# On the regularity theory for mixed local and nonlocal quasilinear parabolic
equations

Prashanta Garain and Juha Kinnunen
Department of Mathematics
Ben-Gurion University of the Negev
P.O.B. 653
Beer Sheva 8410501, Israel
Email<EMAIL_ADDRESS>
Department of Mathematics
Aalto University
P.O. Box 11100, FI-00076, Finland
Email<EMAIL_ADDRESS>
###### Abstract.

We consider mixed local and nonlocal quasilinear parabolic equations of
$p$-Laplace type and discuss several regularity properties of weak solutions
for such equations. More precisely, we establish local boundeness of weak
subsolutions, local Hölder continuity of weak solutions, lower semicontinuity
of weak supersolutions as well as upper semicontinuity of weak subsolutions.
We also discuss the pointwise behavior of the semicontinuous representatives.
Our main results are valid for sign changing solutions. Our approach is purely
analytic and is based on energy estimates and the De Giorgi theory.

###### Key words and phrases:

Mixed local and nonlocal quasilinear parabolic equation, energy estimates,
local boundedness, Hölder continuity, lower and upper semicontinuity,
pointwise behavior, De-Giorgi theory.

###### 2010 Mathematics Subject Classification:

35B65, 35B45, 35K59, 35K92, 35M10, 35R11.

##  1\. Introduction

We discuss regularity properties of weak solutions
$u:\mathbb{R}^{N}\times(0,T)\to\mathbb{R}$ for the mixed local and nonlocal
quasilinear parabolic equation

(1.1)
$\partial_{t}u+\mathcal{L}u(x,t)-\operatorname{div}\mathcal{B}(x,t,u,\nabla
u)=g(x,t,u)\text{ in }\Omega\times(0,T),$

where $T>0$, $\Omega\subset\mathbb{R}^{N}$, with $N\geq 2$, is a bounded
domain (i.e. bounded, open and connected set) and $\mathcal{L}$ is an integro-
differential operator of the form

(1.2)
$\mathcal{L}u(x,t)=\text{P.V.}\,\int_{\mathbb{R}^{N}}\mathcal{A}\big{(}u(x,y,t)\big{)}K(x,y,t)\,dx\,dy\,dt,$

where P.V. denotes the principal value and
$\mathcal{A}:\mathbb{R}^{N}\times\mathbb{R}^{N}\times(0,T)\to\mathbb{R}$
satisfies the growth condition

(1.3)
$C_{1}|u(x,t)-u(y,t)|^{p-2}\big{(}u(x,t)-u(y,t)\big{)}\leq\mathcal{A}\big{(}u(x,y,t)\big{)}\leq
C_{2}|u(x,t)-u(y,t)|^{p-2}\big{(}u(x,t)-u(y,t)\big{)}$

for some positive constants $C_{1}$ and $C_{2}$. Throughout the article, we
assume that $1<p<\infty$ unless otherwise mentioned. The kernel $K$ is
symmetric in $x$ and $y$ such that, for some $0<s<1$ and $\Lambda\geq 1$, we
have

(1.4) $\frac{\Lambda^{-1}}{|x-y|^{N+ps}}\leq
K(x,y,t)\leq\frac{\Lambda}{|x-y|^{N+ps}},$

for every $x,y\in\mathbb{R}^{N}$ and $t\in(0,T)$. Here
$\mathcal{B}(x,t,u,\zeta):\Omega\times(0,T)\times\mathbb{R}^{N+1}\to\mathbb{R}^{N}$
is continuous with respect to $(u,\zeta)$ for almost every
$(x,t)\in\Omega\times(0,T)$ with

(1.5) $\mathcal{B}(x,t,u,\zeta)\zeta\geq
C_{3}|\zeta|^{p}\quad\text{and}\quad|\mathcal{B}(x,t,u,\zeta)|\leq
C_{4}|\zeta|^{p-1}$

for almost every $(x,t)\in\Omega\times(0,T)$ and for every
$(u,\zeta)\in\mathbb{R}^{N+1}$. We assume that the source function $g$
satisfies

(1.6) $|g(x,t,u)|\leq\alpha|u|^{l-1}+|h(x,t)|,$

for every $(x,t,u)\in\Omega\times(0,T)\times\mathbb{R}$, where
$1<l\leq\max\\{2,p(1+\frac{2}{N})\\}$, $\alpha\geq 0$ and $h$ is an integrable
function to be made precise later.

