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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 288, pp. 1--24.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/288\hfil H\"older continuity]
{H\"older continuity of bounded weak solutions to generalized
parabolic $p$-Laplacian equations II: singular case}

\author[S. Hwang, G. M. Lieberman \hfil EJDE-2015/288\hfilneg]
{Sukjung Hwang, Gary M. Lieberman}

\address{Sukjung Hwang \newline
Center for Mathematical Analysis and Computation, 
Yonsei University, Seoul 03722, Korea}
\email{sukjung\_hwang@yonsei.ac.kr}

\address{Gary M. Lieberman
Department of Mathematics,
Iowa State University, Ames, IA 50011, USA}
\email{lieb@iastate.edu}

\thanks{Submitted July 17, 2015. Published November 19, 2015.}
\subjclass[2010]{35B45, 35K67}
\keywords{Quasilinear parabolic equation; singular equation;
\hfill\break\indent  generalized structure; a priori estimate; H\"older continuity}

\begin{abstract}
 Here we generalize quasilinear parabolic $p$-Laplacian type equations
 to obtain the prototype equation
 \[
 u_t - \operatorname{div} \Big(\frac{g(|Du|)}{|Du|} Du\Big) = 0,
 \]
 where $g$ is a nonnegative, increasing, and continuous function trapped
 in between two power functions $|Du|^{g_0 -1}$ and $|Du|^{g_1 -1}$ with
 $1<g_0 \leq g_1 \le 2$. Through this generalization in the setting
 from Orlicz spaces, we provide a uniform proof with a single geometric
 setting that a bounded weak solution is locally H\"older continuous
 with some degree of commonality between degenerate and singular types.
 By using geometric characters, our proof does not rely on any of
 alternatives which is based on the size of solutions.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\allowdisplaybreaks


\section{Introduction}\label{I}

This article is intended as a companion paper to \cite{HL1}, which proved
the H\"older continuity of solutions to degenerate parabolic equations
satisfying a generalized $p$-Laplacian structure. Here, we examine the
same question for singular equations, but we refer the reader to \cite{HL1}
for a more detailed description of the history of this problem.

Our interest here is in the parabolic equation
 \begin{equation} \label{I:gen}
 u_t-\operatorname{div}\mathcal{A}(x,t,u,Du) = 0
 \end{equation}
 when there is an increasing function $g$ such that
\begin{subequations} \label{I:gen-str}
\begin{gather}
\mathcal{A}(x,t,u,\xi)\cdot \xi \ge C_0G(|\xi|),\label{I:gen-str1} \\
|\mathcal{A}(x,t,u,\xi)| \le C_1g(|\xi|)\label{I:gen-str2}
\end{gather}
\end{subequations}
for some positive constants $C_0$ and $C_1$, where $G$ is defined by
\[
G(\sigma)=\int_0^\sigma g(s)\, ds,
\]
and we assume that there are constants $g_0$ and $g_1$ satisfying $1<g_0\le g_1$
such that
\begin{equation} \label{I:DeltaNabla}
g_0G(\sigma) \le \sigma g(\sigma) \le g_1G(\sigma)
\end{equation}
for all $\sigma>0$. For the most part, we are only concerned here with the 
case that $g_1\le2$, but some of our results do not need this additional 
restriction, so we shall always state it explicitly when it is used.
 Our results generalized those of Ladyzhenskaya and Ural'tseva \cite{LaUr62} and Chen and DiBenedetto \cite{ChDB88,ChDB92}, who proved H\"older continuity under the structure conditions
 \begin{equation} \label{EAp}
\mathcal{A}(x,t,u,\xi)\cdot\xi
\ge C_0|\xi|^p,\ |\mathcal{A}(x,u,\xi)| \le C_1|\xi|^{p-1}
 \end{equation}
with $p=2$ and $p<2$, respectively.
The structure \eqref{EAp} is contained in this model as the special case
 $g(s)=s^{p-1}$, in which case we may take $g_0=g_1=p$, but we consider a class
of structure functions $g$ much wider than that of just power functions.
In this way, we obtain a uniform proof of H\"older continuity (with appropriate
uniformity of constants) for all $p\in(1,2]$ at once under the structure
condition \eqref{EAp} as well as a proof of H\"older continuity under more
 general structure conditions.

 In \cite{HL1}, we have discussed our approach for a generalization of the
case $p\ge2$, so we concern ourselves here with the points relevant to the
generalization of the case $p\le2$.
It is known that solutions of this problem generally become zero in finite
time when $p<2$
(see \cite[Sections VII.2 and VII.3]{DB93} for a more complete discussion of
this phenomenon) but not when $p=2$ (because of the Harnack inequality,
first proved by Moser \cite{Mos64}), so our proof needs to take this behavior
into account. In addition, \cite[Section 4]{DBGiVe10} gives a H\"older
exponent which degenerates as $p$ approaches $2$; the proof must be further
 modified for $p$ close to $2$ if the H\"older exponent is to remain positive
 near $p=2$. Our method manages the whole range $1<p\le2$ uniformly for $p$
away from $1$. Although, as the authors point out in \cite{DBGiVe12}, this method
is more complicated analytically, it does handle the whole range easily and
it is quite simple geometrically.

We use the definition of weak solution given in \cite{HL1}, which we now present.
For an arbitrary open set $\Omega\subset \mathbb R^{n+1}$, we introduce the
generalized Sobolev space $W^{1,G}(\Omega)$, which consists of all functions $u$
defined on $\Omega$ with weak derivative $Du$ satisfying
\[
\iint_{\Omega} G(|Du|)\, dx\,dt <\infty.
\]
We say that $u\in C_{\rm loc}(\Omega) \cap W^{1,G}(\Omega)$ is a
\emph{weak supersolution} of \eqref{I:gen} if
\[
0 \leq -\iint_{\Omega} u\varphi_t\, dx\, dt
+ \iint_{\Omega} \mathcal{A}(x,t,u,Du)\cdot D\varphi\, dx\, dt
\]
for all $\varphi\in C^1(\bar\Omega)$ which vanish on the parabolic boundary
 of $\Omega$; a \emph{weak subsolution} is defined by reversing the inequality;
and a \emph{weak solution} is a function which is both a weak supersolution
and a weak subsolution.
In fact, we shall use a larger class of $\varphi$'s which we discuss in a
later section.

Our method of proof uses some recent geometric ideas of Gianazza, Surnachev,
and Vespri \cite{GiSuVe10}, who gave a different proof for the H\"older continuity
in \cite{ChDB88,ChDB92}.
While \cite{ChDB88,ChDB92} examine an alternative based on the size of the
set on which $|u|$ is close to its maximum, the method in \cite{GiSuVe10}
use a geometric approach from regularity theory and Harnack estimates.
Here, we use this geometric approach along with some elements of the
analytic approach in \cite{ChDB92}.


The proof is based on studying two cases separately. Either a bounded weak
solution $u$ is close to its maximum at least half of a cylinder around
$(x_0, t_0)$ or not. In either case, the conclusion is that the essential
oscillation of $u$ is smaller in a subcylinder centered at $(x_0,t_0)$.
Basically, our goal is reached using the geometric character of $u$
with two integral estimates, local and logarithmic estimates
\eqref{S4:LocalE}, \eqref{S4:LogE}.


In the next section, we provide some preliminary results, mostly involving
notation for our geometric setting.
Section \ref{S2} states the main lemma and uses that lemma to prove the
H\"older continuity of the weak solutions.
The main lemma is proved in Section \ref{S3}, based on some integral
inequalities which are proved in Section \ref{S4}.

\section{Preliminaries}\label{S1}

 \subsection*{Notation}
 (1) The parameters $g_0$, $g_1$, $N$, $C_0$, and $C_1$ are the data.
When we make the additional assumption that $g_1\le2$, we use the word ``data''
to denote the constants $g_0$, $N$, $C_0$, and $C_1$.

(2) Let $K_{\rho}^{y}$ denote the $N-$dimensional cube centered at
$y\in \mathbb{R}^{N}$ with the side length $2\rho$, i.e.,
 \[
 K_{\rho}^{y} := \{ x\in \mathbb{R}^{N} : \max_{1 \leq i \leq N} |x^{i} - y^{i}|
< \rho \}.
 \]
(Here, we use superscripts to denote the coordinates of $x$; we'll use subscripts
to indicate different points.) For simpler notation, let $K_{\rho} := K_{\rho}^{0}$.
We also define the spatial distance $|\cdot|_\infty$ by
\[
|x-y|_\infty = \max_{1\le i \le N}|x^i-y^i|.
\]
In fact, all of our work can be recast with the ball
\[
B_\rho^{y}=\{x\in\mathbb R^N: |x-y|<\rho\},
\]
where $|x-y|$ is the usual Euclidean distance, in place of $K_\rho^y$ with
only slight notational changes. There is no significant reason to use cubes
rather than balls in the degenerate case, but the method used in
\cite{ChDB88,ChDB92} requires that cubes be subdivided into congruent smaller
subcubes, and the corresponding decomposition for balls is much more complicated.
In this work, no such decomposition is needed.

(3) For given $(x_0, t_0) \in \mathbb{R}^{N+1}$, and given positive constants
$\theta$, $\rho$ and $k$, we say
 \begin{gather*}
 T_{k, \rho}(\theta) := \theta k^2 G\big(\frac{k}{\rho}\big)^{-1}, \\
 Q_{k, \rho}^{x_0, t_0}(\theta) := K_{\rho}^{x_0} \times [t_0 -T_{k,\rho}, t_0], \\
 Q_{k, \rho}(\theta) := Q_{k, \rho}^{0,0}(\theta).
 \end{gather*}
The point $(x_0,t_0)$ is called the \emph{top-center point of
$Q_{k,\rho}^{x_0,t_0}(\theta)$}.
We also abbreviate
\[
T_{k,\rho}= T_{k,\rho}(1), \quad Q_{k,\rho}^{x_0,t_0} = Q_{k, \rho}^{x_0, t_0}(1),
\quad Q_{k\rho}=Q_{k,\rho}(1).
\]

 \subsection*{Geometry}
We refer the reader to \cite{HL1} for a discussion of our choices of notation,
but we do recall that if $u$ is any function defined on an open set $\Omega$,
then for any positive number $\omega$ and any $(x_0,t_0)\in \Omega$,
there a number $R$ such that
\[
 Q_{\omega, 4R}^{x_0, t_0} \subset \Omega.
 \]


 \subsection*{Useful inequalities}

 Because of the generalized functions $g$ and $G$, we are not able to
apply H\"older's inequality or typical Young's inequality.
Here we present essential inequalities which will be used through out the paper,
all of which were proved in \cite{HL1}.

 \begin{lemma}\label{S1:ineq}
 For a nonnegative and nondecreasing function $g \in C[0,\infty)$, let $G$
be the antiderivative of $g$. Suppose that $g$ and $G$ satisfy
\eqref{I:DeltaNabla}. Then for all nonnegative real numbers $\sigma$,
$\sigma_1$, and $\sigma_2$, we have
 \begin{itemize}
 \item[(a)] $G(\sigma)/\sigma$ is a monotone increasing function.
 \item[(b)] For $\beta > 1$,
 \[
 \beta^{g_0} G(\sigma) \leq G(\beta \sigma) \leq \beta^{g_1} G(\sigma).
 \]
 \item[(c)] For $0< \beta < 1$,
 \[
 \beta^{g_1} G(\sigma) \leq G(\beta \sigma) \leq \beta^{g_0} G(\sigma).
 \]
 \item[(d)]
 $ \sigma_1 g(\sigma_2) \leq \sigma_1 g(\sigma_1) + \sigma_2 g(\sigma_2)$.


