) decides he doesn't want any part of this and bolts."
Deputies searched for the escaped suspect throughout the night without luck,
Then, shortly before 10 a.m. Monday, a citizen spotted a man in handcuffs wearing a plastic bag near the Walmart on Carl Boyer Drive.
"Deputies responded to Carl Boyer and that's when he (suspect) runs again," Mohroff said.
Deputies ran after the handcuffed suspect, catching up with him at Oak and Soledad.
Kealey has appeared in the Santa Clarita Courthouse on four prior misdemeanor charges, including:
– theft of personal property, Oct. 18, 2016. "He pleaded no contest on March 3, to one count of misdemeanor vandalism, under $400 and was sentenced to 15 days in jail and 36 months summary probation," Ricardo Santiago, spokesman for the Los Angeles County District Attorney's Office said.
– possession of a narcotic, Apr. 18, 2017. "He Pleaded no contest on June 7 to one count of misdemeanor possession of cocaine and was sentenced to 90 days in jail and 36 months summary probation," Santiago said.
– vandalism, June 7, 2017.
– possession of a controlled substance, Sept. 23, 2017.
At the time of his arrest Monday, Kealey was scheduled to appear in court Nov. 7 in connection with his arrests a year ago and his arrest in April, arrest papers show.
on Twitter @jamesarthurholt