There are several interesting equations that satisfy the conditions (1.3),
(1.4) and (1.5). The leading example of (1.1) is the mixed local and nonlocal
parabolic $p$-Laplace equation

(1.7) $\partial_{t}u+a(-\Delta_{p})^{s}u-b\Delta_{p}u=0\text{ in
}\Omega\times(0,T),$

with $a>0$ and $b>0$. This equation is obtained by choosing

(1.8)
$\mathcal{A}\big{(}u(x,y,t)\big{)}=|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t)),$
(1.9) $K(x,y,t)=a|x-y|^{-N-ps}\quad\text{and}\quad\mathcal{B}(x,t,u,\nabla
u)=b|\nabla u|^{p-2}\nabla u.$

This kind of equation appears in image processing, Lévy processes etc, see
Dipierro-Valdinoci [25] and the references therein. For $\mathcal{A}$ as in
(1.8) and $a=1$ in (1.9), the operator $\mathcal{L}$ defined by (1.2) becomes
the fractional $p$-Laplace operator $(-\Delta_{p})^{s}$. For $b=1$ in (1.9),

$\operatorname{div}\big{(}\mathcal{B}(x,t,u,\nabla
u)\big{)}=\operatorname{div}(|\nabla u|^{p-2}\nabla u)=\Delta_{p}u$

is the $p$-Laplace operator. We would like to emphasize that (1.1) also
extends the following mixed Finsler and fractional $p$-Laplace equation

(1.10) $\partial_{t}u+(-\Delta_{p})^{s}u=\Delta_{\mathcal{F},p}\,u\text{ in
}\Omega\times(0,T),$

where

(1.11) $\Delta_{\mathcal{F},p}\,u=\text{div}\big{(}\mathcal{F}(\nabla
u)^{p-1}\nabla_{\eta}\mathcal{F}(\nabla u)\big{)},$

is the Finsler $p$-Laplace operator, with $\nabla_{\eta}$ denoting the
gradient operator with respect to the $\eta$ variable. Here
$\mathcal{F}:\mathbb{R}^{N}\to[0,\infty)$ is the Finsler-Minkowski norm, that
is, $\mathcal{F}$ is a nonnegative convex function in
$C^{1}(\mathbb{R}^{N}\setminus\\{0\\})$ such that $\mathcal{F}(\eta)=0$ if and
only if $\eta=0$, and $\mathcal{F}$ is even and positively homogeneous of
degree $1$, so that

(1.12) $\mathcal{F}(t\eta)=|t|\mathcal{F}(\eta)\quad\text{for every
$\eta\in\mathbb{R}^{N}$ and $t\in\mathbb{R}$}.$

Then, it follows that $\mathcal{B}(x,t,u,\nabla u)=\mathcal{F}(\nabla
u)^{p-1}\nabla_{\eta}\mathcal{F}(\nabla u)$ satisfies the hypothesis (1.5),
see Xia [41, Chapter $1$]. Various examples of Finsler-Minkowski norm
$\mathcal{F}$ can be found in, for example in Belloni-Ferone-Kawohl [4], Xia
[41, p. 22–23] and the references therein. A typical example of $\mathcal{F}$
includes the $l^{q}$-norm defined by

(1.13)
$\mathcal{F}(\eta)=\Big{(}\sum_{i=1}^{N}|\eta_{i}|^{q}\Big{)}^{\frac{1}{q}},\quad
q>1,$

where $\eta=(\eta_{1},\eta_{2},\ldots,\eta_{N})$. When $\mathcal{F}$ is the
$l^{q}$-norm as in (1.13), we have

(1.14) $\Delta_{\mathcal{F},p}\,u=\sum_{i=1}^{N}\frac{\partial}{\partial
x_{i}}\bigg{(}\Big{(}\sum_{k=1}^{N}\Big{|}\frac{\partial u}{\partial
x_{k}}\Big{|}^{q}\Big{)}^{\frac{p-q}{q}}\Big{|}\frac{\partial u}{\partial
x_{i}}\Big{|}^{q-2}\frac{\partial u}{\partial x_{i}}\bigg{)}.$

For $q=2$ in (1.14), $\Delta_{\mathcal{F},p}$ becomes the usual $p$-Laplace
operator $\Delta_{p}$. Moreover, for $q=p$ in (1.14), the operator
$\Delta_{\mathcal{F},p}$ reduces to the pseudo $p$-Laplace operator, see
Belloni-Kawohl [5] and therefore, equation (1.10) extends the following mixed
pseudo and fractional $p$-Laplace equation

(1.15)
$\partial_{t}u+(-\Delta_{p})^{s}u=\sum_{i=1}^{N}\frac{\partial}{\partial
x_{i}}\Big{(}\Big{|}\frac{\partial u}{\partial
x_{i}}\Big{|}^{p-2}\frac{\partial u}{\partial x_{i}}\Big{)}\text{ in
}\Omega\times(0,T).$