\item[(e)] \emph{(Young's inequality)} For any $\epsilon \in (0,1)$,
 \[
 \sigma_1 g(\sigma_2) \leq \epsilon^{1-g_1} g_1 G(\sigma_1)
+ \epsilon g_1 G(\sigma_2).
 \]
 \end{itemize}
 \end{lemma}

 \begin{lemma}\label{S1:H-ineq}
 For any $\sigma >0$, let
 \[
 h(\sigma) = \frac{1}{\sigma} \int_{0}^{\sigma} g(s) \,ds,  \quad
  H(\sigma) = \int_{0}^{\sigma} h(s) \,ds.
 \]
 Then we have
\begin{gather*}
 g_0 h(\sigma) \leq g(\sigma) \leq g_1 h(\sigma), \\
 g_0 H(\sigma) \leq G(\sigma) \leq g_1 H(\sigma), \\
 (g_0 - 1) h(\sigma) \leq \sigma h'(\sigma) \leq (g_1 - 1) h(\sigma), \\
 \frac{1}{g_1} \sigma h(\sigma) \leq H(\sigma) \leq \frac{1}{g_0} \sigma h(\sigma), \\
 \beta^{g_0} H(\sigma) \leq H(\beta \sigma) \leq \beta^{g_1} H(\sigma)
\end{gather*}
for any $\beta>1$.
 \end{lemma}

 Our next result concerns some inequalities about integration of a
function over various intervals. We shall use these inequalities in
the proof of the Main Lemma. This lemma is probably well-known, but we
are unaware of any reference for it.

 \begin{lemma} \label{S1:fintegral}
 Let $f$ be a continuous, decreasing, positive function defined on $(0,\infty)$.
Then, for all $\delta$ and $\sigma\in(0,1)$, we have
\begin{equation} \label{S1:fintegral:E2}
\int_0^1 f(\delta+s)\,ds \le \frac 1\sigma\int_0^{\sigma}f(\delta+s)\,ds.
\end{equation}
If, in addition, for all $\beta>1$ and $\sigma>0$, we have
 \begin{equation} \label{S1:fintegral:E}
\beta f(\beta\sigma) \ge f(\sigma), \quad f(\beta\sigma) \le f(\sigma),
\end{equation}
then, for all $\delta\in (0,1)$, we have
\begin{equation} \label{S1:fintegral:E1}
\int_0^\delta f(\delta+s)\, ds \le \frac 2{2+\ln (1/\delta)} \
int_0^1 f(\delta+s)\, ds.
\end{equation}
\end{lemma}

\begin{proof}
To prove \eqref{S1:fintegral:E2}, we define the function
\[
F(\sigma) = \sigma \int_0^1 f(\delta+s)\, ds - \int_0^{\sigma} f(\delta+s)\, ds.
\]
Since
\[
F'(\sigma) = \int_0^1f(\delta+s)\, ds - f(\delta+\sigma),
\]
and $f$ is decreasing, it follows that $F'$ is increasing so $F$ is convex.
Moreover
\[
F(0)=F(1)=0,
\]
so $F(\sigma) \le0$ for all $\sigma\in(0,1)$. Simple algebra then yields
\eqref{S1:fintegral:E2}.

To prove \eqref{S1:fintegral:E1}, we first use a change of variables to see that,
for any $j\ge1$, we have
\[
\int_{j\delta}^{2j\delta} f(\delta+s)\, ds
=\int_0^{j\delta} f((j+1)\delta +\sigma)\, d\sigma
= j\int_0^{\delta} f((j+1)\delta+js))\, ds.
\]
Since $(j+1)\delta+js\le (j+1)(\delta+s)$ and $f$ is decreasing, we have
\[
\int_{j\delta}^{2j\delta} f(\delta+s)\, ds
\ge j \int_0^\delta f((j+1)(\delta+s))\,ds
\]
and then \eqref{S1:fintegral:E} gives
\[
\int_{j\delta}^{2j\delta} f(\delta+s)\, ds
\ge \frac j{j+1}\int_0^{\delta} f(\delta+s)\, ds.
\]
We now let $J$ be the unique positive integer such that
$2^{-J}<\delta \le 2^{1-J}$ and we take $j=2^i$ with $i=0,\dots,J-1$.
Since $j/(j+1)\ge 1/2$, it follows that
\[
\int_0^\delta f(\delta+s)\, ds
\le 2\int_{2^i\delta}^{2^{i+1}\delta} f(\delta+s)\,ds.
\]
Since
\[
\int_0^{2^J\delta}f(\delta+s)\, ds
= \int_0^\delta f(\delta+s)\, ds
+ \sum_{i=0}^{J-1} \int_{2^i\delta}^{2^{i+1}\delta} f(\delta+s)\,ds,
\]
we infer that
\[
\int_0^{2^J\delta} f(\delta+s)\, ds
\ge [1+\frac 12J]\int_0^\delta f(\delta+s)\, ds.
\]
The proof is completed by noting that $J>\ln(1/\delta)$ and that
\[
\int_0^1 f(\delta+s)\, ds \ge\int_0^{2^J\delta} f(\delta+s)\, ds.
\]
\end{proof}

Note that condition \eqref{S1:fintegral:E} is satisfied if
$f(\sigma)=\sigma^{-p}$ with $0\le p \le 1$, in which case this
lemma can be proved by computing the integrals directly.


\section{Basic results and the proof of H\"older continuity} \label{S2}

In this section, we prove the H\"older continuity of solutions of
\eqref{I:gen} for singular equations (that is, equations with $g_1\le 2$) and
for degenerate equations (that is, equations with $g_0\ge 2$).
Our proof is based on some estimates for nonnegative supersolutions of the equation,
and these estimates will be proved in the next section.

Our Main Lemma states that a nonnegative supersolution $u$ of a singular
equation is strictly positive in a subcylinder if $u$ is near to the maximum
value in more than a half of cylinder.

\begin{lemma}[Main Lemma] \label{S2:MainLemma}
Let $\omega$ and $R$ be positive constants. Then there are positive
constants $\delta$ and $\mu$, both less than one and determined only by the
data such that, if $u$ is a nonnegative solution of \eqref{I:gen} in
\[
Q =Q_{\delta\omega,2R}\big(\frac 34\big)
\]
with $g_1\le2$, and
\begin{equation}\label{S2:MainLemma-hyp}
\big| Q\cap \{u \le \frac{\omega}{2}\}\big| \le \frac{1}{2}|Q|,
\end{equation}
then
\begin{equation} \label{S2:MainLemma:estimate}
\operatorname{ess\,inf}_{\mathcal Q} u \geq \mu \omega,
\end{equation}
with
$\mathcal Q= Q_{\mu\omega,R/2}$.
\end{lemma}

We shall prove this lemma in the next section. Here we show first how to
infer a decay estimate for the oscillation of any bounded solution of \eqref{I:gen}.

\begin{lemma} \label{S2:Lmainboth}
Let $C_0$, $C_1$, $g_0$, $g_1$, $\rho$, and $\omega$ be positive constants
 with $C_0\le C_1$ and $1<g_0\le g_1\le 2$. Then there are positive constants
 $\sigma$ and $\lambda$, both less than one and determined only by data such that,
if $u$ is a bounded weak solution of \eqref{I:gen} in $Q_{\omega,\rho}$ with
\[
\operatorname{ess\,osc}_{Q_{\omega,\rho}} u \le \omega,
\]
then
\begin{equation} \label{S2:Lmainbothest}
\operatorname{ess\,osc}_{Q_{\sigma\omega,\lambda\rho}} u \le \sigma\omega.
\end{equation}
\end{lemma}

\begin{proof}
We begin by taking $\delta$ and $\mu$ to be the constants from
Lemma~\ref{S2:MainLemma} and we set
\[
\sigma=1-\mu, \quad \lambda = \frac 1{4}\left(\frac\mu\sigma\right)^{(2-g_0)/g_0}.
\]
From the proof of Lemma~\ref{S2:MainLemma}, it follows that $\mu\le1/4$,
so $\mu/\sigma\le1$.
We also introduce the functions $u_1$ and $u_2$ by
\begin{equation} \label{S2:u12}
u_1=u- \inf_{Q_{\omega,\rho}} u, \quad u_2=\omega-u_1.
\end{equation}
It follows from Lemma~\ref{S1:ineq}(b) that
\[
3\big(\frac {\delta\omega}2\big)^2
G\big(\frac{\delta\omega}\rho\big)^{-1}
\le \omega^2 G\big(\frac {\omega}\rho\big)^{-1},
\]
and hence the cylinder $Q$ from Lemma~\ref{S2:MainLemma} is a subset
of $Q_{\omega,\rho}$ provided $R=\rho/2$.

There are now two cases. First, if
\[
\big|Q\cap \{u_1\le \frac \omega2\}\big| \le \frac 12 |Q|,
\]
then we apply Lemma~\ref{S2:MainLemma} to $u_1$ and hence
\[
\operatorname{ess\,inf}_{\mathcal Q} u_1\ge \mu\omega.
\]
Since
\[
\operatorname{ess\,sup}_{\mathcal Q} u_1\le \omega,
\]
it follows that
\[
\operatorname{ess\,osc}_{\mathcal Q} u
= \operatorname{ess\,osc}_{\mathcal Q} u_1\le (1-\mu)\omega=\sigma\omega.
\]
On the other hand if
\[
\big|Q\cap \{u_1\le \frac \omega2\}\big| \ge \frac 12 |Q|,
\]
then
\[
\big|Q\cap \{u_2\le \frac \omega2\}\big| \le \frac 12 |Q|,
\]
and an application of Lemma~\ref{S2:MainLemma} to $u_2$ implies once again that
\[
\operatorname{ess\,osc}_{\mathcal Q} u \le \sigma \omega.
\]
Since $4\lambda\mu/\sigma\le1$, we infer from Lemma~\ref{S1:ineq}(c) that
\[
(\sigma\omega)^2G\big(\frac {\sigma\omega}{\lambda\rho}\big)^{-1}
\le (\mu\omega)^2G\big( \frac {4\mu\omega}{\rho}\big)^{-1}.
\]
Since $\lambda\le 1/4$, it follows that $Q_{\sigma\omega,\lambda\rho}$
is a subset of the cylinder $\mathcal Q$ from Lemma~\ref{S2:MainLemma},
and \eqref{S2:Lmainbothest} follows.
\end{proof}

For any real number $\tau$ and any function $u$ defined on an open subset $\Omega$ of $\mathbb R^{N+1}$, we define
\begin{subequations} \label{S2:timescale}
\begin{gather}
|\tau|_G= \frac {U}{G^{-1}(U^2/|\tau|)}, \\
\intertext {where}
U= \operatorname{ess\,osc}_{\Omega} u.
\end{gather}
\end{subequations}
With this time scale, we define the parabolic distance between two sets
such $\mathcal{K}_{1}$ and $\mathcal {K}_2$ by
\[
\operatorname{dist}_P (\mathcal{K}_{1};\mathcal{K}_{2})
:= \inf_{\substack {(x,t) \in \mathcal{K}_{1}\\
 (y,s)\in \mathcal{K}_{2},\,  s\le t} }
 \max\{|x-y|_\infty, |t-s|_{G} \}.
\]
(Note that, strictly speaking, this quantity is not a distance because it
is not symmetric with respect to the order in which we write the sets.
Nonetheless, the terminology of distance is useful as a suggestion of the
 technically correct situation.)


