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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 200, pp. 1--21.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/200\hfil H\"older regularity for signed solutions]
{H\"older regularity for signed solutions to singular porous medium type equations}

\author[S. Puglisi \hfil EJDE-2012/200\hfilneg]
{Simona Puglisi} 

\address{Simona Puglisi \newline
Dipartimento di Matematica e Informatica \\
University of Catania \\
Viale A. Doria 6, 95125 Catania, Italy}
\email{spuglisi@dmi.unict.it}

\thanks{Submitted July 17, 2012. Published November 15, 2012.}
\subjclass[2000]{35K67, 35B65, 35B45}
\keywords{Singular parabolic equations; H\"older continuity}

\begin{abstract}
 We prove H\"older regularity for bounded signed solution to singular porous
 medium type equations, whose prototype is
 $$
 u_t-\operatorname{div}m|u|^{m-1}Du=0\quad\text{weakly in }E_T,
 $$
 with $m\in(0,1)$.
\end{abstract}

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

\section{Introduction and statement of main result}

Let $E$ be an open set in $\mathbb{R}^N$, for $T>0$ denote the 
cylindrical domain
$$
E_T=E\times (0,T]
$$
and let $\Gamma=\partial E_T\setminus\bar E\times\{T\}$ be its parabolic boundary.
We consider quasi-linear homogeneous singular parabolic partial differential
 equation
\begin{equation} \label{pm}
u_t-\operatorname{div} \mathcal{A}(x,t,u,Du)=0 \quad \text{weakly in }E_T,
\end{equation}
where $\mathcal{A}:E_T\times\mathbb{R}^{N+1}\to\mathbb{R}^N$ is measurable
and subject to the structure conditions
\begin{equation}\label{struct cond}
\left\{\begin{gathered}
\mathcal{A}(x,t,z,\xi)\cdot\xi\geq C_0m|z|^{m-1}|\xi|^2 \\
|\mathcal{A}(x,t,z,\xi)|\leq C_1m|z|^{m-1}|\xi|
\end{gathered}\right.
\end{equation}
for a.e. $(x,t)\in E_T$, for every $z\in\mathbb{R}$, $\xi\in\mathbb{R}^N$, where $C_0$,
 $C_1$ are given positive constants and $0<m<1$.

The prototype of this class of parabolic equations is the porous medium equation
\[
u_t-\operatorname{div}m|u|^{m-1}Du=0 \quad \text{weakly in }E_T.
\]
The modulus of ellipticity of this class of parabolic equations is $m|u|^{m-1}$.
Whenever $m>1$, such a modulus vanishes when $u$ vanishes, and for this reason 
we say that the equation \eqref{pm}-\eqref{struct cond} is \emph{degenerate}.
 Whenever $0<m<1$, such a modulus approaches infinity as $u\to0$, and for this 
reason we say that the equation \eqref{pm}-\eqref{struct cond} is \emph{singular}.
One also speaks about \emph{slow}, when $m>1$, or \emph{fast diffusion}, 
when $0<m<1$ (see the monograph \cite{jl}).

We are interested only in \emph{local} solutions to \emph{singular} porous
 medium type equation.
The parameters $\{N,m,C_0,C_1\}$ are the data, and we say that a generic 
constant $\gamma=\gamma(N,m,C_0,C_1)$ depends upon the data, if it
 can be quantitatively determined a priori only in terms of the indicated parameters.
 As usual, in the following the constant $\gamma$ may change from line to line.

Let us give the notion of weak solution for this kind of equations as follows.
A function $u\in C_{\rm loc}(0,T;L^2_{\rm loc}(E))$ with 
$|u|^m\in L^2_{\rm loc}(0,T;H^1_{\rm loc}(E))$ is a local weak 
sub(super)-solution to \eqref{pm} if for every compact set
 $\mathcal{K}\subset E$ and every subinterval $[t_1,t_2]\subset(0,T]$
\[
\int_{\mathcal{K}} u\varphi\, dx\big|_{t_1}^{t_2}
+\int_{t_1}^{t_2}\int_{\mathcal{K}}[-u\varphi_t
+\mathcal{A}(x,t,u,Du)\cdot D\varphi]\,dx\,dt\leq(\geq)\,0,
\]
for all non-negative test functions 
$\varphi\in H^1_{\rm loc}(0,T;L^2(\mathcal{K}))
\cap L^2_{\rm loc}(0,T;H^1_0(\mathcal{K}))$.


Our aim is to show that locally bounded, local, weak solutions of variable sign 
to our problem \eqref{pm}-\eqref{struct cond}, with $0<m<1$, are locally H\"older
continuous.

Let us introduce the parabolic $m$-distance of a compact set
 $\mathcal{K}\subset E_T$ from the parabolic boundary $\Gamma$ in the following way
\[
\operatorname{m-dist}(\mathcal{K},\Gamma)
=\inf_{(x,t)\in\mathcal{K}, (y,s)\in\Gamma}
\Big(\|u\|_{\infty,\,E_T}^\frac{1-m}{2}|x-y|+|t-s|^{1/2}\Big).
\]
We can state the main result of this paper as follows.

\begin{theorem}\label{teor}
Let $u$ be a bounded, local, weak solution to \eqref{pm}-\eqref{struct cond}.
Then $u$ is locally H\"older continuous in $E_T$ and there exist constants 
$c>1$ and $\alpha\in(0,1)$ such that for every compact set $\mathcal{K}\subset E_T$
\[
|u(x_1,t_1)-u(x_2,t_2)|\leq c\,\|u\|_{\infty,E_T}
\Big(\frac{\|u\|_{\infty,E_T}^{\frac{1-m}{2}}|x_1-x_2|+|t_1-t_2|^{1/2}}
{\operatorname{m-dist}(\mathcal{K},\Gamma)}\Big)^\alpha,
\]
for every pair of points $(x_1,t_1),(x_2,t_2)\in\mathcal{K}$.
\end{theorem}

The constant $c$ depends only upon the data, the norm
 $\|u\|_{\infty,\mathcal{K}}$ and $\operatorname{m-dist}(\mathcal{K},\Gamma)$;
 the constant $\alpha$ depends only upon the data and the norm
 $\|u\|_{\infty,\,\mathcal{K}}$.

In some physical applications it is natural to consider positive solutions 
to quasi-linear parabolic equations of the form \eqref{pm}, and it is also a
very useful simplification from the mathematical point of view. 
Therefore, most of the papers directly deal with positive solutions.

A H\"older regularity result for signed solutions was obtained first by
 DiBenedetto in \cite{dib86} for degenerate ($p>2$) $p$-laplacian type equations 
and then by Chen and DiBenedetto in \cite{chen dib} for 
singular ($1<p<2$) $p$-laplacian type ones (see also \cite{dib}). 
Later on, in 1993 Porzio and Vespri \cite{porzio vespri} considered
 the case of a degenerate doubly non-linear equation, whose prototype is
\[
u_t-\operatorname{div}\big(|u|^{m-1} |Du|^{p-2} Du \big)=0,
\]
for $p\geq 2$ and $m\geq 1$. Notice that this kind of equations admits as 
a particular case both the degenerate $p$-laplacian type equations 
(for $m=1$ and $p>2$) and the degenerate porous medium type equations 
(for $p=2$ and $m>1$). As a consequence, it only remained open the case 
of the singular porous medium type equations.

We want to point out that the difficulty in our case is due to the presence 
of the term $|u|^{m-1}$ in the modulus of continuity; indeed, the fact that 
$u$ changes sign plays a crucial role here. In the $p$-laplacian case,
 the modulus of continuity is $|Du|^{p-2}$, thus the proof does not change 
if $u$ is positive or if it changes sign.
One could think to follow the lines of \cite{chen dib} with minor modification, 
but at some point it will appear $|u|^{m-1}$ that one cannot control from 
above in a sublevel of the modulus of $u$, being $0<m<1$.

An important point of our strategy is to work with a different equation, 
apparently more complicated, but instead easier to handle, to which we can reduce, 
thanks to a change of variables introduced by Vespri in \cite{vespri}.
We will apply a technique due to DiBenedetto \cite{dib86,dib} via an 
alternative argument; we will write energy estimates for super(sub)-solutions 
and logarithmic estimates. We notice that, due to the change of variables,
 our logarithmic function has to be different by the usual one 
(see for instance \cite{dib}). Then we will use the so-called reduction of 
oscillation procedure: the H\"older continuity of a solution $u$ to the 
transformed equation \eqref{pm2} will be heuristically a consequence of the
following fact: for every $(x_0,t_0)\in E_T$ there exists a family of 
nested and shrinking cylinders in which the essential oscillation of $u$ 
goes to zero in a way that can be quantitatively determined in terms of the data.
Since this result is well known for non-negative solutions (see \cite{dib,dib g v}), 
it will suffice to consider the case in which the infimum of our solution is 
negative and the supremum is positive.

\section{Change of variables}

To justify some of the following calculations, we assume $u$ to be smooth. 
In no way this is a restrictive assumption: indeed the modulus of continuity
 of $u$ will play no role in the forthcoming calculations.
 
Let us consider $n\in\mathbb{N}$ such that
\[
n>\frac1m\,,
\]
and define
\[
|v|^{n-1}v=u,
\]
which is equivalent to
\[
v=|u|^{\frac{1}{n}-1}u.
\]
Notice that
\[
Du=n|v|^{n-1}Dv, \quad Dv=\frac{1}{n}|u|^{\frac{1}{n}-1}Du.
\]
With this substitution equation \eqref{pm} becomes
\[
\big(|v|^{n-1}v\big)_t-\operatorname{div}\widetilde{\mathcal{A}}(x,t,v,Dv)=0 \quad
 \text{weakly in }E_T,
\]
where
\[
\widetilde{\mathcal{A}}(x,t,v,Dv)=\mathcal{A}(x,t,u,Du)\big|_{u=|v|^{n-1}v}.
\]
Now, let us see what the structure conditions become. We have
\begin{align*}
\widetilde{\mathcal{A}}(x,t,v,Dv)\cdot Dv
&=\frac{1}{n}|u|^{\frac{1}{n}-1}\mathcal{A}(x,t,u,Du)\cdot Du\\
&\geq\frac{m}{n}\,C_0|u|^{\frac{1}{n}+m-2}|Du|^2\\
&=nmC_0|v|^{1+nm-2n}|v|^{2(n-1)}|Dv|^2\\
&=nmC_0|v|^{nm-1}|Dv|^2;
\end{align*}
since the exponent is $nm-1>0$, the equation is ``degenerate''.