Before proceeding further, let us discuss some known results. In the purely
local case $a=0$, the equation (1.7) has been studied thoroughly over the last
decades. Local boundedness, Harnack inequalities, Hölder continuity among
several other regularity results are discussed in DiBenedetto [19] and
DiBenedetto-Gianazza-Vespri [20]. For lower semicontinuity and further
properties of weak supersolutions, we refer to Kinnunen-Lindqvist [33], Kuusi
[34], Liao [35] and the references therein. In the purely nonlocal case $b=0$,
existence, uniqueness and asymptotic behavior of strong solutions of (1.7)
were studied by Mazón-Rossi-Toledo [37] and Vázquez [40]. Strömqvist [39]
obtained a local boundedness result for weak subsolutions with $p>2$. Brasco-
Lindgren-Strömqvist [11] obtained local Hölder continuity result based on the
method of discrete differentiation. By an alternative approach, Ding-Zhang-
Zhou [21] established local Hölder continuity along with local boundedness
result. Lower semicontinuity result for doubly nonlinear nonlocal problems can
be found in Banerjee-Garain-Kinnunen [2]. In the steady state case equation
(1.7), with $a=b=1$, reduces to

(1.16) $-\Delta_{p}u+(-\Delta_{p})^{s}u=0.$

Foondun [27] proved Harnack and Hölder continuity results for (1.16) with
$p=2$. For an alternative approach to obtain Harnack inequality for (1.16),
see Chen-Kim-Song-Vondraček [14]. Existence and symmetry results together with
various other qualitative properties of solutions of (1.16) have recently been
studied by Biagi-Dipierro-Valdinoci-Vecchi [6, 7], Dipierro-Proietti Lippi-
Valdinoci [22, 23] and Dipierro-Ros-Oton-Serra-Valdinoci [24]. Much less is
known in the nonlinear case $p\neq 2$ of (1.16). For this, we refer to
Buccheri-da Silva-Miranda [12], da Silva-Salort [17], Biagi-Mugnai-Vecchi [8],
Garain-Kinnunen [28] and Garain-Ukhlov [30]. Concerning mixed parabolic
equation, Barlow-Bass-Chen-Kassmann [3] obtained Harnack inequality for the
linear equation

(1.17) $\partial_{t}u+(-\Delta)^{s}u-\Delta u=0.$

Chen-Kumagai [15] also proved Harnack inequality along with local Hölder
continuity result. Recently, Garain-kinnunen [29] proved a weak Harnack
inequality for (1.17) by analytic means.

The main purpose of this article is to establish local boundedness of weak
subsolutions (Theorem 3.1-Theorem 3.2) and local Hölder continuity of weak
solutions of (1.1) (Theorem 4.1). Further, we provide lower and upper
semicontinuous representatives of weak supersolutions (Theorem 5.1) and weak
subsolutions (Corollary 5.2) of (1.1) respectively. We also investigate the
pointwise behavior of such representatives (Theorem 5.3). Moreover, all of our
main results are valid for sign changing solutions. Shang-Fang-Zhang [38]
recently established the local boundedness and Hölder continuity for

$\partial_{t}u+\mathcal{L}u(x,t)-\Delta_{p}u=0\text{ in }\Omega\times(0,T),$

but our results cover a wider class of equations. As far as we are aware, all
our main results are new, even for the homogeneous case $g\equiv 0$ in (1.1).
In contrast to the approach from probability and analysis introduced in [3,
15], we study the problem (1.1) with a purely analytic approach. To this end,
we adopt the approach from Castro-Kuusi-Palatucci [16], Ding-Zhang-Zhou [21]
to the mixed problem (1.1) to obtain the local boundedness result (Theorem
3.1-Theorem 3.2) and local Hölder continuity result in Theorem 4.1. Due to the
nonlocality, a tail quantity defined by (2.2) appears in our local boundedness
estimate. This captures both the local and nonlocal behavior of the equation
(1.1). We refer to Kassmann [32], Castro-Kuusi-Palatucci [16], Brasco-
Lindgren-Schikorra-Strömqvist [9, 10, 11], Ding-Zhang-Zhou [21] for the purely
nonlocal tail and Garain-Kinnunen [28, 29] for the mixed local and nonlocal
tail. To provide the semicontinuous representatives (Theorem 5.1 and Corollary
5.2) and their pointwise behavior (Theorem 5.3), we use the theory from Liao
[35] and adopt the approach from Banerjee-Garain-Kinnunen [2, 28]. We obtain
energy estimates and De Giorgi type lemmas to prove our main results. In
Section $2$, we discuss the functional setting for the problem (1.1) and state
some useful results. Section $3$ is devoted to prove the local boundedness
result. In Section $4$, we prove the local Hölder continuity result and in
Section $5$, we prove the semicontinuity results and investigate their
pointwise bahavior.