The proof of \cite[Theorem 2.4]{HL1} immediately yields a modulus of continuity
in terms of $G$ and a H\"older continuity estimate.

\begin{theorem}\label{S2:T:NaturalConty}
Let $u$ be a bounded weak solution of \eqref{I:gen} with \eqref{I:gen-str}
in $\Omega$, and suppose $1<g_0\le g_1\le2$. Then $u$ is locally continuous.
 Moreover, there exist constants $\gamma$ and $\alpha\in(0,1)$ depending
only upon the data such that, for any two distinct points
$(x_1,t_1)$ and $(x_2,t_2)$ in any subset $\Omega'$ of $\Omega$ with
$\operatorname{dist}_P(\Omega'; \partial_{p}\Omega)$ positive, we have
\begin{equation} \label{S2:T:NaturalConty:E1}
\left|u(x_1,t_1) - u(x_2,t_2)\right|
\leq \gamma U \Big(\frac{|x_1 - x_2|+ |t_1 - t_2|_{G}}{\operatorname{dist}_P
(\Omega'; \partial_{P}\Omega)}\Big)^{\alpha}.
\end{equation}
In addition (with the same constants),
\begin{equation} \label{S2:T:NaturalConty:E2}
\left|u(x_1,t_1) - u(x_2,t_2)\right|
\leq \gamma U \Big(\frac{|x_1 - x_2|+ |1|_G\max\{|t_1 - t_2|^{1/g_0},
|t_1-t_2|^{1/g_1}\}}{\operatorname{dist}_P(\Omega'; \partial_{P}\Omega)}\Big)^{\alpha}.
\end{equation}
\end{theorem}

For initial regularity, we have the following variant of Lemma~\ref{S2:MainLemma}.
Note that this lemma is essentially the same as
\cite[Proposition IV.13.1]{DB93} (the result is mentioned only indirectly in
\cite{ChDB92}), but the proof is much simpler. To simplify notation, we define
the  cylinders
\[
Q^{+,x_0,t_0}_{k,R}(\theta) = K^{x_0}_R
\times\Big(t_0, t_0+\theta k^2G\big(\frac kR\big)^{-1}\Big), \quad
Q^+_{k,R}(\theta) = Q^{+,0,0}_{k,R}(\theta),
\]
and we set $Q^+_{k,R} = Q^+_{k,R}(1)$. With $\nu_0$ the constant from
Proposition~\ref{S3:prop4a} and $U$ a given constant, we also define $Q_R(U)$
to be the cylinder $Q^+_{U,R}(\nu_0/9)$.

\begin{lemma} \label{S2:Lmainin}
Let $C_0$, $C_1$, $g_0$, $g_1$, $\rho$, and $\omega$ be positive constants
with $C_0\le C_1$ and $1< g_0\le g_1$. Suppose also that $u$ is a bounded weak
solution of \eqref{I:gen} in $Q^+_{\omega,\rho}$ with
\[
\operatorname{ess\,osc}_{Q^+_{\omega,\rho}} u \le \omega.
\]
Then there is a constant $\lambda\in(0,1)$, determined only by data, such that
\begin{subequations} \label{S2:Lmaininest}
\begin{gather}
\operatorname{ess\,osc}_{Q^+_{\omega',\lambda\rho}} u \le \omega', \\
\intertext {where}
\omega'= \max\{\frac 56\omega,3\operatorname{ess\,osc}_{K_{\rho}\times\{0\}} u\}.
\end{gather}
\end{subequations}
\end{lemma}

\begin{proof}
 We begin by setting
\[
\omega_0= \operatorname{ess\,osc}_{K_{\rho}\times\{0\}} u.
\]
If $\omega<3\omega_0$, then $\omega'=3\omega_0$. We now perform some elementary
calculations to show that $Q^+_{\omega',\lambda\rho} \subset Q^+_{\omega,\rho}$
if $\lambda$ is small enough. First, since $\omega\le\omega'$ and $\lambda\le1$,
we can use Lemma~\ref{S1:ineq}(c) to infer that
\[
G\big(\frac \omega\rho\big) \le \big(\frac{\lambda\omega}{\omega'}\big)^{g_0}
G\big(\frac {\omega'}{\lambda\rho}\big) \le \lambda^{g_0}
G\big(\frac {\omega'}{\lambda\rho}\big).
\]
If $\lambda\le 9^{-1/g_0}$, then we have
\[
G\big(\frac \omega\rho\big)\le \frac 19G\big(\frac {\omega'}{\lambda\rho}\big).
\]
Since $\omega_0 \le \omega$, it follows that $\omega\ge \frac 13\omega'$
 and therefore
\[
\omega^2 G\big(\frac \omega\rho\big)^{-1} \ge \frac19(\omega')^2
G\big(\frac \omega\rho\big)^{-1},
\]
so
\[
\omega^2 G\big(\frac \omega\rho\big)^{-1}\ge (\omega')^2
 G\big(\frac {\omega'}{\lambda\rho}\big)^{-1}.
\]
Hence $\lambda\le 9^{-1/g_0}$ implies that
$Q^+_{\omega',\lambda\rho} \subset Q^+_{\omega,\rho}$ and therefore
 \eqref{S2:Lmaininest} is valid.

If $\omega\ge 3\omega_0$, we take $\nu_0$ be the constant from
Proposition \ref{S3:prop4a}. This time, we set $R= \frac 16\rho$,
and we note that $Q^+_{\omega/3, 2R}(\nu_0) \subset Q^+_{\omega,\rho}$.
To proceed, we define
\[
u_1= u-\operatorname{ess\,inf}_{Q^+_{\omega/3,2R}} u, \quad u_2= \omega-u_1,
\]
and we set $k= \omega/3$.

We consider two cases. First, if
\begin{equation} \label{S2:Lmainin-1}
\operatorname{ess\,sup}_{K_{2R}\times\{0\}} u_1 \le \frac 23\omega,
\end{equation}
then $u_1\ge k$ on $K_{2R}\times\{0\}$. We then apply
 Proposition~\ref{S3:prop4a} to $u_1$ in $Q^+_{k,2R}(\nu_0)$ to infer that
\[
\operatorname{ess\,inf}_{Q^+_{k,R}(\nu_0)} u_1 \ge \frac k2.
\]
It follows that
\begin{equation} \label{S2:Lmainin-2}
\operatorname{ess\,osc}_{Q^+_{k,R}(\nu_0)} u \le \omega-\frac k2= \frac 56\omega.
\end{equation}

If \eqref{S2:Lmainin-1} does not hold, then some straightforward algebra shows that
\[
\operatorname{ess\,sup}_{K_{2R}\times\{0\}} u_2 \le \frac 23\omega,
\]
so we can apply Proposition~\ref{S3:prop4a} to $u_2$, again obtaining
\eqref{S2:Lmainin-2}.

To see that \eqref{S2:Lmainin-2} implies \eqref{S2:Lmaininest}, we examine
separately the cases $\omega'=\frac 56\omega$ and $\omega'=\omega_0$.
In both cases, we see that $\lambda \rho\le R$ if $\lambda\le\frac 13$.

In the first case, we observe that $\lambda\le \frac 13$ implies that
$5/(6\lambda) \ge \frac 52\ge1$. If, in addition,
\[
\lambda \le \frac 56 \big(\frac {4\nu_0}{25}\big)^{1/g_0},
\]
we conclude that
\[
G\big(\frac \omega\rho\big) \le \big(\frac {6\lambda}5\big)^{g_0}
G\big( \frac {5\omega}{6\lambda}\big)
\le \frac {4\nu_0}{25}G\big( \frac {5\omega}{6\lambda}\big),
\]
and hence $Q^+_{\omega',\lambda\rho} \subset Q^+_{k,R}(\nu_0)$ in this case.
Combining this observation with \eqref{S2:Lmainin-2} gives \eqref{S2:Lmaininest}.

In the second case, we observe that $\omega'\ge \frac 56\omega$, so
$\lambda\le \frac 56$ implies that
\[
G\big(\frac \omega\rho\big)
\le \big( \frac {\lambda\omega}{\omega'} \big)^{g_0}
G\big(\frac {\omega'}{\lambda\rho}\big)
\]
If also
\[
\lambda \le \frac 56(\frac {\nu_0}{17})^{1/g_0},
\]
Then we have
\begin{align*}
G\big(\frac \omega\rho\big)
&\le \big(\frac {\lambda \omega}{\omega'}\big)^{g_0}
 G\big(\frac {\omega'}{\lambda\rho}\big) \\
&\le \big(\frac {5\lambda }{6}\big)^{g_0} G\big(\frac {\omega'}{\lambda\rho}\big) \\
&\le \frac 19\big( \frac 56\big)^2\nu_0 G\big(\frac {\omega'}{\lambda\rho}\big)
\end{align*}
since $(5/6)^2/9\ge 1/17$. It follows again that
$Q^+_{\omega',\lambda\rho} \subset Q^+_{k,R}(\nu_0)$ and hence we
 obtain \eqref{S2:Lmaininest}.
Combining all these cases, we see that the result is true with
\[
\lambda = \min \{\frac 19, \frac 56(\frac {\nu_0}{17})^{1/g_0}\}
\]
since $9^{-1/g_0}\ge 1/9$.
\end{proof}

From this lemma, we infer a continuity estimate near the initial surface.
We recall from \cite{Lie96} that $B\Omega$ is the set of all
$(x_0,t_0)\in\partial_P\Omega$ such that, for some positive numbers
 $r$ and $s$, the cylinder
\[
K_r^{x_0}\times (t_0,t_0+s)
\]
is a subset of $\Omega$. We also define the \emph{initial surface}
$B'\Omega$ of $\Omega$ as in \cite{HL1} to be the set of all
$(x_0,t_0)\in B\Omega$ such that $K_r\times\{t_0\}\subset\partial_P\Omega$
 for some $r'>0$. As noted in \cite{HL1}, $B'\Omega$ need not be the same
as $B\Omega$.

For $(x_0,t_0)\in B'\Omega$ and $\omega>0$, we write
$\operatorname{dist}_B(x_0,t_0)$ for the supremum of the set of all numbers
$r$ such that $Q^{+,x_0,t_0}_{\omega,r}\subset \Omega$ and
$K_r^{x_0,t_0}\times\{0\}\subset \partial_P\Omega$.

\begin{theorem} \label{S2:thminitial}
Let $u$ be a bounded weak solution of \eqref{I:gen} in $\Omega$, and
suppose $1< g_0\le g_1\le2$. Suppose also that the restriction of $u$
to $B'\Omega$ is continuous at some $(x_0,t_0)\in B'\Omega$.
Then $u$ is locally continuous up to $(x_0,t_0)$. Specifically,
if there is a continuous increasing function $\tilde\omega$ defined on
$[0,\operatorname{dist}_B(x_0,t_0))$ with $\tilde\omega(0)=0$,
\begin{equation} \label{S2:Lthmininitial-o}
\frac 56 \tilde\omega(2r)\le \tilde\omega(r)
\end{equation}
for all $r\in(0,\operatorname{dist}_B(x_0,t_0)/2)$, and with
\[
|u(x_0,t_0)-u(x_1,t_0)| \le \tilde\omega(|x_0-x_1|)
\]
for all $x_1$ with $|x_0-x_1| < \operatorname{dist}_B(x_,t_0)$, then there exist constants $\gamma$ and $\alpha\in(0,1)$ depending only upon the data such that, for any $(x,t)\in\Omega$ with $t\ge t_0$, we have
\begin{equation} \label{S2:thminitial:E1}
\begin{split}
\left|u(x_0,t_0) - u(x,t)\right|
&\leq \gamma U
\Big(\frac{|x_0 - x|+ |t_0 - t|_{G}}{\operatorname{dist}_B(x_0,t_0)}\Big)^{\alpha} \\
&\quad{}+3\tilde\omega\left(\gamma|x_0-x|_\infty
 +\gamma \operatorname{dist}_B(x_0,t_0)^{1-\alpha}|t_0-t|_G^\alpha\right).
\end{split}
\end{equation}
\end{theorem}

\begin{proof}
We start by taking $\omega_0=U$ and $\rho_0=\operatorname{dist}_B(x_0,t_0)$.
If $(x,t)\notin Q^{+,x_0,t_0}_{\omega_0,\rho_0}$, then the result is immediate
for any $\alpha$ as long as $\gamma \ge1$. With $\lambda$ as in
Lemma~\ref{S2:Lmainin}, we set $\sigma=5/6$, and
\[
\varepsilon=\min\{\lambda, \frac 12\sigma^{(2-g_0)/g_0}\}.
\]