In the same way
\begin{align*}
|\widetilde{\mathcal{A}}(x,t,v,Dv)|
&=|\mathcal{A}(x,t,u,Du)|\leq mC_1|u|^{m-1}|Du|\\
&=mC_1|v|^{n(m-1)}n|v|^{n-1}|Dv|
=nmC_1|v|^{nm-1}|Dv|.
\end{align*}
If we denote our variable with $u$ again, we are then led to consider 
equations of the type
\[
(|u|^{n-1}u)_t-\operatorname{div}\widetilde{\mathcal{A}}(x,t,u,Du)=0 \quad
 \text{weakly in }E_T,
\]
with structure conditions
\begin{equation}\label{struct cond2}
\left\{\begin{gathered}
\widetilde{\mathcal{A}}(x,t,z,\xi)\cdot\xi\geq nmC_0|z|^{nm-1}|\xi|^2\\
|\widetilde{\mathcal{A}}(x,t,z,\xi)|\leq nmC_1|z|^{nm-1}|\xi|,
\end{gathered}\right.
\end{equation}
for a.e. $(x,t)\in E_T$ and for every $z\in\mathbb{R}$, $\xi\in\mathbb{R}^N$.

Without loss of generality, we can assume $n$ to be odd; in this case
\[
|u|^{n-1}u=u^n,
\]
and we can rewrite the equation as
\begin{equation} \label{pm2}
(u^n)_t-\operatorname{div}\widetilde{\mathcal{A}}(x,t,u,Du)=0 \quad
 \text{weakly in }E_T.
\end{equation}
Hence we have reduced problem \eqref{pm}-\eqref{struct cond} to \eqref{pm2}
with structure conditions \eqref{struct cond2}.

Let us now see what the notion of weak solution becomes in this new setting.
A function $u$ such that $u^n\in C_{\rm loc}(0,T;L^2_{\rm loc}(E))$ 
with $|u|^{nm}\in L^2_{\rm loc}(0,T;H^1_{\rm loc}(E))$ is a 
local weak sub(super)-solution to \eqref{pm2} if for every compact set 
$\mathcal{K}\subset E$ and every subinterval $[t_1,t_2]\subset(0,T]$
\[
\int_{\mathcal{K}} u^n\varphi\,dx\big|_{t_1}^{t_2}
+\int_{t_1}^{t_2}\int_{\mathcal{K}}[-u^n\varphi_t
+\widetilde{\mathcal{A}}(x,t,u,Du)\cdot D\varphi]\,dx\,dt\leq(\geq)\,0,
\]
for all non-negative test functions 
$\varphi\in H^1_{\rm loc}(0,T;L^2(\mathcal{K}))\cap L^2_{\rm loc}
(0,T;H^1_0(\mathcal{K}))$.

\section{Preliminaries}

Let $r,s\geq1$ and let us consider the Banach spaces
\begin{gather*}
V^{r,s}(E_T)=L^\infty\big(0,T;L^r(E)\big)\cap L^s\big(0,T;W^{1,s}(E)\big),\\
V_0^{r,s}(E_T)=L^\infty\big(0,T;L^r(E)\big)\cap L^s\big(0,T;W_0^{1,s}(E)\big),
\end{gather*}
both equipped with the norm
\[
\|v\|_{V^{r,s}(E_T)}=\operatornamewithlimits{ess\,sup}_{0<t<T}\|v(\cdot,t)\|_{r,E}+\|Dv\|_{s,E_T};
\]
when $r=s$, let $V^{r,r}(E_T)=V^r(E_T)$ and $V_0^{r,r}(E_T)=V_0^r(E_T)$.
Both spaces are embedded in $L^q(E_T)$, for some $q>s$ 
(for a proof one can see \cite{dib}).

\begin{proposition}\label{prop1}
If $v\in V^{r,s}_0(E_T)$, then there exists a positive constant $\gamma$,
depending only upon $N,r,$ and $s$, such that
\[
\iint_{E_T}|v|^q \,dx\, dt\leq\gamma^q\Big(\iint_{E_T}|Dv|^s \,dx\, dt\Big)
\Big(\operatornamewithlimits{ess\,sup}_{0<t<T}\int_E|v|^r dx\Big)^{s/N}
\]
with $q=s\frac{N+r}{N}$.
In particular 
\[
\|v\|_{q,E_T}\leq\gamma\|v\|_{V^{r,s}(E_T)}.
\]
\end{proposition}

Note that, taking $r=s$ in the previous proposition, and applying
 H\"older inequality, one obtains the following result.

\begin{proposition}\label{cor1}
If $v\in V^r_0(E_T)$, then there exists a positive constant $\gamma$ depending 
only upon $N$ and $r$, such that
\[
\|v\|^r_{r,E_T}\leq\gamma\,\big|\{|v|>0\}\big|^\frac{r}{N+r}\,\|v\|^r_{V^r(E_T)}.
\]
\end{proposition}

Given $(y,s)\in E_T$, and $\lambda,R>0$, we will denote by $K_R(y)$ the cube 
centered at $y$ with edge $2R$; i.e.,
\[
K_R(y)=\Big\{x\in\mathbb{R}^N:\max_{1\leq i\leq N}|x_i-y_{i}|<R\Big\},
\]
and let $\partial K_R(y)$ be its boundary. Let $(y,s)+Q_R(\lambda)$ be 
the generic cylinder
\[
(y,s)+Q_\rho(\lambda)=K_{\rho}(y)\times[s-\lambda,s].
\]
If $k\in\mathbb{R}$, introduce the truncated functions
\[
(u-k)_\pm=\max\{\pm(u-k),0\}.
\]
The following lemma, proved in \cite{degiorgi}, will be very useful in the sequel.

\begin{lemma}
Let $v\in W^{1,1}(K_\rho(y))$ and let $k,l\in\mathbb{R}$, with $k<l$.
There exists a constant $\gamma=\gamma(N,p)$ independent of $k,l,v,y,\rho$ such that
\begin{equation}\label{lemma degiorgi}
(l-k)|\{v>l\}|\leq\gamma\,\frac{\rho^{N+1}}{|\{v<k\}|}\int_{\{k<v<l\}}|Dv|\,dx.
\end{equation}
\end{lemma}

Let us state now a lemma on fast geometric convergence one can 
find in \cite{degiorgi}; for a simple proof see again \cite{dib} and 
 \cite{lad sol ural}.


\begin{lemma}\label{geom conv}
Let $\{Y_n\}_{n\in\mathbb{N}}$ be a sequence of positive numbers satisfying
\[
Y_{n+1}\leq Cb^nY_n^{1+\alpha},
\]
being $C,b>1$ and $\alpha>0$. If
\[
Y_0\leq C^{-\frac1\alpha}b^{-\frac1{\alpha^2}},
\]
then $Y_n$ converges to 0, as $n$ tends to $+\infty$.
\end{lemma}

Let us prove energy estimates we will need later.
We start with estimates for super-solutions, then we will state the 
analogous ones for sub-solutions.


\begin{proposition}[Energy estimates for super-solutions] 
Let $u$ be a local, weak super-solution to \eqref{struct cond2}-\eqref{pm2} in $E_T$.
 There exists a positive constant $\gamma$, depending only upon the data, 
such that for every cylinder $(y,s)+Q_R(\lambda)\subset E_T$, every level $k\in\mathbb{R}$
and every non-negative, piecewise smooth cutoff function $\zeta$ vanishing on 
$\partial K_R(y)$,
\begin{equation}\label{ee super}
\begin{split}
&\operatornamewithlimits{ess\,sup}_{s-\lambda<t\leq s}
 \int_{K_R(y)}\Big(\int_u^k(k-s)_+s^{n-1}ds\Big)\zeta^2(x,t)\,dx\\
&+\iint_{(y,s)+Q_R(\lambda)}|u|^{nm-1}\big|D[(u-k)_-\zeta]\big|^2\,dx\,d\tau\\
&\leq\gamma \Big\{\int_{K_R(y)}\Big(\int_u^k(k-s)_+s^{n-1}ds\Big)\zeta^2(x,s-\lambda)
 \,dx\\
&\quad +\iint_{(y,s)+Q_R(\lambda)}\Big(\int_u^k(k-s)_+s^{n-1}ds\Big)\zeta
 |\zeta_\tau|\,dx\,d\tau\\
&\quad +\iint_{(y,s)+Q_R(\lambda)}|u|^{nm-1}(u-k)_-^2|D\zeta|^2\,dx\,d\tau\Big\}.
\end{split}
\end{equation}
\end{proposition}

\begin{proof}
After a translation we may assume that $(y,s)$ coincides with the origin 
and it suffices to prove \eqref{ee super} for the cylinder $Q_R(\lambda)$. 
In the weak formulation of \eqref{pm2}, take the test function
\[
\varphi=-(u-k)_-\zeta^2
\]
over $Q_t=K_R\times(-\lambda\,,t],$ where $-\lambda<t\leq0$.