##  2\. Functional setting and auxiliary results

In this section, we discuss the functional setting of weak solutions for the
problem (1.1) and state some useful results.

###  2.1. Notation

The following notation will be used throughout the paper. We denote the
positive and negative parts of $a\in\mathbb{R}$ by $a_{+}=\max\\{a,0\\}$ and
$a_{-}=\max\\{-a,0\\}$ respectively. The conjugate exponent of $t>1$ is
written as $t^{\prime}=\frac{t}{t-1}$. The Lebesgue measure of a set $S$ is
denoted by $|S|$. The barred integral sign $\fint_{S}f$ denotes the integral
average of $f$ over $S$. We write $C$ to denote a constant which may vary from
line to line or even in the same line. The dependencies of the constant $C$ on
the parameters $r_{1},r_{2},\ldots,r_{k}$ are indicated as
$C=C(r_{1},r_{2},\ldots,r_{k})$. For $(r,\theta)\in(0,\infty)\times\mathbb{R}$
and $(x_{0},t_{0})\in\mathbb{R}^{N+1}$, we consider an open ball
$B_{r}(x_{0})$ of radius $r$ with centre at $x_{0}$ and a cylinder
$\mathcal{Q}_{r,\theta}(x_{0},t_{0})=B_{r}(x_{0})\times(t_{0}-\theta
r^{p},t_{0}+\theta r^{p})$. If $\theta=1$, we write
$\mathcal{Q}_{r}(x_{0},t_{0})=\mathcal{Q}_{r,\theta}(x_{0},t_{0})$. Moreover,
we write $\Omega_{T}=\Omega\times(0,T)$ with $T>0$.

###  2.2. Sobolev spaces

Let $1<p<\infty$ and assume that $\Omega\subset\mathbb{R}^{N}$ is an open
connected set. Recall that the Lebesgue space $L^{p}(\Omega)$ is the set of
measurable functions $u:\Omega\to\mathbb{R}$ such that
$\|u\|_{L^{p}(\Omega)}<\infty$, where

$\|u\|_{L^{p}(\Omega)}=\left(\int_{\Omega}|u|^{p}\,dx\right)^{\frac{1}{p}}.$

If $0<p\leq 1$, we denote by $L^{p}(\Omega)$ to be the set of measurable
functions $u:\Omega\to\mathbb{R}$ such that $\int_{\Omega}|u|^{p}\,dx<\infty$.
We say that $u\in L^{p}_{\mathrm{loc}}(\Omega)$ if $u\in
L^{p}(\Omega^{\prime})$ for every $\Omega^{\prime}\Subset\Omega$.

For $1<p<\infty$, the classical Sobolev space $W^{1,p}(\Omega)$ is defined by

$W^{1,p}(\Omega)=\\{u\in L^{p}(\Omega):\|u\|_{W^{1,p}(\Omega)}<\infty\\},$

where

$\|u\|_{W^{1,p}(\Omega)}=\left(\int_{\Omega}|u(x)|^{p}\,dx+\int_{\Omega}|\nabla
u(x)|^{p}\,dx\right)^{\frac{1}{p}}.$

The Sobolev space $W_{0}^{1,p}(\Omega)$ with zero boundary value is defined by

$W_{0}^{1,p}(\Omega)=\\{u\in W^{1,p}(\Omega):u=0\text{ in
}\mathbb{R}^{N}\setminus\Omega\\}.$

We present some known theory of fractional Sobolev spaces, see Di Nezza-
Palatucci-Valdinoci [18] for more details.