If $(x,t)\in Q^{+,x_0,t_0}_{\omega_0,\rho_0}$, then we define
$\rho_n=\lambda^n\rho_0$. We also define $\omega'_n$ for $n>0$
inductively as $\omega'_{n+1}=\max\{\frac 56\omega'_n, 3\omega^*(\rho_n)\}$,
and we set
\[
Q_n= Q^{+,x_0,t_0}_{\omega'_n,\rho_n}.
\]

It follows from Lemma~\ref{S2:Lmainin} that
$\operatorname{ess\,osc}_{Q_n} u \le \omega'_n$, but this estimate must
 be improved. To this end, we set
\[
\omega_n= \max\big\{ \big(\frac56\big)^n\omega_0,3\tilde\omega(\rho_{n-1})\big\},
\]
and infer from the proof of \cite[Theorem 2.6]{HL1} that $\omega'_n\le\omega_n$
for $n>0$. Hence
\[
\operatorname{ess\,osc}_{Q_n} u \le \omega_n.
\]
 As before, we assume that $x\neq x_0$ and $t\neq t_0$, so there are nonnegative
integers $n$ and $m$ such that
\[
\rho_{n+1} \le |x_0-x|_\infty < \rho_n,
\]
and
\[
\omega_{m+1}^2G\Big( \frac {\omega_{m+1}}{\rho_{m+1}}\Big)^{-1}
\le |t_0-t|< \omega_m^2G\big( \frac {\omega_m}{\rho_{m}}\big)^{-1} .
\]
With $\alpha_1= \log_{1/2}(5/6)$, it follows that
\[
\big(\frac56\big)^n \le \Big(\frac {2|x_0-x|_\infty}{\rho_0}\Big)^{\alpha_1},
\quad \tilde\omega(\rho_n) \le \tilde\omega(\frac 1\lambda|x_0-x|_\infty).
\]
Moreover, if we set $\beta= \varepsilon \sigma^{(2-g_0)/g_0}$ and
$\widehat{\omega}_m= \beta^m\omega_0$, it follows that
$\widehat{\omega}_{m+1} \le \omega'_{m+1}$, so (as in the proof of Theorem 2.4)
\[
|t_0-t|_G \ge \beta^{m+1}\rho_0.
\]
For $\alpha_2= \log_\beta\sigma$, we infer again that
\[
\widehat{\omega}_{m} \le \Big( \frac {|t_0-t|_G}{\rho_0}\Big)^{\alpha_2}\omega_0.
\]
In addition, for $\alpha_3=\log_\beta\lambda$ (which is in the interval $(0,1]$),
we infer that
\[
\rho_{m} \le \Big( \frac {|t_0-t|_G}{\beta\rho_0}\Big)^{\alpha_3}\rho_0.
\]
Therefore,
\[
\bar\omega(\rho_m) \le \bar\omega\Big( \rho_0^{1-\alpha_3}|t_0-t|_G^{\alpha_3}\Big).
\]
And the proof is complete by combining all these inequalities and taking
$\alpha=\min\{\alpha_1,\alpha_2,\alpha_3\}$.
\end{proof}

As in \cite[Theorem 2.5]{HL1}, condition \eqref{S2:Lthmininitial-o}
involves no loss of generality in that any modulus of continuity for the
restriction of $u$ to $B'\Omega$ is controlled by one satisfying this condition.

\section{Proof of the main lemma}\label{S3}

Throughout this section, $u$ is a bounded nonnegative weak solution of 
\eqref{I:gen} with \eqref{I:gen-str}. The proof of Lemma~\ref{S2:MainLemma}
is composed of four steps under the assumption that $u$ is large at least half
 of a cylinder $Q_{\omega, 2R}$. First, Proposition~\ref{S3:prop1} gives
spatial cube at some fixed time level on which $u$ is away from its minimum
(zero value) on arbitrary fraction of the spatial cube. From the spatial cube,
 positive information spread in both later time and over the space variables
 with time limitations (Proposition~\ref{S3:prop2} and Proposition~\ref{S3:prop3a}).
 Controlling the positive quantity $\theta >0$ in $T_{k,\rho}(\theta)$ is key to
overcoming those time restrictions. Once we have a subcylinder centered at $(0, 0)$
 in $Q_{\omega, 4R}$ with arbitrary fraction of the subcylinder, we finally apply
modified De Giorgi iteration (Proposition~\ref{S3:prop4}) to obtain strictly
positive infimum of $u$ in a smaller cylinder around $(0,0)$.

\subsection{Basic results}

Our first proposition shows that if a nonnegative function is large on part
of a cylinder, then it is large on part of a fixed cylinder.
Except for some minor variation in notation, our result is
\cite[Lemma 7.1, Chapter III]{DB93}; we refer the reader to
\cite[Proposition 3.1]{HL1} for a proof using the present notation.

\begin{proposition} \label{S3:prop1}
Let $k$, $\rho$, and $T$ be positive constants. If $u$ is a measurable nonnegative
function defined on $Q=K_\rho\times(-T,0)$ and if there is a constant
$\nu_1\in [0,1)$ such that
\[
|Q\cap \{u\le k\}| \le (1-\nu_1)|Q|,
\]
then there is a number
\[
\tau_1\in \Big(- T, -\, \frac {\nu_1}{2-\nu_1}T\Big)
\]
for which
\[
\left| \{ x\in K_\rho: u(x,\tau_1)\le k\}\right|
\le \big(1-\frac {\nu_1}2\big) |K_\rho|.
\]
\end{proposition}

Our next proposition is similar to \cite[Lemma IV.10.2]{DB93}.

\begin{proposition}\label{S3:prop2}
Let $\nu$, $k$, $\rho $, and $\theta$ be given positive constants with $\nu<1$.
If $g_1\le 1$, then, for any $\epsilon \in (0,1)$, there exists a constant
$\delta =\delta (\nu,\epsilon, \theta, \text{data})$ such that,
if $u$ is a nonnegative supersolution of \eqref{I:gen} in
$K_\rho\times(-\tau,0)$ with
\begin{equation}\label{S3:prop2-hyp}
\left|\{x\in K_{\rho}: u(x,-\tau) < k \}\right|< (1-\nu) \left|K_{\rho}\right|
\end{equation}
for some
\begin{equation}
\tau \leq \theta(\delta k)^2 G\Big(\frac{\delta k}{\rho}\Big)^{-1}, \label{S3:prop2-tauge}
\end{equation}
then
\[
\left|\{ x\in K_{\rho}: u(x, -t) < \delta k \}\right|
< \left(1- (1-\epsilon)\nu\right)|K_{\rho}|
\]
for any $-t \in (-\tau, 0]$.
\end{proposition}

\begin{proof}
The proof is almost identical to that of \cite[Proposition 3.2]{HL1}.
With $\Psi$ defined as
\[
\Psi= \ln^+ \Big( \frac {k}{(1+\delta)k-(u-k)_-}\Big),
\]
we note that $\delta k |\Psi'| \leq 1$. It follows that
\begin{align*}
|\Psi'|^2 G\big(\frac{|D\zeta|}{\Psi'}\big)
&\leq \left( \delta k |\Psi'|\right)^{2-g_1} \left( \delta k \right)^{-2}
  G\left( \delta k |D\zeta| \right) \\
&\leq \sigma^{-g_1} \left(\delta k\right)^{-2}
  G\left( \frac{\delta k}{\rho} \right).
\end{align*}
Arguing as in the proof of \cite[Proposition 3.2]{HL1}
(and noting that $2^{g_1} \ge1$) then yields
\begin{align*}
& \int_{-\tau}^{-s}\int_{K_{\rho}} h(\Psi^2)|\Psi||\Psi'|^{2}
 G\big(\frac{|D\zeta|}{|\Psi'|}\big) \,dx\,dt \\
&\leq 2^{g_1}\theta h\left(j^2 (\ln 2)^2 \right) ( j \ln 2 )
 \sigma^{-g_1} |K_{\rho}| \\
&\leq 2^{g_1}\theta \frac{H(j^2 (\ln 2)^2)}{j \ln 2} \sigma^{-g_1} |K_{\rho}|
\end{align*}
for any $s\in (0,\tau)$. This inequality is the same as \cite[(3.4)]{HL1}.

Since the remainder of the proof of \cite[Proposition 3.2]{HL1} is valid for
the full range $1<g_0\le g_1$, we do not repeat it here.
\end{proof}


Our next step should be a proposition concerning the spread of positivity over
 space analogous to \cite[Proposition 3.3]{HL1}; however, because we need a
 much stronger result here, we defer its discussion to the next subsection.
Instead, we present a modified DeGiorgi iteration with generalized
structure conditions \eqref{I:gen-str}, which was proved as
\cite[Proposition 3.4]{HL1}. Basically, our
Proposition~\ref{S3:prop4} is equivalent to
\cite[Lemmata III.4.1, III.9.1, IV.4.1]{DB93}. We point out in particular
that \cite[Lemma IV.4.1]{DB93}, which is the same as
\cite[Lemma 3.1]{ChDB88}, follows from our proposition by taking $\theta=1$,
$k= \omega/2^m$ and $\rho =(2^{m+1}/\omega)^{(2-p)/p}R$.

\begin{proposition}\label{S3:prop4}
For a given positive constant $\theta$, there exists
$\nu_0 = \nu_{0} (\theta, \text{data}) \in (0,1)$ such that, if $u$
is a nonnegative supersolution of \eqref{I:gen} in $Q_{k,2\rho}(\theta)$ with
\begin{equation} \label{S3:prop4:hyp}
|\{(x,t)\in Q_{k,2\rho}(\theta): u(x,t) < k \}|
< \nu_0 |Q_{k,2\rho}(\theta)|
\end{equation}
for some positive constants $k$ and $\rho$, then
\[
\operatorname{ess\,inf}_{Q_{k,\rho}(\theta)} u(x,t) \geq \frac{k}{2}.
\]
\end{proposition}

We also recall \cite[Proposition 3.5]{HL1}, which will be critical
in our proof of initial regularity.

\begin{proposition}\label{S3:prop4a}
There exists $\nu_0\in (0,1)$, determined only by the data, such that,
if $u$ is a nonnegative supersolution of \eqref{I:gen} in $Q_{k,2\rho}(\theta)$ with
\begin{subequations} \label{S3:prop4a:SC}
\begin{gather}
|\{(x,t)\in Q_{k,2\rho}(\theta): u(x,t) < k \}|
< \frac {\nu_0}{\theta} |Q_{k,2\rho}(\theta)| \\
\intertext{for some positive constants $k$, $\rho$, and $\theta$ and if}
u(x,-T_{k,2\rho}(\theta))\ge k
\end{gather}
\end{subequations}
for all $x\in K_{2\rho}$, then
\[
\operatorname{ess\,inf}_{K_\rho\times(-T_{k,2\rho}(\theta),0)} u \geq \frac{k}{2}.
\]
\end{proposition}


\subsection{Expansion of positivity in space}

Throughout this subsection, $\nu$, $\nu_0$, $\rho$, and $k$ are given positive
constants with $\nu, \nu_0<1$.
Also, to simplify notation, we set
\[
T=\big( \frac k2\big)^2 G\big( \frac {k}{2\rho}\big)^{-1}.
\]

We assume that $u$ is a nonnegative supersolution of \eqref{I:gen}
in $K_{2\rho}\times (-T ,0)$ such that
\begin{equation} \label{E:51large}
\big| \{x\in K_{2\rho}: u(x,t) \le \frac k2\}\big|
\le (1-\nu_0)|K_{2\rho}|
\end{equation}
for all $t\in (-T,0)$.