Taking into account that
\[
\frac{\partial}{\partial\tau}\Big(\int_u^k(k-s)_+s^{n-1}ds\Big)
=-u^{n-1}(u-k)_-u_\tau,
\]
and estimating the various terms separately, we have first
\begin{align*}
-\iint_{Q_t}(u^n)_\tau(u-k)_-\zeta^2\,dx\,d\tau
&=n\iint_{Q_t}\frac{\partial}{\partial\tau}
 \Big(\int_u^k(k-s)_+s^{n-1}ds\Big)\zeta^2\,dx\,d\tau\\
&\geq n\int_{K_R}\Big(\int_u^k(k-s)_+s^{n-1}ds\Big)\zeta^2(x,t)\,dx\\
&\quad -n\int_{K_R}\Big(\int_u^k(k-s)_+s^{n-1}ds\Big)\zeta^2(x,-\lambda)\,dx\\
&\quad -2n\iint_{Q_t}\Big(\int_u^k(k-s)_+s^{n-1}ds\Big)\zeta|\zeta_\tau|\,dx\,d\tau.
\end{align*}
From the structure conditions \eqref{struct cond2} and Young's inequality 
it follows that
\begin{align*}
&-\iint_{Q_t}\widetilde{\mathcal{A}}(x,\tau,u,Du)D\big[(u-k)_-\zeta^2\big]\,dx\,d\tau\\
&= -\iint_{Q_t}\widetilde{\mathcal{A}}(x,\tau,u,Du)D(u-k)_-\zeta^2\,dx\,d\tau\\
&\quad -2\iint_{Q_t}\widetilde{\mathcal{A}}(x,\tau,u,Du) (u-k)_- \zeta D\zeta\,dx\,d\tau\\
&\geq nmC_0\iint_{Q_t}|u|^{nm-1}|D(u-k)_-|^2 \zeta^2\,dx\,d\tau\\
&\quad -2nmC_1\iint_{Q_t}|u|^{nm-1}|D(u-k)_-| (u-k)_- \zeta|D\zeta|\,dx\,d\tau\\
&\geq nm\frac{C_0}2
\iint_{Q_t}|u|^{nm-1}\big|D[(u-k)_-\zeta]\big|^2\,dx\,d\tau\\
&\quad -2nm\frac{C_1^2}{C_0}\iint_{Q_t}|u|^{nm-1} (u-k)_-^2|D\zeta|^2\,dx\,d\tau.
\end{align*}
Combining these estimates and taking the supremum over $t\in(-\lambda,0]$,
completes the proof.
\end{proof}

\begin{proposition}[Energy estimates for sub-solutions]
Let $u$ be a local, weak sub-solution to \eqref{struct cond2}-\eqref{pm2}
 in $E_T$. There exists a positive constant $\gamma$, depending only upon the data, 
such that for every cylinder $(y,s)+Q_R(\lambda)\subset E_T$, every level 
$k\in\mathbb{R}$ and every non-negative, piecewise smooth cutoff function $\zeta$
vanishing on $\partial K_R(y)$,
\begin{equation}\label{ee sub}
\begin{split}
&\operatornamewithlimits{ess\,sup}_{s-\lambda<t\leq s}\int_{K_R(y)}
\Big(\int_k^u(s-k)_+s^{n-1}ds\Big)\zeta^2(x,t)\,dx\\
&+\iint_{(y,s)+Q_R(\lambda)}|u|^{nm-1}\big|D[(u-k)_+\zeta]\big|^2\,dx\,d\tau\\
&\leq\gamma \Big\{\int_{K_R(y)}\Big(\int_k^u(s-k)_+s^{n-1}ds\Big)\zeta^2
 (x,s-\lambda)\,dx\\
&\quad +\iint_{(y,s)+Q_R(\lambda)}\Big(\int_k^u(s-k)_+s^{n-1}ds\Big)|\zeta_\tau|\,dx\,d\tau\\
&\quad +\iint_{(y,s)+Q_R(\lambda)}|u|^{nm-1}(u-k)_+^2|D\zeta|^2\,dx\,d\tau\Big\}.
\end{split}
\end{equation}
\end{proposition}

\begin{proof}
The proof is analogous to the previous one; we just need to take the test function
$\varphi=(u-k)_+\zeta^2$
and observe that
\[
\frac{\partial}{\partial\tau}\Big(\int_k^u(s-k)_+s^{n-1}ds\Big)
=u^{n-1}(u-k)_+u_\tau. \qedhere
\]
\end{proof}

Let us introduce the logarithmic function
\[
\psi(H^n,(u^n-k^n)_+,\nu^n)=\log^+\Big(\frac{H^n}{H^n-(u^n-k^n)_++\nu^n}\Big),
\]
where
\[
H^n=\operatornamewithlimits{ess\,sup}_{(y,s)+Q_{R}(\lambda)}(u^n-k^n)_+,\quad
0<\nu^n<\min\{1,H^n\},
\]
and for $s>0$
\[
\log^+s=\max\{\log s,0\}.
\]

\begin{proposition}[Logarithmic estimates]
Let $u$ be a local, weak solution to \eqref{struct cond2}-\eqref{pm2} in $E_T$. 
There exists a positive constant $\gamma$, depending only upon the data, 
such that for every cylinder $(y,s)+Q_R(\lambda)\subset E_T$, every level
 $k\in\mathbb{R}$ and every non-negative, piecewise smooth cutoff function $\zeta=\zeta(x)$
\begin{equation}\label{log estim}
\begin{split}
&\operatornamewithlimits{ess\,sup}_{s-\lambda<t\leq s}\int_{K_R(y)}\psi^2\big(H^n,(u^n-k^n)_+,\nu^n\big)(x,t)\,
 \zeta^2(x)\,dx\\
&\leq\int_{K_R(y)}\psi^2\left(H^n,(u^n-k^n)_+,\nu^n\right)(x,s-\lambda) \zeta^2(x)
 \,dx\\
&\quad +\gamma\iint_{(y,s)+Q_R(\lambda)}|u|^{n(m-1)} \psi
 \left(H^n,(u^n-k^n)_+,\nu^n\right)|D\zeta|^2\,dx\,d\tau.
\end{split}
\end{equation}
\end{proposition}

\begin{proof}
Again we assume that $(y,s)$ coincides with the origin. Put $v=u^n$ and, 
in the weak formulation of \eqref{pm2}, take the test function
\[
\varphi=\frac{\partial\psi^2}{\partial v}\zeta^2=2\psi\psi'\zeta^2,
\]
over $Q_t=K_R\times(-\lambda,t]$, where $-\lambda<t\leq0$.
\\By direct calculation
\begin{equation}\label{psiqdrosecondo}
\left(\psi^2\right)''=2(1+\psi)(\psi')^2\in L^\infty_{\rm loc}(E_T),
\end{equation}
which implies that such a $\varphi$ is an admissible testing function.
Estimating the various terms separately, we have
\begin{align*}
\iint_{Q_t}v_\tau\frac{\partial\psi^2}{\partial v} \zeta^2\,dx\,d\tau
&=\iint_{Q_t}\frac{\partial}{\partial\tau}\psi^2 \zeta^2\,dx\,d\tau\\
&=\int_{K_R}\psi^2(x,t) \zeta^2(x)\,dx-\int_{K_R}\psi^2(x,-\lambda) \zeta^2(x)\,dx;
\end{align*}
using \eqref{psiqdrosecondo} and the structure conditions \eqref{struct cond2}
\begin{align*}
&\iint_{Q_t}\widetilde{\mathcal{A}}(x,\tau,u,Du)
D\Big(\frac{\partial\psi^2}{\partial v} \zeta^2\Big)\,dx\,d\tau\\
&= \iint_{Q_t}\widetilde{\mathcal{A}}(x,\tau,u,Du)Dv(\psi^2)'' \zeta^2\,dx\,d\tau
+2\iint_{Q_t}(\psi^2)' \zeta \widetilde{\mathcal{A}}(x,\tau,u,Du)D\zeta\,dx\,d\tau\\
&= 2n\iint_{Q_t}u^{n-1}\widetilde{\mathcal{A}}(x,\tau,u,Du)Du 
 (1+\psi)(\psi')^2 \zeta^2\,dx\,d\tau\\
&\quad +4\iint_{Q_t}\psi \psi' \zeta \widetilde{\mathcal{A}}(x,\tau,u,Du)D\zeta\,dx\,d\tau\\
&\geq 2n^2mC_0\iint_{Q_t}u^{n-1}|u|^{nm-1}|D(u-k)_+|^2(1+\psi)(\psi')^2\zeta^2\,dx\,d\tau\\
&\quad -4nmC_1\iint_{Q_t}|u|^{nm-1}|D(u-k)_+| \zeta|D\zeta| \psi\psi'\,dx\,d\tau.
\end{align*}
Applying Young's inequality, we obtain
\begin{align*}
&\iint_{Q_t}\widetilde{\mathcal{A}}(x,\tau,u,Du)D
\Big(\frac{\partial\psi^2}{\partial v} \zeta^2\Big)\,dx\,d\tau\\
&\geq 2nm(nC_0-C_1\varepsilon^2)\iint_{Q_t}|u|^{nm-1}|u|^{n-1}|D(u-k)_+|^2\psi(\psi')^2\zeta^2\,dx\,d\tau\\
&\quad -2nm\,\frac{C_1}{\varepsilon^2}\iint_{Q_t}|u|^{n(m-1)}|D\zeta|^2\psi\,dx\,d\tau.
\end{align*}
Combining these estimates, discarding the term with the gradient on 
the left-hand side, and taking the supremum over $t\in(-\lambda\,,0]$, proves 
the proposition.
\end{proof}

\section{Reduction of the oscillation}

To obtain the H\"older regularity, we argue as usual with this kind of 
estimate by a reduction-of-oscillation procedure. Let us state the basic result.

\begin{theorem}\label{red}
Let $(y,s)\in E_T$, and $\rho,\,\omega>0$ such that
\[
(y,s)+ Q_{2\theta\rho}\Big( \frac{(2\rho)^2}{\omega^{nm-1}}\Big)\subset E_T\,, \quad
\operatornamewithlimits{ess\,osc}_{(y,s)+Q_{2\theta\rho}\Big( \frac{(2\rho)^2}{\omega^{nm-1}}\Big)} u
\leq \omega\,,
\]
where 
\[\theta=\omega^{\frac{1-n}2}.
\]
Then, there exist $\eta_*,\,c_0\in (0,1)$, depending only upon data, such that
\[
\operatornamewithlimits{ess\,osc}_{\mathcal{Q}^*} u\leq \eta_* \omega\,,
\]
where
\[
\mathcal{Q}^* = (y,s)+Q_{\theta\rho} \left( \theta_* \rho^2 \right)\,, \quad
\theta_* = \frac{c_0}2\, \omega^{1-nm}\,.
\]
\end{theorem}

As we show at the end, the local H\"older continuity of locally bounded
solutions is a straightforward consequence of Theorem \ref{red}.
The proof of this theorem splits into two alternatives.