We wish to prove the following proposition, which is a
generalization of \cite[Lemma IV.5.1]{DB93}. In fact, this lemma is not the
complete first alternative as described in that source; we single it out
as the crucial step in that alternative.

\begin{proposition} \label{S3:prop3a}
Let $\nu\in (0,1)$ and $\nu_0\in (0,1]$ be constants.
If $g_1\le2$ and if $u$ is a nonnegative supersolution of \eqref{I:gen} in $K_{2\rho}\times (-T,0)$ which satisfies \eqref{E:51large}, then there is a constant $\delta^*$ determined only by $\nu$, $\nu_0$, and the data such that
\begin{equation} \label{S3:prop3aestimate}
\big| \{ x\in K_\rho: u(x,t)\le \frac {\delta^*k}2\}\big| \le \nu |K_{2\rho}|
\end{equation}
for all $t\in (-T_1,0)$, where
\begin{equation} \label{S3:prop3aT1}
T_1=\big( \frac k2\big)^2 G\big( \frac {k}{\rho}\big)^{-1}.
\end{equation}
\end{proposition}

Our proof follows that of \cite[Lemma IV.5.1]{DB93} rather closely with a
few modifications based on ideas from \cite[Section 4]{Lie91}.
In addition, our proof shows much more easily that the constants in
\cite[Chapter IV]{DB93} are stable as $p\ne2$.

Our first step is as in \cite[Section IV.6]{DB93}. We show that $u$ satisfies
an additional integral inequality, which is the basis of the proof of
Proposition \ref{S3:prop3a}.
Before stating our inequalities, we introduce some notation. For positive
constants $\kappa$ and $\delta$ with $\kappa\le k/2$ and $\delta<1$,
we define two functions $\Phi_\kappa$ and $\Psi_\kappa$ as follows:
\begin{subequations} \label{S3:kfunctions}
\begin{gather}
\Phi_\kappa(\sigma) = \int_0^{(\sigma-\kappa)_-}
\frac {(1+\delta)\kappa-s}{G\big( \frac {(1+\delta)\kappa-s}{2\rho}\big)}\, ds, \\
\Psi_\kappa(\sigma)
= \ln\big[ \frac {(1+\delta)\kappa}{(1+\delta)\kappa-(\sigma-\kappa)_-}\big].
\end{gather}
\end{subequations}
We also note that there are two Lipschitz functions, $\zeta_1$ defined
on $K_{2\rho}$ and $\zeta_2$ defined on $[-T,0]$, such that
\begin{subequations} \label{S3:zeta}
\begin{gather}
\zeta_1 =0 \text { on the boundary of }K_{2\rho}, \\
\zeta_1 =1 \text { in } K_\rho, \\
|D\zeta_1| \le \frac 1\rho \text { in } K_{2\rho}, \\
\{x\in K_{2\rho}: \zeta_1(x)>\varepsilon\} \text { is convex for all }\varepsilon\in(0,1), \\
\zeta_2(-T)=0, \\
\zeta_2=1 \text { on } (-T_1,0), \\
0 \le \zeta_2' \le \big(\frac 2k\big)^2G\big(\frac k \rho\big) \text { on } (-T,0). \label{S3:zetat}
\end{gather}
\end{subequations}
Let us note that it's easy to arrange that $\zeta_2'\ge0$ and that
\[
\frac 1{\zeta_2'}\ge \big(\frac k2\big)^2G\big(\frac k{2\rho}\big)^{-1}
-\big(\frac k2\big)^2G\big(\frac {k}{\rho}\big)^{-1}.
\]
Since Lemma~\ref{S1:ineq}(b) implies that
\[
G\big(\frac {k}{\rho}\big) \ge 2^{g_0}G\big(\frac {k}{2\rho}\big)
\ge 2G\big(\frac {k}{2\rho}\big),
\]
we infer the second inequality of \eqref{S3:zetat}.

Also, we introduce the notation $D^-$ to denote the derivative
\[
D^-f(t)= \limsup_{h\to0^+} \frac {f(t)-f(t-h)}h.
\]

With these preliminaries, we can now state our integral inequality.
Our proof of this inequality is essentially the same as that
for \cite[Lemma IV.6.1]{DB93}; the new ingredient is a more careful
estimate of the integral involving $\zeta_t$ (which we denote by $I_4$).
In this way, we obtain an estimate which does not depend on $p-2$
being bounded away from zero, which was the case in
\cite[(6.9) Chapter IV]{DB93}.

\begin{lemma} \label{S3:integralinequality}
If $g_1\le2$ and if $u$ is a weak supersolution of \eqref{I:gen} in
$K_{2\rho}\times (-T,0)$ satisfying \eqref{E:51large},
then there are positive constants $\gamma$ and $\gamma_0$, determined only
by $\nu$, $\nu_0$, and the data such that
\begin{equation} \label{S3:Eintegralinequality}
D^-\Big( \int_{K_{2\rho}} \Phi_\kappa(u(x,t))\zeta^q(x,t)\, dx\Big)
+ \gamma_0\int_{K_{2\rho}} \Psi_\kappa^{g_0}(u(x,t))\zeta^q(x,t)\, dx
 \le \gamma |K_{2\rho}|
\end{equation}
for all $t\in (-T,0)$, where
\begin{equation} \label{S3:integralinequalityq}
q= g_0/(g_0-1).
\end{equation}
\end{lemma}

\begin{proof}
With
\[
u^*=\frac {(1+\delta)\kappa-(u-\kappa)_-}{2\rho},
\]
we use the test function
\[
\frac {\zeta^q((1+\delta)\kappa-(u-\kappa)_-)}{G(u^*)}
\]
in the weak form of the differential inequality satisfied by $u$
to infer that, for all sufficiently small positive $h$, we have
\[
I_1 +I_2\le I_3+I_4
\]
with
\begin{gather*}
I_1 =\int_{K_{2\rho}} \Phi_\kappa(u(x,t))\zeta^q(x,t)\,dx
-\int_{K_{2\rho}} \Phi_\kappa(u(x,t-h))\zeta^q(x,t-h)\,dx,
\\
\begin{aligned}
I_2 &=\int_{t-h}^h\int_{K_{2\rho}} \zeta^q(x,\tau) D(u-\kappa)_-(x,\tau)
 A \frac 1{G(u^*(x,\tau))}\\
&\quad\times \Big[1- \frac{u^*(x,\tau) g(u^*(x,\tau))}{G(u^*(x,\tau))}\Big]
 \, dx\,d\tau,
\end{aligned}\\
 I_3 =q\int_{t-h}^t \int_{K_{2\rho}} D\zeta(x,\tau)
 A \zeta^{q-1}(x,\tau) \frac {(1+\delta)\kappa-(u-\kappa)_-}
 {G\left(u^*(x,\tau)\right)} \,dx\, d\tau,
\\
 I_4=q\int_{t-h}^t \int_{K_{2\rho}} \Phi_\kappa(u(x,\tau))
\zeta^{q-1}(x,\tau)\zeta_t(x,\tau)\,dx\,d\tau,
\end{gather*}
and $A$ evaluated at $(x,\tau,u(x,\tau),Du(x,\tau))$ in $I_2$ and $I_3$.
We now use \eqref {I:gen-str1}
and the first inequality in \eqref{I:DeltaNabla} to see that
\[
I_2 \ge C_0(g_0-1) \int_{t-h}^t \int_{K_{2\rho}} \zeta^q(x,\tau)
 \frac {G(|D(u-\kappa)_-(x,\tau)|)}{G(u^*(x,\tau))}\,dx\,d\tau.
\]
Also, \eqref {I:gen-str2} and Lemma~\ref{S1:ineq}(e)
(with $\sigma_1= (qC_1/C_0)|D\zeta(x,\tau)|\rho u^*(x,\tau)$,
 $\sigma_2= |D(u-\kappa)_-(x,\tau)|$, and $\epsilon = \zeta(x,\tau)(g_0-1)/(2g_1)$)
imply that
\[
qD\zeta(x,\tau) \cdot A (x,\tau,u,Du) \zeta^{q-1}(x,\tau)
\frac {(1+\delta)\kappa-(u-\kappa)_-}{G\left(u^*(x,\tau)\right)} \le J_1+J_2
\]
with
\begin{gather*}
J_1 = g_1^{g_1}(\frac 2{g_0-1})^{g_1-1}\zeta^{q-g_1}
\frac {G(q(C_1/C_0)|D\zeta|\rho u^*)}{G(u^*)},\\
J_2 = \frac 12 C_0(g_0-1)\zeta^q \frac {G(|D(u-\kappa )_-|)}{G(u^*)}.
\end{gather*}
From our conditions on $\zeta$ and because $q\ge2\ge g_1$, we conclude
that there is a constant $\gamma_1$, determined only by data, such that
$J_1 \le \gamma_1$,
so
\[
I_3 \le \gamma _1h|K_{2\rho}|+ \frac 12I_2.
\]
Next, we estimate $\Phi_\kappa$. Since $\kappa\le k/2$ and
$\delta\in (0,1)$, it follows that, for all $s\in (0,(u-\kappa)_-)$,
we have $(1+\delta)\kappa-s \le 2k$ and hence
\[
G\Big( \frac {(1+\delta)\kappa-s}{2\rho}\Big)
\ge \Big(\frac {(1+\delta)\kappa-s}{2k}\Big)^2 G\big(\frac k \rho\big).
\]
It follows that
\[
\Phi_\kappa(u)
\le 4k^2G\big(\frac k \rho\big)^{-1} \int_0^{(u-\kappa)_-}
[(1+\delta)\kappa-s]^{-1}\, ds 
=4k^2G\big(\frac k \rho\big)^{-1} \Psi_\kappa(u),
\]
and therefore
\[
I_4\le 16q\int_{t-h}^t\int_{K_{2\rho}}
\Psi_\kappa(u(x,\tau))\zeta^{q-1}(x,\tau)\, dx\, d\tau.
\]
Combining all these inequalities and setting
\begin{gather*}
I_{21} = \int_{t-h}^t \int_{K_{2\rho}} \zeta^2(x,\tau)
\frac {G(|D(u-\kappa)_-(x,\tau)|)}{G(u^*(x,\tau))}\,dx\,d\tau, \\
I_{41} = \int_{t-h}^t\int_{K_{2\rho}}\Psi_\kappa(u(x,\tau))
\zeta^{q-1}(x,\tau)\, dx\, d\tau
\end{gather*}
yields
\begin{equation} \label{S3:Lintegralinequality1}
I_1+\frac 12C_0(g_0-1) I_{21} \le \gamma_1 h|K_{2\rho}| + 16qI_{41}.
\end{equation}
Our next step is to compare
\[
I_{22} = \int_{t-h}^t\int_{K_{2\rho}}\zeta^q(x,\tau)
\Psi_\kappa^{g_0}(u(x,\tau))\,dx\,d\tau
\]
with $I_{21}$. To this end, we first use
Lemma~\ref{Lpoin} with $\varphi=\zeta_1^q$, $v=(u-\kappa)_-$,
and $p=g_0$ to conclude that there is a constant $\gamma_2$ determined
only by the data and $\nu_0$ such that, for almost all
$\tau \in (t-h,t)$, we have
\begin{equation} \label{S3:I21}
\int_{K_{2\rho}} \zeta^q(x,\tau)\Psi_\kappa^{g_0}(u(x,\tau))\, dx
\le \gamma_2\rho^{g_0}\int_{K_{2\rho}}\zeta^q(x,\tau)
|D\Psi_\kappa(u(x,\tau))|^{g_0}\, dx.
\end{equation}
(Of course, we have multiplied \eqref{Lpoin:E} by $\zeta_2^q(\tau)$ here.)
Now we use the explicit expression for $\Psi_\kappa$ to infer that
\[
\rho|D\Psi_\kappa(u(x,\tau))| = \frac {|D(u-\kappa)_-(x,\tau)|}{2u^*(x,\tau)}
\le\frac {|D(u-\kappa)_-(x,\tau)|}{u^*(x,\tau)}.
\]
Whenever $|D(u-\kappa)_-(x,\tau)| \le u^*(x,\tau)$, we conclude that
\[
\rho^{g_0}|D\Psi_\kappa(u(x,\tau))|^{g_0}\le 1
\]
and, wherever $|D(u-\kappa)_-(x,\tau)| > u^*(x,\tau)$, we infer
from Lemma~\ref{S1:ineq} that
\[
\rho^{g_0}|D\Psi_\kappa(u(x,\tau))|^{g_0}
\le \frac {G(|D(u-\kappa)_-(x,\tau)|)}{G(u^*(x,\tau))}.
\]
It follows that, for any $(x,\tau)$, we have
\[
\rho^{g_0}|D\Psi_\kappa(u(x,\tau))|^{g_0}
\le 1+\frac {G(|D(u-\kappa)_-(x,\tau)|)}{G(u^*(x,\tau))}.
\]
Inserting this inequality into \eqref{S3:I21} and integrating
the resultant inequality with respect to $\tau$ yields
\[
I_{22} \le \gamma_2(I_{21}+h|K_{2\rho}|).
\]
By invoking \eqref{S3:Lintegralinequality1}, we conclude that
\[
I_1+ \frac {1}{2\gamma_2}C_0(g_0-1)I_{22}
\le \Big(\gamma_1+ \frac {1}{2\gamma_2}C_0(g_0-1)\Big)h|K_{2\rho}| +16qI_{41}.
\]
We now note that
\[
\Psi_\kappa(u)\zeta^{q-1}= \left(\Psi_\kappa^{g_0}(u) \zeta^q\right)^{1/g_0},
\]
so Young's inequality shows that
\[
\Psi_\kappa(u)\zeta^{q-1} \le \varepsilon \Psi_\kappa^{g_0}(u) \zeta^q
+ \varepsilon^{-q}
\]
for any $\varepsilon\in(0,1)$.
By choosing $\varepsilon$ sufficiently small, we see that there are constants
$\gamma_0$ and $\gamma$ such that
\[
I_1+\gamma_0 I_2 \le \gamma h|K_{2\rho}|.
\]
To complete the proof, we divide this inequality by $h$ and take the
limit superior as $h\to0^+$.
\end{proof}


Our next step is to estimate the integral of $\zeta^q$ over suitable $N$-dimensional
sets with $q$ defined by \eqref{S3:integralinequalityq}. Specifically, for
each positive integer $n$ and a number $\delta\in(0,1)$ to be further
specified, we define the set
\[
K_{\rho,n}(t) = \{x\in K_{2\rho}: u(x,t)<\delta^nk\}
\]
and we introduce the quantities
\[
A_n(t) = \frac 1{|K_{2\rho}|} \int_{K_{\rho,n}(t)} \zeta^q(x,t)\, dx, \quad
Y_n = \sup_{-T<t<0}A_n(t)
\]
We shall show that, for a suitable choice of $\delta$ (which will require
at least that $\delta\le1/2$) and $n$, we can make $Y_n$ small.
In fact, based on the discussion in \cite[Section 7 Chapter IV]{DB93},
we shall find $n_0$ and $\delta$ so that $Y_{n_0} \le \nu$.
In fact, our method is to estimate $A_{n+1}(t)$ in terms of $Y_n$ for each $n$.

We first estimate $A_{n+1}(t)$ if
\begin{equation} \label{S3:Dge}
D^- \Big( \int_{K_{2\rho}} \zeta^q(x,t) \Phi_{\delta^nk} (u(x,t))\, dx\Big) \ge 0.
\end{equation}
(This is the case \cite[(7.5) Chapter IV]{DB93}.) Our estimate now takes the
following form.

\begin{lemma} \label{S3:L7.5}
Let $\nu$ and $\nu_0$ be constants in $(0,1)$.
If \eqref{S3:Dge} holds, then there is a constant $\delta_0$, determined
only by $\nu$, $\nu_0$, and the data, such that $\delta\le \delta_0$ implies that
\begin{equation} \label{S3:L7.5:Yn1}
A_{n+1}(t) \le \nu.
\end{equation}
\end{lemma}

\begin{proof}
On $K_{\rho,n+1}(t)$, we have
\begin{align*}
\Psi_{\delta^nk}(u)
&= \ln \big[ \frac {(1+\delta)\delta^nk}{(1+\delta)\delta^nk - (u-\delta^nk)_-} \big]\\
& \ge \ln \big[ \frac {(1+\delta)\delta^nk}{(1+\delta)\delta^nk
 - (\delta^{n+1}k-\delta^nk)_-} \big] \\
&= \ln \frac {1+\delta}{2\delta}.
\end{align*}
It follows that
\[
\Big( \ln \frac {1+\delta}{2\delta} \Big)^{g_0}
\int_{K_{\rho,n+1}(t)}\zeta^q(x,t)\, dx
 \le \int_{K_{\rho,n+1}(t)} \zeta^q(x,t) \Psi_{\delta^nk}(u(x,t))\, dx.
\]
By invoking \eqref{S3:Eintegralinequality} and \eqref{S3:Dge}, we conclude that
\[
\int_{K_{\rho,n+1}(t)}\zeta^q(x,t)\, dx
\le \frac \gamma{\gamma_0}
\Big( \ln \frac {1+\delta}{2\delta} \Big)^{-g_0}|K_{2\rho}|.
\]
By choosing $\delta_0$ sufficiently small, we infer \eqref{S3:L7.5:Yn1}.
\end{proof}

Our estimate when \eqref{S3:Dge} fails is more complicated, as shown for the
power case in \cite[Section IV.8]{DB93}.

\begin{lemma} \label{S3:L7.6}
Let $\nu$ and $\nu_0$ be constants in $(0,1)$.
There are positive constants $\delta_1$ and $\sigma<1$, determined only
by $\nu$, $\nu_0$, and the data, such that if
\begin{equation} \label{S3:Dless}
D^-\Big( \int_{K_{2\rho}} \zeta^q(x,t)\Phi_{\delta^nk} (u(x,t))\, dx \Big)<0
\end{equation}
for some $\delta\in(0,\delta_1)$ and if $Y_n>\nu$, then
\begin{equation} \label{S3:L7.6equation}
A_{n+1}(t) \le \sigma Y_n.
\end{equation}
\end{lemma}

\begin{proof}
In this case, we define
\[
t_*=\sup\Big\{\tau\in(-T,t): D^-
\Big( \int_{K_{2\rho}} \zeta^q(x,\tau)\Phi_{\delta^nk} (u(x,\tau))\, dx \Big)
\ge0 \Big\}
\]
(and note that this set is nonempty). From the definition of $t_*$, we have that
\begin{equation} \label{S3:L7.6:Phiint}
\int_{K_{2\rho}} \zeta^q(x,t) \Phi_{\delta^nk} u(x,t)\, dx
\le \int_{K_{2\rho}} \zeta^q(x,t_{*}) \Phi_{\delta^nk} u(x,t_{*})\, dx.
\end{equation}
It follows from Lemma~\ref{S3:integralinequality} and the definition of $t_*$ that
\[
\int_{K_{2\rho}} \zeta^q(x,t_{*}) \Psi^{g_0}_{\delta^nk} u(x,t_{*})\, dx\le C|K_{2\rho}|,
\]
with $C=  \gamma/\gamma_0$.
Now we set
\[
K_*(s)=\{x\in K_{2\rho}: (u-\delta^nk )_-(x,t_*) >s\delta^nk\}
\]
for $s\in (0,1)$, and
\[
I_1(s)= \int_{K_*(s)} \zeta^q(x,t_*)\, dx.
\]
As in the proof of Lemma~\ref{S3:L7.5}, we have that
\[
\Phi_{s\delta^nk}(u(x,t_*)) \ge \ln\frac {1+\delta}{1+\delta-s},
\]
so
\begin{equation} \label{S3:L7.6:1}
I_1(s)\le C\Big(\ln\frac {1+\delta}{1+\delta-s}\Big)^{-g_0}|K_{2\rho}|.
\end{equation}
Moreover, if $x\in K_*(s)$, then
\[
u(x,t_*) <(1-s)\delta^nk \le \delta^nk,
\]
and hence $K_*(s)\subset K_{\rho,n}$, so
\begin{equation} \label{S3:L7.6:2}
I_1(s)\le Y_n|K_{2\rho}|.
\end{equation}
We now define
\[
s_*= \Big[1-\exp\Big(- \big( \frac {2C}{\nu}\big)^{1/g_0}\Big)\Big](1+\delta_*),
\]
with $\delta_*\in(0,1)$ chosen so that $s_*<1$.
Since $Y_n>\nu$, a simple calculation shows that
\begin{equation} \label{S3:L7.6:CYn}
C\Big(\ln \frac {1+\delta}{1+\delta -s}\Big)^{-g_0}\le \frac 12Y_n
\end{equation}
 for $s>s_*$ provided $\delta\le \delta_*$.