Let $\epsilon\in(0,1),R>0$, and $(y,s)\in E_T$. Consider the cylinder
\[
Q_\epsilon:=K_{R^{1-\epsilon\frac{n-1}{2}}}(y)
\times(s-R^{2-\epsilon(nm-1)},s]\subset E_T,
\]
and set
\[
\mu_+\geq\operatornamewithlimits{ess\,sup}_{(y,s)+Q_\epsilon}u\,,\quad
\mu_-\leq\operatornamewithlimits{ess\,inf}_{(y,s)+Q_\epsilon}u\,, \quad
\omega=\mu_+-\mu_-\,.
\]
Let us recall that, without loss of generality, we can assume 
$\mu_+>0$, $\mu_-<0$ and
\[
\mu_+\geq|\mu_-|.
\]
Indeed, otherwise just change the sign of $u$ and work with the new function.

If we take $2\rho<R$, and assume without loss of generality
\begin{equation}\label{assump omega}
\omega>R^\epsilon,
\end{equation}
then we guarantee that
\[
(y,s)+Q_{2\theta\rho}\Big(\frac{(2\rho)^2}{\omega^{nm-1}}\Big)\subset Q_\epsilon.
\]

\section{The first alternative}

We distinguish two alternatives; the first of them consists in assuming
\begin{equation}\label{lem 1alt}
\Big|\Big\{u<\mu_-+\frac{\omega}{2}\Big\}\cap
\Big\{(y,s)+Q_{2\theta\rho}\Big(\frac{(2\rho)^2}{\omega^{nm-1}}\big)\Big\} \Big|
\leq c_0\Big|Q_{2\theta\rho}\Big(\frac{(2\rho)^2}{\omega^{nm-1}}\Big)\Big|,
\end{equation}
being $c_0\in (0,1)$ a constant to be determined later.

Let us prove now the following De Giorgi type lemma.

\begin{lemma} There exists a number $ c_0\in (0,1)$, depending only upon data, 
such that if \eqref{lem 1alt} holds, then
\begin{equation} \label{tesi lem1}
u\geq \mu_-+\frac{\omega}{4}\quad\text{a.e. in }(y,s)+Q_{\theta\rho}
\Big(\frac{\rho^2}{\omega^{nm-1}}\Big).
\end{equation}
\end{lemma}

\begin{proof}
Without loss of generality we may assume $(y,s)=(0,0)$ and for $k=0,1,\ldots$, set
\[
\rho_k=\rho+\frac{\rho}{2^k},\quad
\widetilde{K}_k=K_{\theta\rho_k},\quad
\widetilde{Q}_k=\widetilde{K}_k\times\big(-\frac{\rho_k^2}{\omega^{nm-1}},0\big].
\]
Let $\zeta_k$ be a piecewise smooth cutoff function in $\widetilde{Q}_k$ 
vanishing on the parabolic boundary of $\widetilde{Q}_k$ such that 
$0\leq\zeta_k\leq1$, $\zeta_k=1$ in $\widetilde{Q}_{k+1}$ and
\[
|D\zeta_k|\leq\frac{2^{k+2}}{\rho}\omega^\frac{n-1}{2},\quad
0\leq \zeta_{k,t}\leq\frac{2^k}{\rho^2}\,\omega^{nm-1}.
\]
We consider the following levels
\begin{equation}\label{levels}
\begin{gathered}
h_k=\mu_-+\frac{\omega}{4}+\frac{\omega}{2^{k+2}}
  \quad \text{if }\mu_-\geq-\frac{\omega}{8},\\
h_k=\mu_-+\frac{\omega}{2^5}+\frac{\omega}{2^{k+5}}
  \quad \text{if }\mu_-<-\frac{\omega}{8}.
\end{gathered}
\end{equation}
We first treat the least favorable case in which $u$ might be close 
to zero; i.e.,
 we assume first that
\begin{equation}\label{1caso}
\mu_-\geq-\frac{\omega}{8}.
\end{equation}
Write down the energy estimates \eqref{ee super} for $(u-h_k)_-$ 
over the cylinder $\widetilde{Q}_k$, to obtain
\begin{align*}
&\operatornamewithlimits{ess\,sup}_{-\frac{\rho_k^2}{\omega^{nm-1}}<t\leq 0}\int_{\widetilde{K}_k}
\Big(\int_u^{h_k}(h_k-s)_+s^{n-1}ds\Big)\zeta_k^2(x,t)\,dx\\
&+ \iint_{\widetilde{Q}_k}|u|^{nm-1}\big|D[(u-h_k)_-\zeta_k]\big|^2\,dx\,d\tau\\
&\leq\gamma \Big\{\iint_{\widetilde{Q}_k}\Big(\int_u^{h_k}(h_k-s)_+s^{n-1}ds\Big)
 |\zeta_{k,\tau}|\,dx\,d\tau\\
&\quad + \iint_{\widetilde{Q}_k}|u|^{nm-1}(u-h_k)_-^2|D\zeta_k|^2\,dx\,d\tau\Big\}.
\end{align*}
Let us introduce the truncation
\[
v=\max \Big(u,\frac{\omega}{2^4}\Big),
\]
in order to estimate the terms with the integral over $[u,h_k]$; we have
\begin{equation}\label{stima int1}
\begin{split}
\int_u^{h_k}(h_k-s)_+s^{n-1}ds
&\geq\int_v^{h_k}(h_k-s)_+s^{n-1}ds\\
&\geq v^{n-1}\frac{(v-h_k)^2_-}{2}
 \geq\Big(\frac{\omega}{2^4}\Big)^{n-1}\frac{(v-h_k)_-^2}{2}\,.
\end{split}
\end{equation}
On the other hand, as $(u-h_k)_-\leq\omega$ and $-\frac\omega8\leq\mu_-<0$, we have
\begin{equation}\label{stima int2}
\int_u^{h_k}(h_k-s)_+s^{n-1}ds\leq h_k^{n-1}\,\frac{(u-h_k)^2_-}{2}\leq\frac{\omega^{n+1}}{2}\,.
\end{equation}
By the definition of $v$, we obtain
\begin{equation}\label{Dv}
\begin{split}
&\iint_{\widetilde{Q}_k}v^{nm-1}\big|D[(v-h_k)_-\zeta_k]\big|^2\,dx\,d\tau\\
&= \iint_{\widetilde{Q}_k\cap\left\{u>\frac{\omega}{2^4}\right\}}|u|^{nm-1}
 \big|D[(u-h_k)_-\zeta_k]\big|^2\,dx\,d\tau\\
&\quad +\iint_{\widetilde{Q}_k\cap\left\{u\leq \frac{\omega}{2^4}\right\}}
\Big(\frac{\omega}{2^4}\Big)^{nm-1}\left(\frac{\omega}{2^4}-h_k\right)_-^2|D\zeta_k|^2\,dx\,d\tau\\
&\leq \iint_{\widetilde{Q}_k}|u|^{nm-1}\big|D[(u-h_k)_-\zeta_k]\big|^2\,dx\,d\tau
+\frac{2^{2(k+1)}}{\rho^2}\,\omega^{n(m+1)}|A_k|\,,
\end{split}
\end{equation}
where
\[
A_k=\{u<h_k\}\cap \widetilde{Q}_k.
\]
Let us observe that 
\begin{equation}\label{inclusion}
A_k=\widetilde A_k:=\{v<h_k\}\cap \widetilde{Q}_k.
\end{equation}
Indeed, the inclusion $A_k\supseteq\widetilde A_k$ follows by the definition of $v$; 
let us now prove the other one:
 if $v=u$ there is nothing to prove; if $v=\frac{\omega}{2^4}$, 
by \eqref{1caso} we have
\[
h_k=\mu_-+\frac{\omega}{4}+\frac{\omega}{2^{k+2}}\geq\frac{\omega}{8}
+\frac{\omega}{2^{k+2}}\geq\frac{\omega}{2^4}.
\]
Taking into account that $|u|\leq\omega$, \eqref{stima int1}-\eqref{inclusion} yield
\begin{align*}
&\Big(\frac{\omega}{2^4}\Big)^{n-1}
 \operatornamewithlimits{ess\,sup}_{-\frac{\rho_k^2}{\omega^{nm-1}}<t\leq 0}
 \int_{\widetilde{K}_k}\! (v-h_k)_-^2 \zeta_k^2(x,t)\,dx
 +\iint_{\widetilde{Q}_k}\!  v^{nm-1}\big|D[(v-h_k)_-\zeta_k]\big|^2\,dx\,d\tau\\
&\leq\gamma\frac{2^{2k}}{\rho^2}\,\omega^{n(m+1)}|\widetilde{A}_k|,
\end{align*}
and again, thanks to the definition of $v$, it follows that
\begin{equation}\label{1 lem1}
\begin{split}
&\operatornamewithlimits{ess\,sup}_{-\frac{\rho_k^2}{\omega^{nm-1}}<t\leq 0}
 \int_{\widetilde{K}_k}(v-h_k)_-^2 \zeta_k^2(x,t)\,dx
+\Big(\frac{\omega}{2^4}\Big)^{n(m-1)}
 \iint_{\widetilde{Q}_k}\big|D[(v-h_k)_-\zeta_k]\big|^2\,dx\,d\tau\\
&\leq\gamma\frac{2^{2k}}{\rho^2}\,\omega^{nm+1}|\widetilde{A}_k|.
\end{split}
\end{equation}
The change of variables
\[
\bar{x}=x\,\theta^{-1}, \quad \bar{t}=\omega^{nm-1}\tau
\]
maps the cube $\widetilde{K}_k$ into $K_{\rho_k}$,
and the cylinder $\widetilde{Q}_k$ into
$Q_k=K_{\rho_k}\times(-\rho_k^2,0]$.
With $(\bar{x},\bar{t})\to u(\bar{x},\bar{t})$ denoting again
the transformed function, the assumption \eqref{lem 1alt} of the lemma implies
\begin{equation}\label{lem 1alt camb}
\Big|\Big\{v<\mu_-+\frac{\omega}{2}\Big\}\cap Q_0\Big|\leq c_0|Q_0|.
\end{equation}
Performing such a change of variables in \eqref{1 lem1}, we have
\begin{align*}
&\operatornamewithlimits{ess\,sup}_{-\rho_k^2<t\leq 0}\int_{K_{\rho_k}}(v-h_k)_-^2 \zeta_k^2(\bar{x},t)\,d\bar{x}
+\iint_{Q_k}\big|D[(v-h_k)_-\zeta_k]\big|^2d\bar{x}d\bar{t}\\
&\leq\gamma\frac{2^{2k}}{\rho^2}\,\omega^2|\bar{A}_k|,
\end{align*}
where
\[
\bar{A}_k=\{v<h_k\}\cap Q_k.
\]
This implies
\begin{equation}\label{stimaV2}
\big\|(v-h_k)_-\zeta_k\big\|_{V^2(Q_k)}^2
\leq \gamma\frac{2^{2k}}{\rho^2}\,\omega^2|\bar{A}_k|.
\end{equation}
Then from Proposition \ref{cor1} with $r=2$ and \eqref{stimaV2}, one obtains
\begin{align*}
\iint_{Q_{k+1}}(v-h_k)_-^2d\bar{x}d\bar{t}
&\leq\iint_{Q_k}(v-h_k)_-^2\zeta_k^2\,d\bar{x}d\bar{t}\\
&\leq \gamma|\{v<h_k\}\cap Q_k|^\frac{2}{N+2}\,\big\|(v-h_k)_-
 \zeta_k\big\|^2_{V^2(Q_k)}\\
&\leq\gamma\frac{2^{2k}}{\rho^2}\,\omega^2|\bar{A}_k|^{1+\frac{2}{N+2}};
\end{align*}
the left-hand side is estimated by
\begin{align*}
\iint_{Q_{k+1}}(v-h_k)_-^2d\bar{x} d\bar{t}
&=\iint_{Q_{k+1}\cap\{v<h_k\}}(h_k-v)^2d\bar{x} d\bar{t}\\
&\geq \iint_{Q_{k+1}\cap\{v<h_{k+1}\}}(h_k-v)^2d\bar{x} d\bar{t}\\
&\geq(h_k-h_{k+1})^2|\bar{A}_{k+1}|\\
&=\left(\frac{\omega}{2^{k+3}}\right)^2|\bar{A}_{k+1}|.
\end{align*}
Combining the previous estimates yields
\[
|\bar{A}_{k+1}|\leq\gamma\frac{2^{4k}}{\rho^2}|\bar{A}_k|^{1+\frac{2}{N+2}},
\]
and setting
\[
Y_k=\frac{|\bar{A}_k|}{|Q_k|}\,,
\]
it follows that
\[
Y_{k+1} \leq\gamma\,2^{4k}\,Y_k^{1+\frac{2}{N+2}}.
\]
Thanks to Lemma \ref{geom conv}, we deduce that $Y_k$ tends to zero as
 $k\to\infty$, provided
\[
Y_0=\frac{|\{v<h_0\}\cap Q_0|}{|Q_0|}
=\frac{|\left\{v<\mu_-+\frac{\omega}{2}\right\}\cap Q_0|}{|Q_0|}
\leq \gamma^{-\frac{N+2}{2}}\,2^{-(N+2)^2},
\]
that is \eqref{lem 1alt camb}, with $c_0:=\gamma^{-\frac{N+2}{2}}\,2^{-(N+2)^2}$.