Next, we set
\[
I_2 =\int_{K_{2\rho}} \zeta^q(x,t_*)\Phi_{\delta^nk}(u(x,t_*))\, dx
\]
and use Fubini's theorem to conclude that
\begin{align*}
I_2
&=\int_{K_{2\rho}} \zeta^q(x,t_*)
\Big( \int_0^{\delta^nk} \frac { \chi_{\{(\delta^nk-u)+>s \}}
 ((1+\delta)\delta^nk-s) }{G\big( \frac {(1+\delta)\delta^nk-s}{2\rho}\big)}\, ds
\Big)\, dx \\
&= \int_0^{\delta^nk} \frac {(1+\delta)\delta^nk-s}
 {G\big( \frac {(1+\delta)\delta^nk-s}{2\rho}\big)}
 \Big( \int_{K_{2\rho}} \zeta^q(x,t_*) \chi_{\{(\delta^nk-u)+>s \}}\, dx\Big)\, ds
\end{align*}
Using the change of variables $\tau= s/(\delta^{n}k)$, we see that
\begin{align*}
I_2
&=\int_0^1 \frac {(1+\delta)-\tau}
 {G\big( \frac {\delta^nk(1+\delta-\tau)}{2\rho}\big)}
 \Big( \int_{K_{2\rho}} \zeta^q(x,t_*) \chi_{\{(\delta^nk-u)+>\delta^nk\tau \}}\,
  dx\Big)\, d\tau \\
&= \int_0^1 \frac {(1+\delta)-\tau}
 {G\big( \frac {\delta^nk(1+\delta-\tau)}{2\rho}\big)}I_1(\tau)\, d\tau.
\end{align*}
Combining this equation with \eqref{S3:L7.6:1}, \eqref{S3:L7.6:2}, and
\eqref{S3:L7.6:CYn} then yields
\[
I_2\le Y_n|K_{2\rho}|\Big[\int_0^{s_*} \frac {(1+\delta)-\tau}
 {G\big( \frac {\delta^nk(1+\delta-\tau)}{2\rho}\big)}\, d\tau
 + \frac 12\int_{s_*}^1 \frac {(1+\delta)-\tau}
 {G\big( \frac {\delta^nk(1+\delta-\tau)}{2\rho}\big)}\, d\tau\Big].
\]
We now define the function
\[
f(\tau)= \frac {\tau}{G\big( \frac {\delta^nk\tau}{2\rho}\big)}
\]
and we set $\sigma_*= 1-s_*$. Using the change of variables $s=1-\tau$
then yields
\[
I_2 \le Y_n|K_{2\rho}|\mathcal K,
\]
with
\[
\mathcal K= \int_{\sigma_*}^1 f(\delta+s)\, ds
+ \frac 12\int_0^{\sigma_*}f(\delta+s)\, ds.
\]
Since
\[
\mathcal K = \int_0^1f(\delta+s)\, ds -\frac 12 \int_0^{\sigma_*}
 f(\delta+s)\, ds,
 \]
it follows from \eqref{S1:fintegral:E1} that
\[
\mathcal K \le \big(1-\frac {\sigma_*}2\big) \int_0^1f(\delta+s)\, ds,
\]
and therefore
\begin{equation} \label{S3:L7.6:Phi1}
I_2 \le Y_n|K_{2\rho}|\left(1-\frac {\sigma_*}2\right) \int_0^1f(\delta+s)\, ds.
\end{equation}

Our next step is to obtain a lower bound for $I_2$.
Taking into account \eqref{S3:L7.6:Phiint}, we have
\[
I_2 \ge \int_{K_{\rho,n+1}(t)} \zeta^q(x,t)\Phi_{\delta^nk}(u(x,t))\, dx.
\]
Next, for $z<\delta^{n+1}k$, we have
\begin{align*}
\Phi_{\delta^nk}(z)
&= \int_0^{(z-\delta^nk)_-}\frac {(1+\delta)\delta^nk -s}
 {G\big( \frac {(1+\delta)\delta^nk-s}{2\rho}\big)}\, ds \\
&\ge \int_0^{\delta^nk(1-\delta)}\frac {(1+\delta)\delta^nk -s}
{G\big( \frac {(1+\delta)\delta^nk-s}{2\rho}\big)}\, ds \\
&= \int_0^{1-\delta} f(\delta+s)\, ds \\
&\ge \Big(1- \frac 2{2+\ln\delta}\Big) \int_0^1 f(\delta+s)\,ds
\end{align*}
by  \eqref{S1:fintegral:E2}, so
\[
\Phi_{\delta^nk}(u(x,t)) \ge \Big(1- \frac 2{2+\ln(1/\delta)}\Big)
\int_0^\delta f(\delta+s)\, ds
\]
for all $x\in K_{\rho,n+1}(t)$ and hence
\[
I_2 \ge \Big(1- \frac 2{2+\ln(1/\delta)}\Big)
\Big(\int_{K_{\rho,n+1}(t)} \zeta^q(x,t)\, dx\Big)
\Big(\int_0^\delta f(\delta+s)\, ds\Big).
\]
In conjunction with \eqref{S3:L7.6:Phi1}, this inequality implies that
\[
A_{n+1}(t) \le \frac {1-(\sigma_*/2)}{1- ( 2/(2+\ln(1/\delta))}Y_n.
\]
By taking $\delta_2$ sufficiently small, we can make sure that
\[
\sigma=\frac {1-(\sigma_*/2)}{1- ( 2/(2+\ln(1/\delta_2))}
\]
is in the interval $(0,1)$. If we take $\delta_1=\min\{\delta_*,\delta_2\}$,
 we then infer \eqref{S3:L7.6equation} for $\delta\le\delta_1$.
 \end{proof}
 As shown in \cite{DB93}, if $t_*$ and $t$ are equal in this proof,
we can infer \eqref{S3:L7.5:Yn1} very simply.

We are now ready to prove Proposition~\ref{S3:prop3a}.

\begin{proof}[Proof of Proposition \ref{S3:prop3a}]
Since $Y_{n+1} \le Y_n$, it follows from Lemmata~\ref{S3:L7.5}
and \ref{S3:L7.6} that, for all positive integers $n$, we have
\[
A_{n+1}(t) \le \max \{\nu,\sigma Y_n\}
\]
for all $t\in (-T,0)$ and hence
$Y_{n+1}\le \max\{\nu,\sigma Y_n\}$.
Induction implies that
\[
Y_n \le \max\{\nu,\sigma^{n-1} Y_1\}
\]
for all $n$. In addition $Y_1\le1$, so there is a positive integer $n_0$,
 determined by $\nu$, $a_0$, and the data such that $Y_{n_0}\le\nu$.

Next, we recall that $\zeta=1$ on $K_\rho\times(-T_1,0)$, and hence, for all
$t\in(-T_1,0)$, we have
\begin{align*}
\big| \{ x\in K_\rho:u(x,t)\le \delta^{n_0}k\}\big|
&= \int_{\{x\in K_\rho: u(x,t)\le \delta^{n_0}k\} }\zeta^q(x,t)\, dx \\
&\le \int_{\{x\in K_{2\rho}: u(x,t)\le \delta^{n_0}k\}} \zeta^q(x,t)\, dx
\le Y_{n_0}.
\end{align*}
The proof is complete by using the inequality $Y_{n_0} \le \nu$ and taking
 $\delta^*=\delta^{n_0}$.
\end{proof}

\subsection{Proof of main lemma}

\begin{proof}
With $\delta$ to be chosen, we use Proposition~ \ref{S3:prop1} with
 $k=\omega/2$, $\rho=2R$, $\nu_1=\frac 12$, and
\[
T= 3\big(\frac {\delta \omega}2\big)^2G\big( \frac {\delta \omega}{2R}\big)^{-1}
\]
to infer that there is a $\tau_1\in (-T, -T/3)$ such that
\[
\left|\{ x\in K_{2R}: u(x,\tau_1)\le \frac \omega2\}\right|
\le \frac 34|K_{2R}|.
\]

Next, we set $\nu=\frac 14$, $\rho=2R$, $k=\omega/2$, $\tau=\tau_1$, and
$\theta=3$. Since $\tau_1\le T$ and
\[
T = 3(\delta k)^2 G\big( \frac {2\delta k}\rho\big)^{-1}
 \le 3(\delta k)^2G\big( \frac {\delta k}{\rho}\big)^{-1},
 \]
 it follows that \eqref{S3:prop2-hyp} is satisfied, so
 Proposition~ \ref{S3:prop2} implies that
 \begin{equation} \label{S2:MainLemmaE1}
 \big| \{x\in K_{2R}: u(x,t)\le \frac {\delta \omega}2\}\big|
\le \frac 78|K_{2R}|
 \end{equation}
 for all $t\in (\tau_1,0)$ provided we take $\delta$ to be the constant from that proposition. (In particular, $\delta$ is determined only by the data.)
 Since $\tau_1\ge T/3$, it follows that
 \[
 \tau_1 \ge \big(\frac {\delta \omega}2\big)^2
G\big( \frac {\delta \omega}{2R}\big)^{-1},
 \]
 and hence \eqref{S2:MainLemmaE1} holds for all
 \[
 t\in\Big( -\big(\frac {\delta \omega}2\big)^2
G\big( \frac {\delta \omega}{2R}\big)^{-1},0\Big).
 \]

 Now we use Proposition~\ref{S3:prop3a}, with $\omega=\delta \omega$ and
$\nu$ to be chosen, to infer that there is a constant $\delta^*\in(0,1)$,
determined only by the data and $\nu$ such that
\begin{equation} \label{S2:MainLemmaE2}
\big|\{ x\in K_R: u(x,t) \le \frac {\delta^*\delta \omega}2\} \big|
\le \nu |K_{2R}|
\end{equation}
for all
\[
t\in \Big( - \big( \frac {\delta \omega}2\big)^2
G\big( \frac {\delta \omega}R\big)^{-1}, 0\Big).
\]
Since 
$G\big( \frac {\delta \omega/2}R\big) \le G\big(\frac {\delta \omega}R\big)$
and $\delta^*\le1$, it follows that
\[
\big(\frac {\delta \omega}2\big)^2
G\big(\frac {\delta \omega}R\big)^{-1}
\ge \big(\frac {\delta \omega}2\big)^2
G\big(\frac {\delta \omega/2}R\big)^{-1} 
\ge \Big( \frac {\delta^*\delta \omega/2}R\Big)^2
G\Big( \frac {\delta^*\delta \omega/2}R\Big)^{-1}.
\]
We now take $\nu_0$ to be the constant corresponding to $\theta=1$
in Proposition~\ref{S3:prop4}, and we set $\nu=2^{-N}\nu_0$, which determines
$\delta^*$. Then \eqref{S3:prop4:hyp} is satisfied for
$k=\delta^*\delta \omega/2$ and $\rho=R/2$. Proposition~\ref{S3:prop4}
then yields \eqref{S2:MainLemma:estimate} with $\mu =\delta^*\delta/4$.
\end{proof}

\section{Auxiliary theorems}\label{S4}

We now present the basic results used in the previous sections of the paper.
Since the results are either well-known or were proved in \cite{HL1},
we just state the results here.

 \subsection{Local energy estimate}
The local energy estimate is a fundamental inequality playing an important
role in the proofs of several results, especially Proposition~\ref{S3:prop1},
Proposition~\ref{S3:prop2},
 and Proposition~\ref{S3:prop3a}. We refer the reader to
\cite[Proposition 4.1]{HL1} for a proof.

 \begin{proposition}
 Let $G$ satisfy structure conditions \eqref{I:gen-str} in a cylinder
$Q_{\rho}:=K_{\rho}\times [t_0,t_1]$, and let $\zeta$ be a cutoff function
on the cylinder $Q_\rho$, vanishing on the parabolic boundary of
$Q_\rho$ with $0\le\zeta\le1$. Define constants $r$, $s$, and $q$ by
 \begin{equation}\label{S4:LocalE-rsq}
 r=1- \frac{1}{g_1}, \quad s=\frac{g_0}{g_1}, \quad \text{and} \quad q= 2g_1.
 \end{equation}
If $u$ is a locally bounded weak supersolution of \eqref{I:gen},
then there exist constants $c_0$, $c_1$, and $c_2$ depending on data such that
 \begin{equation}\label{S4:LocalE}
\begin{split}
 &\int_{K_{\rho}\times \{t_1\}} G^{r-1}
\Big(\frac{\zeta (u-k)_{-}}{\rho}\Big) (u-k)_{-}^{s+2} \zeta^{q} \,dx \\
& + c_0\iint_{Q_{\rho}} G\left(|D(u-k)_{-}|\right) G^{r-1}
 \Big(\frac{\zeta (u-k)_{-}}{\rho}\Big) (u-k)_{-}^{s} \zeta^{q} \,dx\,dt \\
&\leq c_1\iint_{Q_{\rho}} G^{r-1} \Big(\frac{\zeta (u-k)_{-}}{\rho}\Big)
  (u-k)_{-}^{s+2} \zeta^{q-1}| \zeta_{t}| \,dx\,dt \\
 &\quad + c_2 \iint_{Q_{\rho}} G\left(|D\zeta|
\zeta(u-k)_{-}\right) G^{r-1} \Big(\frac{\zeta (u-k)_{-}}{\rho}\Big)
(u-k)_{-}^{s} \,dx\,dt
 \end{split}
\end{equation}
for any constant $k$.
 \end{proposition}

We refer the reader to \cite[Proposition 4.1]{HL1} for the corresponding
result about nonpositive subsolutions.