Therefore,
\[
v\geq \mu_-+\frac{\omega}{4}\quad \text{a.e. in }K_\rho\times(-\rho^2,0].
\]
Returning to the variables $x,t$, we have
\begin{equation} \label{quasi tesi lem1}
v\geq \mu_-+\frac{\omega}{4}\quad \text{a.e. in }
Q_{\theta\rho}\Big(\frac{\rho^2}{\omega^{nm-1}}\Big);
\end{equation}
this implies that $u=v$ in $Q_{\theta\rho}\big(\frac{\rho^2}{\omega^{nm-1}}\big)$ and,
consequently, \eqref{tesi lem1}. In fact, by contradiction, if there were a
point $(x,t)\in Q_{\theta\rho}\big(\frac{\rho^2}{\omega^{nm-1}}\big)$ such that
$v(x,t)=\frac{\omega}{2^4}$, by \eqref{quasi tesi lem1} and \eqref{1caso},
we would obtain
\[
\frac{\omega}{2^4}\geq\mu_-+\displaystyle\frac{\omega}{4}\geq\frac{\omega}{8}\,.
\]
Assume now that \eqref{1caso} is violated; that is,
$\mu_-<-\frac{\omega}{8}$. Choosing the levels $h_k$ according to \eqref{levels},
we have
\[
h_k=\mu_-+\frac{\omega}{2^5}+\frac{\omega}{2^{k+5}}
 <-\frac{\omega}{8}+\frac{\omega}{2^5}+\frac{\omega}{2^{k+5}}\leq-\frac{\omega}{2^5}\,.
\]
Thus on the set $\{u\leq h_k\}$, one has
\[
|u|^{nm-1}\geq\Big(\frac{\omega}{2^5}\Big)^{nm-1}.
\]
It follows that $|u|^{nm-1}$ can be estimate above and below by $\omega^{nm-1}$
up to a constant, depending only upon the data; the proof can be repeated
 as before, but in this case there is no need to introduce the truncated function $v$.
\end{proof}

Therefore under assumption \eqref{lem 1alt},
\[
-\operatornamewithlimits{ess\,inf}_{Q_{\theta\rho}(\frac{\rho^2}{\omega^{nm-1}})}u
\leq -\mu_--\frac{\omega}{4};
\]
adding
$\operatornamewithlimits{ess\,sup}\limits_{Q_{\theta\rho}(\frac{\rho^2}{\omega^{nm-1}})}u$,
gives
\[
\operatornamewithlimits{ess\,osc}_{Q_{\theta\rho}(\frac{\rho^2}{\omega^{nm-1}})}u
\leq\operatornamewithlimits{ess\,sup}_{Q_{\theta\rho}(\frac{\rho^2}{\omega^{nm-1}})}u
-\mu_--\frac{\omega}{4}\leq\frac{3}{4}\,\omega.
\]

\section{The second alternative}

Let us recall the two fundamental hypotheses we assume, namely
\[
\mu_+>0\,,\quad \mu_-<0\,,\quad \mu_+\geq|\mu_-|.
\]
Throughout this new section, let us assume that \eqref{lem 1alt} does not hold; i.e.,
\[
\Big|\big\{u\geq\mu_-+\frac{\omega}{2}\big\}\cap
\Big\{(y,s)+Q_{2\theta\rho} \Big(\frac{(2\rho)^2}{\omega^{nm-1}}\Big)\Big\} \Big|
<(1-c_0)\Big|Q_{2\theta\rho}\Big(\frac{(2\rho)^2}{\omega^{nm-1}}\Big)\Big|.
\]
For simplicity in the following we assume $(y,s)=(0,0)$.

\begin{lemma}
There exists a time level $t^*$ in the interval
$\big(-\frac{(2\rho)^2}{\omega^{nm-1}},
 -\frac{c_0}{2}\frac{(2\rho)^2}{\omega^{nm-1}}\big)$
such that
\begin{equation}\label{K 2alt}
\Big|\big\{u(\cdot,t^*)<\mu_-+\frac{\omega}{2}\big\}\cap K_{2\theta\rho}\Big|
>\frac{c_0}{2}|K_{2\theta\rho}|.
\end{equation}
This in turn implies
\begin{equation}\label{Kreverse 2alt}
\Big|\big\{u(\cdot,t^*)\geq\mu_+-\frac{\omega}{4}\big\}\cap K_{2\theta\rho}\Big|
\leq\Big(1-\frac{c_0}{2}\Big)|K_{2\theta\rho}|.
\end{equation}
\end{lemma}

\begin{proof}
By contradiction, suppose that \eqref{K 2alt} does not hold for any $t^*$ 
in the indicated range; then
\begin{align*}
\Big|\big\{u<\mu_-+\frac{\omega}{2}\big\}\cap Q_{2\theta\rho}
\Big(\frac{(2\rho)^2}{\omega^{nm-1}}\Big)\Big|
&=\int_{-\frac{(2\rho)^2}{\omega^{nm-1}}}
 ^{-\frac{c_0}{2}\frac{(2\rho)^2}{\omega^{nm-1}}}
\Big|\big\{u(\cdot,t^*)<\mu_-+\frac\omega2\big\}\cap K_{2\theta\rho}\Big|dt^*\\
&\quad +\int_{-\frac{c_0}{2}\frac{(2\rho)^2}{\omega^{nm-1}}}^{0}
\Big|\big\{u(\cdot,t^*)<\mu_-+\frac\omega2\big\}\cap K_{2\theta\rho}\Big|dt^*\\
&\leq\frac{c_0}{2}|K_{2\theta\rho}|\Big(1-\frac{c_0}{2}\Big)
 \frac{(2\rho)^2}{\omega^{nm-1}}+|K_{2\theta\rho}|
 \frac{c_0}{2}\frac{(2\rho)^2}{\omega^{nm-1}}\\
&<c_0\big|Q_{2\theta\rho}\big(\frac{(2\rho)^2}{\omega^{nm-1}}\big)\big|.
\end{align*}
This proves \eqref{K 2alt}; \eqref{Kreverse 2alt} follows by the fact 
that \eqref{K 2alt} is equivalent to
\[
\Big|\big\{u(\cdot,t^*)\geq\mu_-+\frac{\omega}{2}\big\}\cap K_{2\theta\rho}\Big|
<\Big(1-\frac{c_0}{2}\Big)|K_{2\theta\rho}|,
\]
and $\mu_-+\frac{\omega}{2}\leq\mu_+-\frac{\omega}{4}$.
\end{proof}

The next lemma asserts that a property similar to \eqref{Kreverse 2alt} 
continues to hold for all time levels from $t^*$ up to zero.

\begin{lemma}
There exists a positive integer $j^*$, depending upon the data and $c_0$, such that
\[
\Big|\big\{u(\cdot,t)>\mu_+-\frac{\omega}{2^{j^*}}\big\}\cap K_{2\theta\rho}\Big|
<\big(1-\frac{c_0^2}{4}\big)|K_{2\theta\rho}|,
\]
for all times $t^*<t<0$.
\end{lemma}

\begin{proof}
Consider the logarithmic estimates \eqref{log estim} written over the 
cylinder $K_{2\theta\rho}\times (t^*,0)$ for the function $(u^n-k^n)_+$
 and for the level $k=\big(\mu_+^n-(\frac{\omega}{4})^n\big)^{1/n}$.
Notice that, thanks to our assumptions, $\mu_+>\frac{\omega}{4}$, so $k>0$.
The number $\nu$ in the definition of the logarithmic function is taken
 as $\nu=\frac{\omega}{2^{j+2}}$, where $j$ is a positive number to be chosen. 
Thus we have
\[
\psi\left(H^n,(u^n-k^n)_+,\nu^n\right)
=\log^+\Big(\frac{H^n}{H^n-(u^n-k^n)_++\frac{\omega^n}{2^{(j+2)n}}}\Big),
\]
where
\[
H^n=\operatornamewithlimits{ess\,sup}_{K_{2\theta\rho}\times(t^*,0)}
\left[u^n-\big(\mu_+^n-(\frac{\omega}{4})^n\big)\right]_+.
\]
The cutoff function $x\to\zeta(x)$ is taken such that
\[
\zeta =1 \quad \text{on $K_{(1-\sigma)2\theta\rho}$  for }\sigma\in(0,1),\quad
|D\zeta|\leq\frac{1}{\sigma\theta\rho}\,.
\]
With these choices, inequality \eqref{log estim} yields
\begin{equation}\label{1 up to 0}
\begin{split}
&\int_{K_{(1-\sigma)2\theta\rho}}\psi^2(x,t)\,dx\\
&\leq\int_{K_{2\theta\rho}}\psi^2(x,t^*)\,dx
+\gamma\int_{t^*}^0\int_{K_{2\theta\rho}}|u|^{n(m-1)} \psi|D\zeta|^2\,dx\,d\tau,
\end{split}
\end{equation}
for all $t^*\leq t\leq 0$.
Let us observe that
\[
\psi\leq\log\bigg(\frac{\frac{\omega^n}{2^{2n}}}{\frac{\omega^n}{2^{(j+2)n}}}\bigg)
=jn\log2.
\]
To estimate the first integral on the right-hand side of \eqref{1 up to 0}, observe that $\psi$  vanishes on the set $\{u^n<k^n\}$ and that $\mu_+^n-\big(\frac{\omega}{4}\big)^n\geq\left(\mu_+-\frac{\omega}{4}\right)^n$; therefore by \eqref{Kreverse 2alt}
\[
\int_{K_{2\theta\rho}}\psi^2(x,t^*)\,dx\leq
 j^2n^2\log^22\Big(1-\frac{c_0}{2}\Big)|K_{2\theta\rho}|.
\]
The remaining integral is estimated as follows
\begin{align*}
&\gamma\int_{t^*}^0\int_{K_{2\theta\rho}}|u|^{n(m-1)} \psi|D\zeta|^2\,dx\,d\tau\\
&\leq\frac{\gamma}{(\sigma\theta\rho)^2}\,jn\log2\,
 \frac{(2\rho)^2}{\omega^{nm-1}}\,\omega^{n(m-1)}|K_{2\theta\rho}|
=\frac{\gamma}{\sigma^2}\,jn|K_{2\theta\rho}|.
\end{align*}
Combining the previous estimates,
\begin{equation}\label{2 up to 0}
\int_{K_{(1-\sigma)2\theta\rho}}\psi^2(x,t)\,dx
\leq\left\{ j^2n^2\log^22\Big(1-\frac{c_0}{2}\Big)
+\frac{\gamma}{\sigma^2}\,jn\right\}|K_{2\theta\rho}|
\end{equation}
for all $t^*\leq t\leq0$. The left-hand side of \eqref{2 up to 0}
 is estimated below by integrating over the smaller set
\[
\Big\{u^n>\mu_+^n-\frac{\omega^n}{2^{(j+2)n}}\Big\};
\]
on such a set, since $\psi$ is a decreasing function of $H^n$, we have
\[
\psi^2\geq\log^2\Big(\frac{\frac{\omega^n}{2^{2n}}}{\frac{\omega^n}{2^{(j+1)n}}}\Big)
=(j-1)^2n^2\log^22;
\]
hence, for all $t^*\leq t\leq0$, we obtain
\[
\Big|\Big\{u^n(\cdot,t)>\mu_+^n-\frac{\omega^n}{2^{(j+2)n}}\Big\}
\cap K_{(1-\sigma)2\theta\rho}\Big|
\leq \Big\{\big(\frac{j}{j-1}\big)^2\Big(1-\frac{c_0}{2}\Big)
+\frac{\gamma}{\sigma^2j}\Big\}|K_{2\theta\rho}|.
\]
On the other hand,
\begin{align*}
&\Big|\Big\{u^n(\cdot,t)>\mu_+^n-\frac{\omega^n}{2^{(j+2)n}}\Big\}
\cap K_{2\theta\rho}\Big|\\
&\leq \Big|\Big\{u^n(\cdot,t)>\mu_+^n-\frac{\omega^n}{2^{(j+2)n}}\Big\}
 \cap K_{(1-\sigma)2\theta\rho}\Big|+|K_{2\theta\rho}\setminus K_{(1-\sigma)2\theta\rho}|\\
&\leq \Big|\Big\{u^n(\cdot,t)>\mu_+^n-\frac{\omega^n}{2^{(j+2)n}}\Big\}
 \cap K_{(1-\sigma)2\theta\rho}\Big|+N\sigma|K_{2\theta\rho}|.
\end{align*}
Then
\[
\Big|\Big\{u^n(\cdot,t)>\mu_+^n-\frac{\omega^n}{2^{(j+2)n}}\Big\}\cap
K_{2\theta\rho}\Big|
\leq\Big\{\Big(\frac{j}{j-1}\Big)^2\Big(1-\frac{c_0}{2}\Big)
 +\frac{\gamma}{\sigma^2j}+N\sigma\Big\}|K_{2\theta\rho}|.
\]
for all $t^*\leq t\leq0$.
 Now choose $\sigma$ so small and then $j$ so large as to obtain
\begin{align*}
\Big| &\big\{u(\cdot,t)>\Big(\mu_+^n-\frac{\omega^n}{2^{(j+2)n}}\Big)
^{1/n}\big\}\cap K_{2\theta\rho}\Big|
\leq\big(1-\frac{c_0^2}{4}\big)|K_{2\theta\rho}|&\forall& t^*\leq t\leq0.
\end{align*}
Notice that our hypotheses imply $\mu_+\geq\frac{\omega}{2},\,\mu_+<\omega$;
therefore,
\begin{align*}
\Big(\mu_+^n-\frac{\omega^n}{2^{(j+2)n}}\Big)^{1/n}
&<\Big(\mu_+^n-\frac{\mu_+^n}{2^{(j+2)n}}\Big)^{1/n}
 =\mu_+\Big(1-\frac{1}{2^{(j+2)n}}\Big)^{1/n}\\
&\leq \mu_+\Big(1-\frac{1}{2^{(j+2)n}\,n}\Big)
 \leq\mu_+-\frac{\omega}{2^{(j+2)n+1}\,n}\,.
\end{align*}
The proof is completed once we choose $j^*$ as the smallest integer such that
\[
\mu_+-\frac{\omega}{2^{(j+2)n+1}\,n}\leq\mu_+-\frac{\omega}{2^{j^*}}. \qedhere
\] 
\end{proof}


\begin{corollary}\label{coroll}
For all $j\geq j^*$ and for all times 
$-\frac{c_0}{2}\frac{(2\rho)^2}{\omega^{nm-1}}<t<0$,
\begin{equation}\label{cor degiorgi}
\Big|\Big\{u(\cdot,t)>\mu_+-\frac{\omega}{2^{j}}\Big\}
 \cap K_{2\theta\rho}\Big|<\Big(1-\frac{c_0^2}{4}\Big)|K_{2\theta\rho}|.
\end{equation}
\end{corollary}

Motivated by Corollary \ref{coroll}, introduce the cylinder
\[
Q_*=K_{2\theta\rho}\times\big(-\theta_*(2\rho)^2,0\big], \quad
\text{with } \theta_*=\frac{c_0}{2}\omega^{1-nm}.
\]


\begin{lemma}\label{lem Q_*}
For every $\nu_*\in(0,1)$, there exists a positive integer 
$q_*=q_*(\text{data},\nu_*)$ such that
\[
\Big|\Big\{u\geq\mu_+-\frac{\omega}{2^{j_*+q_*}}\Big\}\cap Q_* \Big|\leq\nu_*|Q_*|.
\]
\end{lemma}

\begin{proof}
Write down the energy estimates \eqref{ee sub} for the truncated 
functions $(u-k_j)_+$, with $k_j=\mu_+-\frac{\omega}{2^j}$, 
for $j=j_*,\ldots,j_*+q_*$ over the cylinder
\[
\widetilde{Q}=K_{4\theta\rho}\times\Big(-c_0\frac{(2\rho)^2}{\omega^{nm-1}},0\Big]
\supset Q_*\,;
\]
the cutoff function $\zeta$ is taken to be one on $Q_*$, vanishing on the 
parabolic boundary of $\widetilde{Q}$ and such that
\[
|D\zeta|\leq\frac{1}{\theta\rho}\,,\quad 
0\leq \zeta_t\leq\frac{\omega^{nm-1}}{c_0\rho^2}\,.
\]
Thanks to these choices, the energy estimates \eqref{ee sub} take the form
\begin{align*}
&\iint_{\tilde Q}|u|^{nm-1}|D(u-k_j)_+|^2\zeta^2\,dx\,d\tau\\
&\leq \gamma\Big\{\frac{\omega^{nm-1}}{c_0\rho^2}\iint_{\tilde Q}
\Big(\int_{k_j}^u(s-k_j)_+s^{n-1}ds\Big)\,dx\,d\tau\\
&\quad +\frac{\omega^{n-1}}{\rho^2}\iint_{\tilde Q}|u|^{nm-1}(u-k_j)_+^2\,dx\,d\tau\Big\}.
\end{align*}
Estimating
\[
\begin{split}
\int_{k_j}^u(s-k_j)_+s^{n-1}ds
\leq u^{n-1}\frac{(u-k_j)_+^2}{2}
\leq\omega^{n-1}\frac{(u-k_j)_+^2}{2}\,,
\end{split}
\]
and taking into account that $(u-k_j)_+\leq\frac{\omega}{2^j}$, yields
\[
\iint_{\tilde{Q}}|u|^{nm-1}|D(u-k_j)_+|^2\zeta^2\,dx\,d\tau
\leq\gamma\Big(\frac{\omega}{2^j}\Big)^2\omega^{n-1}
\frac{\omega^{nm-1}}{ c_0\,\rho^2}|Q_*|.
\]
Note that $u>k_j\geq\frac{\omega}{4}$: indeed the second inequality is 
equivalent to
\[
\mu_+\geq|\mu_-|\Big(\frac14+\frac1{2^j}\Big)\Big(\frac34-\frac1{2^j}\Big)^{-1},
\]
and this is implied by our assumptions.
Thus we can estimate
\begin{align*}
\iint_{\widetilde{Q}}|u|^{nm-1}|D(u-k_j)_+|^2\zeta^2\,dx\,d\tau
&\geq\iint_{Q_*}|u|^{nm-1}|D(u-k_j)_+|^2\,dx\,d\tau\\
&\geq\Big(\frac{\omega}{4}\Big)^{nm-1}\iint_{Q_*}|D(u-k_j)_+|^2\,dx\,d\tau;
\end{align*}
it follows that
\begin{equation}\label{en ineq}
\iint_{Q_*}|D(u-k_j)_+|^2\,dx\,d\tau
\leq\gamma\Big(\frac{\omega}{2^j}\Big)^2\omega^{n-1}\frac{1}{c_0\rho^2}|Q_*|.
\end{equation}
Next, apply the isoperimetric inequality \eqref{lemma degiorgi} to the 
function $u(\cdot, t)$, for $t$ in the range $(-\theta_*(2\rho)^2,0]$,
 over the cube $K_{2\theta\rho}$, and for the levels
\[
k=k_j<l=k_{j+1};
\]
in this way $(l-k)=\frac{\omega}{2^{j+1}}$\,.