 \subsection{Logarithmic energy estimate}

With the functions $h$ and $H$ defined in Lemma~\ref{S1:H-ineq},
the logarithmic energy estimate was proved as \cite[Proposition 4.2]{HL1},
which also contains the corresponding result for nonpositive weak subsolutions.

 \begin{proposition}
 Assume that $G$ satisfies \eqref{I:gen-str} in a cylinder $K_{R}\times [t_0, t_1]$.
 Let $q\ge g_1$ and $\delta\in (0,1)$ be constants, and let $\zeta$ be a cut-off
function which is independent of the time variable.
 Let $u$ be a nonnegative weak supersolution of \eqref{I:gen}and let $k$ be
 a positive constant. Then
 \begin{equation}\label{S4:LogE}\begin{split}
&\int_{K_{R}\times \{t_1\}} H(\Psi^2) \zeta^{q} \,dx
 + C_0 (4 g_0 - 2) \int_{t_0}^{t_1}\int_{K_{R}} G(|Du|) h(\Psi^2) (\Psi')^{2} \zeta^{q}\,dx\,dt \\
 &\leq \int_{K_{R}\times \{t_0\}} H(\Psi^2) \zeta^{q} \,dx 
  +C^* \int_{t_0}^{t_1}\int_{K_{R}} h(\Psi^2)
 \Psi (\Psi')^2 G\Big(\frac{|D\zeta|}{|\Psi'|}\Big) \zeta^{q-g_1}\,dx\,dt
 \end{split}\end{equation}
 where
\[
C^*=\frac{C_0}{g_1} \left(\frac{2 q g_1 C_1}{C_0}\right)^{g_1},\quad
 \Psi(u) = \ln^{+} \left[\frac{k}{(1+ \delta) k - (u-k)_{-}}\right].
 \]
 \end{proposition}


 \subsection{A Poincar{\'e} type inequality}

For our proofs, we shall need the following result which is
\cite[Proposition I.2.1]{DB93}.

\begin{lemma} \label{Lpoin}
Let $\Omega$ be a bounded convex subset of $\mathbb R^N$ and let $\varphi$
be a nonnegative continuous function on $\overline {\Omega}$ such that
 $\varphi\le1$ in $\Omega$ and such that the sets $\{x\in\Omega: \varphi(x)>k\}$
are convex for all $k\in(0,1)$. Then, for any $p\ge1$, there is a constant $C$
determined only by $N$ and $p$ such that
\begin{equation} \label{Lpoin:E}
\begin{aligned}
&\Big( \int_\Omega \varphi|v|^p\, dx\Big)^{1/p}\\
&\le C\frac {(\operatorname{diam}\Omega)^N}{|\{x\in \Omega:v(x)=0,\
 \varphi(x)=1\}|^{(N-1)/N}}
\Big( \int_\Omega \varphi|Dv|^p\,dx \Big)^{1/p}
\end{aligned}
\end{equation}
for all $v\in W^{1,p}$.
\end{lemma}

Note that if the set $\{x\in \Omega:v(x)=0,\ \varphi(x)=1\}$ has measure zero,
then \eqref{Lpoin:E} is true because the right hand side is infinite.


 \subsection{Embedding theorem}

Our next result is a variation on the Sobolev imbedding theorem, which is
just \cite[Theorem 4.4]{HL1}.

 \begin{theorem}\label{S4:Embedding-theorem}
 For a nonnegative function $v\in W_{0}^{1,1}(Q)$ where
$Q=K\times[t_0, t_1]$, $K\subset \mathbb{R}^{N}$, we have
 \begin{equation}\label{EB00}
\begin{split}
\iint_{Q} v \,dx\,dt 
&\leq C(N) |{Q} \cap \{v>0\}|^{\frac{1}{N+1}} \\
&\times  \Big[\operatorname{ess\,sup}_{t_0 \leq t \leq t_1} \int_{K} v \,dx
 \Big]^{\frac{1}{N+1}}
\Big[\iint_{Q} |Dv| \,dx\,dt \Big]^{\frac{N}{N+1}}.
 \end{split}
\end{equation}
 \end{theorem}

\subsection{Iteration}

Finally, we recall \cite[Lemma I.4.1]{DB93}.

 \begin{lemma}\label{S4:Iteration-lemma}
 Let $\{Y_n\}$, $n=0,1,2,\ldots$, be a sequence of positive numbers,
satisfying the recursive inequalities
 \begin{equation} \label{S4:Iteration-lemma:E}
 Y_{n+1} \leq C b^{n} Y_{n}^{1+\alpha}
 \end{equation}
 where $C,b >1$ and $\alpha >0$ are given numbers. If
 \[
 Y_0 \leq C^{-\frac{1}{\alpha}}b^{-\frac{1}{\alpha^2}},
 \]
 then $\{Y_n\}$ converges to zero as $n\rightarrow \infty$.
 \end{lemma}

\subsection*{Acknowledgments}
 This work is based on the first author's thesis at Iowa State University. 
The first author was partially supported by EPSRC grant EP/J017450/1 
and NRF grant 2015 R1A5A 1009350.

\begin{thebibliography}{00}

\bibitem{Chen84} Chen, Y. Z.;
 \emph{H\"older estimates for solutions of uniformly degenerate
 quasilinear parabolic equations},
 A Chinese summary appears in Chinese Ann. Math. Ser. A \textbf{5}
 (1984), no. 5, 663,
 Chinese Ann. Math. Ser. B,  \textbf{5}  (1984),  no. 4,  661--678.


 \bibitem{ChDB88}  Chen, Y. Z.;  DiBenedetto, E.;
 \emph{On the local behavior of solutions of singular parabolic  equations},
 Arch. Rational Mech. Anal.,
 \textbf{103}  (1988), no. 4,  319--345.

\bibitem{ChDB92} Chen, Y. Z.;  DiBenedetto, E.;
 \emph{H\"older estimates of solutions of singular parabolic equations
 with measurable coefficients},
 Arch. Rational Mech. Anal.,  \textbf{118}  (1992),  no. 3,  257--271.

 \bibitem{DG57}  De Giorgi, E.;
 \emph{Sulla differenziabilit\`a e l'analiticit\`a delle estremali degli
 integrali multipli regolari} (in Italian),
 Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. (3),
 \textbf{3}  (1957),  25--43.

\bibitem{DB86j}  DiBenedetto, E.;
 \emph{A boundary modulus of continuity for a class of singular parabolic equations},
 J. Differential Equations,  \textbf{6} (1986),  no. 3,  418--447.

 \bibitem{DB86}  DiBenedetto, E.;
 \emph{On the local behaviour of solutions of degenerate parabolic
 equations with measurable coefficients},
 Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4),
 \textbf{13}  (1986),  no. 3,  487--535.

 \bibitem{DB93} DiBenedetto, E.;
 \emph{Degenerate parabolic equations}, Universitext,
 Springer-Verlag, New York  (1993).

 \bibitem{DBGiVe10}  DiBenedetto, E.;  Gianazza, U.;  Vespri, V.;
 \emph{A new approach to the expansion of positivity set of non-negative
 solutions to certain singular parabolic partial differential equations},
 Proc. Amer. Math. Soc.,  \textbf{138}  (2010),  no. 10,  3521--3529.

\bibitem{DBGiVe12}  DiBenedetto, E.;  Gianazza, U.;  Vespri, V.;
 \emph{Harnack's inequality for degenerate and singular parabolic  equations},
 Springer Monographs in Mathematics, Springer, New York  (2012),

 \bibitem{GiSuVe10}  Gianazza, U.; Surnachev, M.;  Vespri, V.;
 \emph{A new proof of the H\"older continuity of solutions to $p$-Laplace
 type parabolic equations},
 Adv. Calc. Var.,  \textbf{3}  (2010), no. 3,  263--278.

\bibitem{Hwang} Hwang, S.;
 \emph{H\"older regularity of solutions of generalised $p$-Laplacian
type parabolic equations}, Ph. D. thesis, Iowa State University,
 (2012),  Paper 12667. http: //lib.dr.iastate.edu/etd/12667.

\bibitem{HL1} Hwang. S.; Lieberman, G. M.;
\emph{H\"older continuity of a bounded weak solution of generalized 
parabolic $p$-Laplacian equation I: degenerate case},
Electron. J. Diff. Equ., \textbf{2015} (2015), no. 287, 1--32.

 \bibitem{KrRu61}  Krasnosel'ski\u\i, M. A.;  Ruticki\u\i, Ja. B.;
 \emph{Convex functions and Orlicz spaces},
 Translated from the first Russian edition by Leo F. Boron,
 P. Noordhoff Ltd., Groningen  (1961).

 \bibitem{LaUr61}  Lady\v{z}enskaja, O. A.;  Ural'ceva, N. N.;
 \emph{Quasilinear elliptic equations and variational problems in several
 independent variables} (in Russian),
 Uspehi Mat. Nauk,  \textbf{16},  (1961),  no. 1 (97), 19--90.
 Russian Math. Surveys, \textbf{16}, (1961), 17--92.

 \bibitem{LaUr62}
 Lady\v{z}enskaja, O. A.;  Ural'ceva, N. N.;
 \emph{A boundary value problem for linear and quasilinear parabolic equations},
(in Russian),  Izv. Akad. Nauk SSSR Ser. Mat.,  \textbf{26}  (1962),  5--52.
Amer. Math. Soc. Transl. (2), \textbf{47} (1965), 217--267.

\bibitem{LaSoUr67} Lady\v{z}enskaja, O. A.;  Solonnikov, V. A.;  Ural'ceva, N. N.;
 \emph{Linear and quasilinear equations of parabolic type}
(in Russian), Translated from the Russian by S. Smith. Translations of
 Mathematical Monographs, Vol. 23, American Mathematical Society,
 Providence, R.I.,  (1967).

 \bibitem{Lie91}  Lieberman, G. M.;
 \emph{The natural generalization of the natural conditions of
 Ladyzhenskaya and Ural'tseva for elliptic equations},
 Comm. Partial Differential Equations,
 \textbf{16}  (1991), no. 2-3, 311--361.

\bibitem{Lie96}  Lieberman, G. M.;
 \emph{Second order parabolic differential equations},
 publisher={World Scientific Publishing Co. Inc.},
River Edge, NJ  (1996).

 \bibitem{Mos64}  Moser, J.;
 \emph{A Harnack inequality for parabolic differential equations},
 Comm. Pure Appl. Math.,  \textbf{17}  (1964),  101--137.
	
\bibitem{RaRe91}  Rao, M. M.;  Ren, Z. D.;
 \emph{Theory of Orlicz spaces},
 Monographs and Textbooks in Pure and Applied Mathematics,
 \textbf{146},  Marcel Dekker Inc., New York,  (1991).

 \bibitem{Urb08}  Urbano, J. M.;
 \emph{The method of intrinsic scaling},
 Lecture Notes in Mathematics,  \textbf{1930},
 A systematic approach to regularity for degenerate and singular
 PDEs, Springer-Verlag, Berlin  (2008).

\end{thebibliography}

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