Taking into account \eqref{cor degiorgi}, this gives
\begin{align*}
&\frac{\omega}{2^{j+1}}|\{u(\cdot,t)>k_{j+1}\}\cap K_{2\theta\rho}|\\
&\leq \frac{(2\theta\rho)^{N+1}}{|\{u(\cdot,t)<k_j\}
 \cap K_{2\theta\rho}|}\int_{\{k_j<u(\cdot,t)<k_{j+1}\}\cap K_{2\theta\rho}}|Du|\,dx\\
&\leq \frac{8\theta\rho}{c_0^2}\int_{\{k_j<u(\cdot,t)<k_{j+1}\}
 \cap K_{2\theta\rho}}|Du|\,dx;
\end{align*}
integrating in $dt$ over the indicated interval and applying the H\"older inequality, 
one gets
\[
\frac{\omega}{2^{j+1}}|A_{j+1}|
\leq\frac{8\theta\rho}{c_0^2}\Big(\iint_{Q_*}|D(u-k_j)_+|^2\,dx\,dt\Big)^{1/2}
(|A_j|-|A_{j+1}|)^{1/2},
\]
where
\[
A_j=\{u>k_j\}\cap Q_*.
\]
Square both sides of this inequality and estimate above the term 
containing $|D(u-k_j)_+|$ by inequality \eqref{en ineq}, to obtain
\[
|A_{j+1}|^2\leq\,\frac{\gamma}{\,c_0^5\,}|Q_*|\,\left(|A_j|-|A_{j+1}|\right).
\]
Add these recursive inequalities for $j=j_*+1,\ldots,j_*+q_*-1$, 
where $q_*$ is to be chosen. Majorizing the right-hand side with the 
corresponding telescopic series, gives
\[
(q_*-2)|A_{j_*+q_*}|^2\leq\sum_{j=j_*+1}^{j_*+q_*-1}|A_{j+1}|^2
\leq \frac{\gamma}{c_0^5}|Q_*|^2.
\]
From this
\[
|A_{j_*+q_*}|\leq\frac{1}{\sqrt{q_*-2}}\,\sqrt{\frac{\gamma}{\,c_0^5\,}\,}|Q_*|.
\]
The number $\nu_*$ being fixed, choose $q_*$ from
\[
\frac{1}{\sqrt{q_*-2}}\,\sqrt{\frac{\gamma}{\,c_0^5\,}}=\nu_*. \qedhere
\]
\end{proof}

Now let $\xi\in(0,\frac{1}{2})$, $a\in(0,1)$ be fixed numbers.

\begin{lemma}\label{lem1sub}
There exists a number $c_*\in (0,1)$, depending upon the data, $\xi$, and $a$, 
such that if
\begin{equation}\label{lem 2alt}
|\left\{u\geq\mu_+-\xi\omega\right\}\cap Q_* |\leq c_*|Q_*|,
\end{equation}
then
\[
u\leq \mu_+-a\xi\omega\quad \text{a.e. in }Q_{\theta\rho}(\theta_*\rho^2).
\]
\end{lemma}

\begin{proof}
For $k=0,1,\ldots$, set
\[
\rho_k=\rho+\frac{\rho}{2^k},\quad
K_k=K_{\theta\rho_k}, \quad Q_k=K_k\times(-\theta_*\rho_k^2,0].
\]
Let $\zeta(x,t)=\zeta_1(x) \zeta_2(t)$ be a piecewise smooth cutoff 
function in $Q_k$ such that
\begin{gather*}
\zeta_1=\begin{cases}
1&\text{in }K_{k+1}\\
0&\text{in }\mathbb{R}^N\setminus K_k,
\end{cases}
\quad |D\zeta_1|\leq \frac{2^{k+2}}{\theta\rho},
\\
\zeta_2= \begin{cases}
1&\text{if }t\geq-\frac{\rho_{k+1}^2}{\omega^{nm-1}}\\
0&\text{if }t<-\frac{\rho_k^2}{\omega^{nm-1}},
\end{cases}
\quad  0\leq\zeta_{2,t}\leq \frac{2^k}{\theta_*\rho^2}.
\end{gather*}
Choose the sequence of truncating levels
\[
h_k=\mu_+-\xi_k\omega,\quad \text{where } \xi_k=a\xi+\frac{1-a}{2^k}\,\xi,
\]
and write down the energy estimates \eqref{ee sub} for $(u-h_k)_+$ 
over the cylinder $Q_k$,
\begin{align*}
&\operatornamewithlimits{ess\,sup}_{-\frac{\rho_k^2}{\omega^{nm-1}}<t\leq0}
 \int_{K_k}\Big(\int_{h_k}^u(s-h_k)_+s^{n-1}ds\Big)\zeta^2(x,t)\,dx\\
&+\iint_{Q_k}|u|^{nm-1}\big|D[(u-h_k)_+\zeta]\big|^2\,dx\,d\tau\\
&\leq \gamma\,\Big\{\iint_{Q_k}\Big(\int_{h_k}^u(s-h_k)_+s^{n-1}ds\Big)
 |\zeta_t|\,dx\,d\tau\\
&\quad +\iint_{Q_k}|u|^{nm-1}(u-h_k)_+^2|D\zeta|^2\,dx\,d\tau\Big\}.
\end{align*}
Let us estimate
\begin{gather*}
\int_{h_k}^u(s-h_k)_+s^{n-1}ds \geq h_k^{n-1}\frac{(u-h_k)^2_+}{2}\,,\\
\int_{h_k}^u(s-h_k)_+s^{n-1}ds \leq u^{n-1}\frac{(u-h_k)_+^2}{2}
 \leq\omega^{n-1}\frac{(u-h_k)_+^2}{2}\,.
\end{gather*}
Taking into account that $(u-h_k)_+\leq\xi\omega$ and the definition of 
$\theta$ and $\theta_*$, one has
\begin{align*}
&\operatornamewithlimits{ess\,sup}_{-\frac{\rho_k^2}{\omega^{nm-1}}<t\leq0} h_k^{n-1}
 \int_{K_k}\frac{(u-h_k)^2_+}{2} \zeta^2(x,t)\,dx\
 +\iint_{Q_k}|u|^{nm-1}\big|D[(u-h_k)_+\zeta]\big|^2\,dx\,d\tau\\
&\leq \gamma (\xi\omega)^2\Big\{\omega^{n-1}\frac{2^k}{\theta_*\rho^2}
+\omega^{nm-1}\frac{2^{2k}}{(\theta\rho)^2}\Big\}|A_k|\\
&=\gamma \frac{2^{2k}}{\rho^2} (\xi\omega)^2\omega^{n-1}\,\omega^{nm-1}|A_k|,
\end{align*}
where
\[
A_k=\{u<h_k\}\cap Q_k.
\]
Note that $u>h_k\geq\Big(\frac{1}{2}-\xi\Big)\omega$: indeed the last 
inequality is equivalent to
\[
\mu_+\geq|\mu_-|\Big(\frac{1}{2}-\xi+\xi_k\Big)
\Big(\frac{1}{2}+\xi-\xi_k\Big)^{-1},
\]
and this follows by our hypotheses.
Therefore, we obtain
\begin{equation}\label{sup grad}
\begin{gathered}
\operatornamewithlimits{ess\,sup}_{-\frac{\rho_k^2}{\omega^{nm-1}}<t\leq0}
\int_{K_k}(u-h_k)^2_+ \zeta^2(x,t)\,dx
\leq\gamma \frac{2^{2k}}{\rho^2}\Big(\frac{1}{2}-\xi\Big)^{1-n}
 (\xi\omega)^2\,\omega^{nm-1}|A_k|,
\\
\iint_{Q_k}|D[(u-h_k)_+\zeta]|^2\,dx\,d\tau
\leq\gamma\,\frac{2^{2k}}{\rho^2}\Big(\frac{1}{2}-\xi\Big)^{1-nm}
 (\xi\omega)^2\omega^{n-1}|A_k|.
\end{gathered}
\end{equation}
By $(u-h_k)_+\geq\frac{1-a}{2^{k+1}}\,\xi\omega$, applying the H\"older inequality,
 and then Proposition \ref{prop1}, \eqref{sup grad} yields
\begin{align*}
\frac{(1-a)^2}{2^{2(k+1)}} (\xi\omega)^2|A_{k+1}|
&\leq\iint_{Q_{k+1}}(u-h_k)_+^2\,dx\,d\tau
 \leq\iint_{Q_k}(u-h_k)_+^2\zeta^2\,dx\,d\tau\\
&\leq\Big(\iint_{Q_k}[(u-h_k)_+\zeta]^\frac{2(N+2)}{N}\,dx\,d\tau\Big)
^{N/(N+2)}|A_k|^{2/(N+2)}\\
&\leq\gamma\Big(\iint_{Q_k}\big|D[(u-h_k)_+\zeta]\big|^2\,dx\,d\tau\Big)
^{N/(N+2)}\\
&\quad\times \bigg(\operatornamewithlimits{ess\,sup}_{-\frac{\rho_k^2}{\omega^{nm-1}}<t\leq0}
 \int_{K_k}[(u-h_k)_+ \zeta]^2dx\bigg)^{2/(N+2)} |A_k|^{2/(N+2)}\\
&\leq\gamma \frac{2^{2k}}{\rho^2} (\xi\omega)^2\omega
^{\frac{2(nm-1)+N(n-1)}{N+2}} \Big(\frac{1}{2}-\xi\Big)
^{\frac{N(1-nm)+2(1-n)}{N+2}} |A_k|^{1+\frac{2}{N+2}}.
\end{align*}
It follows that 
\[
|A_{k+1}|\leq\gamma\,\frac{2^{4k}}{(1-a)^2\rho^2}\omega^\frac{2(nm-1)+N(n-1)}{N+2}
\Big(\frac{1}{2}-\xi\Big)^\frac{N(1-nm)+2(1-n)}{N+2}|A_k|^{1+\frac{2}{N+2}}.
\]
Setting
\[
Y_k=\frac{|A_k|}{|Q_k|},
\]
we obtain
\begin{align*}
Y_{k+1}&\leq \gamma \frac{2^{4k}}{(1-a)^2\rho^2}
\omega^\frac{2(nm-1)+N(n-1)}{N+2}\Big(\frac{1}{2}-\xi\Big)
^\frac{N(1-nm)+2(1-n)}{N+2}\rho^2 (\theta^N\theta_*)^\frac{2}{N+2}
 Y_k^{1+\frac{2}{N+2}}\\
&= \gamma\frac{2^{4k}}{(1-a)^2}\Big(\frac{1}{2}-\xi\Big)
^\frac{N(1-nm)+2(1-n)}{N+2}Y_k^{1+\frac{2}{N+2}}.
\end{align*}
Applying Lemma \ref{geom conv}, $Y_k$ tends to zero as $k\to\infty$, provided
\begin{align*}
Y_0&= \frac{|\{u>h_0\}\cap Q_0|}{|Q_0|}
 =\frac{|\{u>\mu_+-\xi\omega\}\cap Q_0|}{|Q_0|}\\
& \leq \frac{\gamma^{-\frac{N+2}{2}}}{(1-a)^{-(N+2)}}
 \Big(\frac{1}{2}-\xi\Big)^{\frac{N(nm-1)+2(n-1)}{2}} 2^{-(N+2)^2},
\end{align*}
which  is \eqref{lem 2alt} with 
$c_*:=\frac{\gamma^{-\frac{N+2}{2}}}{(1-a)^{-(N+2)}}
\big(\frac{1}{2}-\xi\big)^{\frac{N(nm-1)+2(n-1)}{2}}\,2^{-(N+2)^2}$.
This completes the proof.
\end{proof}

Thanks to Lemma \ref{lem Q_*}, we can apply Lemma \ref{lem1sub} with 
$\xi=\frac{1}{2^{j_*+q_*}}$ and $a=\frac{1}{2}$, getting
\[
u \leq \mu_+-\frac{\omega}{2^{j_*+q_*+1}}\quad
\text{a.e. in }Q_{\theta\rho}\big(\theta_*\rho^2\big),
\]
which implies
\[
\operatornamewithlimits{ess\,sup}_{Q_{\theta\rho}(\theta_*\rho^2)}u
\leq\mu_+-\frac{\omega}{2^{j_*+q_*+1}}\,.
\]
Hence
\[
\operatornamewithlimits{ess\,osc}_{Q_{\theta\rho}(\theta_*\rho^2)}u
\leq\mu_+-\operatornamewithlimits {ess\,inf}_{Q_{\theta\rho}(\theta_*\rho^2)}u
 -\frac{\omega}{2^{j_*+q_*+1}}\leq\omega\Big(1-\frac{1}{2^{j_*+q_*+1}}\Big).
\]


\section{Conclusion}

The two alternatives just discussed can be combined to prove Theorem \ref{red}.

\begin{proof}[Proof of Theorem \ref{red}]
The concluding statement of the first alternative says that
\[
\operatornamewithlimits{ess\,osc}_{Q_{\theta\rho}(\frac{\rho^2}{\omega^{nm-1}})}u\leq \frac{3}{4}\,\omega;
\]
analogously, the conclusion of the second alternative is that
\[
\operatornamewithlimits{ess\,osc}_{\mathcal{Q}^*} u =\operatornamewithlimits{ess\,osc}_{Q_{\theta\rho}(\theta_*\rho^2)}u
\leq \omega\Big(1-\frac{1}{2^{j_*+q_*+1}}\Big).
\]
Recalling the definition of $\theta_*$, we observe that
\[
\mathcal{Q}^* =Q_{\theta\rho}(\theta_*\rho^2)
\subset Q_{\theta\rho}\Big(\frac{\rho^2}{\omega^{nm-1}}\Big).
\]
The thesis follows by defining
\[
\eta_* := 1-\frac{1}{2^{j_*+q_*+1}}\,.
\]
\end{proof}

We are now ready to prove the local H\"older regularity.

\begin{proof}[Proof of Theorem \ref{teor}]
Let us remind that we fixed $\epsilon\in(0,1),\,R>0$, $(y,s)\in E_T$, and we
considered the cylinder
\[
Q_\epsilon=K_{R^{1-\epsilon\frac{n-1}{2}}}(y)\times(s-R^{2-\epsilon(nm-1)},s]\subset E_T.
\]
Let now $\beta,\delta\in(0,1)$ to be chosen, and let us introduce the sequences
\[
R_k:= \beta^kR, \quad \omega_k:=\delta^k\omega, \quad
\theta_k:=\omega_k^{\frac{1-n}2}, \quad 
\mathcal Q_k:= (y,s)+Q_{\theta_k R_k}\Big(\frac{R_k^2}{\omega_k^{nm-1}}\Big),
\]
for $k\in\mathbb{N}$.
The thesis follows by standard arguments once we prove that
\begin{equation}
\begin{gathered}
\mathcal Q_{k+1} \subset\mathcal Q_k \subset Q_\epsilon \subset E_T \quad
  \forall k\in\mathbb{N},  \\
\operatornamewithlimits{ess\,osc}_{\mathcal Q_k} u \leq \omega_k.
\end{gathered} \label{star}
\end{equation}
The inclusion $\mathcal Q_0\subset Q_\epsilon$ immediately follows by
assumption \eqref{assump omega},
while $\mathcal Q_{k+1}\subset\mathcal Q_k$ is equivalent to
\[
\beta\leq\min\{\delta^{\frac{n-1}2},\,\delta^{\frac{nm-1}2}\}
=\delta^{\frac{n-1}2}.
\]
To prove \eqref{star}, we will argue by induction. The validity for $k=0$
is true by construction since
\[
\operatornamewithlimits{ess\,osc}_{\mathcal Q_0}u\leq\operatornamewithlimits{ess\,osc}_{Q_\epsilon}u\leq\omega.
\]
Assume that \eqref{star} holds for $k$ and apply Theorem~\ref{red}
 taking $\rho = \frac{R_k}2$ and $\omega=\omega_k$; thanks to these choices
\[
\theta=\theta_k\,, \quad
(y,s)+ Q_{2\theta\rho}\Big( \frac{(2\rho)^2}{\omega^{nm-1}}\Big) = \mathcal Q_k\,.
\]
The assumptions of Theorem~\ref{red} are satisfied because \eqref{star} holds
for $k$; hence, we have
$\operatorname{ess\,osc}_{\mathcal{Q}^*} u\leq \eta_* \omega_k$,
where in this setting
\[
\mathcal{Q}^* = (y,s)+Q_{\theta_k\frac{R_k}2}
\left( \frac{c_0}{8}\, \omega_k^{1-nm} R_k^2 \right).
\]
This leads us to choose $\delta=\eta_*\in (0,1)$, so that
 $\eta_* \omega_k=\omega_{k+1}$. It remains only to check
$\mathcal Q_{k+1}\subset \mathcal{Q}^*$, which by a simple calculation
 is equivalent to
\[
\beta\leq\min\Big\{ \frac 12 \,\delta^{\frac{n-1}2}\,,
 \sqrt{\frac{c_0}8}\, \delta^{\frac{nm-1}2}\Big\}.
\]
We conclude by choosing $\beta$ small enough.
\end{proof}


\subsection*{Acknowledgments}
The author wishes to warmly thank Ugo Gianazza for his help and
 support while considering the problem. The author also would like
 to thank the anonymous referee for the helpful comments.

\begin{thebibliography}{0}

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

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

\bibitem{dib86} E. DiBenedetto;
\emph{On the local behaviour of solutions of degenerate parabolic equations 
with measurable coefficients}, Ann. Scuola Norm. Sup. Pisa Cl. Sci.
 (4) 13 (1986), no. 3, 487-535.

\bibitem{dib} E. DiBenedetto;
\emph{Degenerate Parabolic Equations}. Universitext. Springer, New York, 1993.

\bibitem{dib g v} E. DiBenedetto, U. Gianazza, V. Vespri;
\emph{Harnack's inequality for degenerate and singular equations}. 
Springer Monographs, Springer-Verlag, New York, 2012.

\bibitem{lad sol ural} O. A. Lady\v{z}enskaja, V. A. Solonnikov,
 N. N. Ural'tzeva;
\emph{Linear and quasilinear equations of parabolic type}.
Translations of Mathematical Monographs, Vol. 23 American Mathematical Society,
 Providence, R.I. 1967.

\bibitem{porzio vespri} M. M. Porzio, V. Vespri;
\emph{H\"older estimates for local solutions of some doubly nonlinear 
degenerate parabolic equations}, J. Differential Equations 103 (1993),
 no. 1, 146-178.


\bibitem{jl} J. L. V\'azquez;
\emph{The Porous Medium Equation}. 
Mathematical Theory, Oxford Math. Monogr., Oxford University Press, Oxford, 2007.

\bibitem{vespri} V. Vespri;
\emph{On the local behaviour of solutions of a certain class of doubly 
nonlinear parabolic equations}, Manuscripta Math. 75 (1992), no. 1, 65-80.

\end{thebibliography}

\end{document}