######  Definition 2.1.

Let $1<p<\infty$ and $0<s<1$. Assume that $\Omega$ is an open and connected
set in $\mathbb{R}^{N}$. The fractional Sobolev space $W^{s,p}(\Omega)$ is
defined by

$W^{s,p}(\Omega)=\\{u\in L^{p}(\Omega):\|u\|_{W^{s,p}(\Omega)}<\infty\\},$

where

$\|u\|_{W^{s,p}(\Omega)}=\left(\int_{\Omega}|u(x)|^{p}\,dx+\int_{\Omega}\int_{\Omega}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+ps}}\,dx\,dy\right)^{\frac{1}{p}}.$

The fractional Sobolev space with zero boundary value is defined by

$W_{0}^{s,p}(\Omega)=\\{u\in W^{s,p}(\mathbb{R}^{N}):u=0\text{ in
}\mathbb{R}^{N}\setminus\Omega\\}.$

For $0<s\leq 1$, the space $W^{s,p}_{\mathrm{loc}}(\Omega)$ is defined by
requiring that a function belongs to $W^{s,p}(\Omega^{\prime})$ for every
$\Omega^{\prime}\Subset\Omega$. Here $\Omega^{\prime}\Subset\Omega$ denotes
that $\overline{\Omega^{\prime}}$ is a compact subset of $\Omega$. The Sobolev
spaces $W^{s,p}(\Omega)$ and $W_{0}^{s,p}(\Omega)$, with $1<p<\infty$ and
$0<s\leq 1$, are reflexive Banach spaces, see [18, 36, 26].

The next result asserts that the classical Sobolev space is continuously
embedded in the fractional Sobolev space, see [18, Proposition 2.2].

######  Lemma 2.2.

Let $1<p<\infty$, $0<s<1$ and assume that $\Omega$ is a smooth bounded domain
in $\mathbb{R}^{N}$. There exists a constant $C=C(N,p,s)$ such that

$||u||_{W^{s,p}(\Omega)}\leq C||u||_{W^{1,p}(\Omega)}$

for every $u\in W^{1,p}(\Omega)$.

The following result for the fractional Sobolev spaces with zero boundary
value follows from [12, Lemma 2.1].

######  Lemma 2.3.

Let $1<p<\infty$, $0<s<1$ and assume that $\Omega$ be a bounded domain in
$\mathbb{R}^{N}$. There exists a constant $C=C(N,p,s,\Omega)$ such that

$\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+ps}}\,dx\,dy\leq
C\int_{\Omega}|\nabla u(x)|^{p}\,dx$

for every $u\in W_{0}^{1,p}(\Omega)$. Here we consider the zero extension of
$u$ to the complement of $\Omega$.

Next we introduce a tail space and a mixed parabolic tail that will be used
throughout the paper. We refer to Brasco-Lindgren-Schikorra [10] for tail
space and to Garain-Kinnunen [29] for mixed parabolic tail with $p=2$.

######  Definition 2.4.

Let $m_{1}>0$ and $m_{2}>0$. We define a tail space
$L_{m_{1}}^{m_{2}}(\mathbb{R}^{N})$ by

(2.1) $L_{m_{1}}^{m_{2}}(\mathbb{R}^{N})=\bigg{\\{}u\in
L^{m_{2}}_{\mathrm{loc}}(\mathbb{R}^{N}):\int_{\mathbb{R}^{N}}\frac{|u(y)|^{m_{2}}}{1+|y|^{N+m_{1}}}\,dy<\infty\bigg{\\}},$

endowed with the norm

$\|u\|_{L_{m_{1}}^{m_{2}}(\mathbb{R}^{N})}=\left(\int_{\mathbb{R}^{N}}\frac{|u(y)|^{m_{2}}}{1+|y|^{N+m_{1}}}\,dy\right)^{\frac{1}{m_{2}}}.$

######  Definition 2.5.

Let $(x_{0},t_{0})\in\mathbb{R}^{N}\times(0,T)$ and the interval
$I=[t_{0}-T_{1},t_{0}+T_{2}]\subset(0,T)$. We define the parabolic tail by

(2.2)
$\begin{split}\mathrm{Tail}_{\infty}(u;x_{0},r,I)&=\mathrm{Tail}_{\infty}(u;x_{0},r,t_{0}-T_{1},t_{0}+T_{2})\\\
&=\bigg{(}r^{p}\operatorname*{ess\,sup}_{t\in I}\int_{\mathbb{R}^{N}\setminus
B_{r}(x_{0})}\frac{|u(y,t)|^{p-1}}{|y-x_{0}|^{N+ps}}\,dy\bigg{)}^{\frac{1}{p-1}}.\end{split}$

From the definitions above, it is clear that for any $v\in
L^{\infty}(I,L^{p-1}_{ps}(\mathbb{R}^{N}))$, the parabolic tail
$\mathrm{Tail}_{\infty}(v;x_{0},r,I)$ is well defined. Now we are ready to
define the notion of weak solutions for the problem (1.1) as follows:

######  Definition 2.6.

Let $g$ satisfies the hypothesis (1.6) for
$1<l\leq\max\\{2,p(1+\frac{2}{N})\\}$ and $h\in
L^{l^{\prime}}_{\mathrm{loc}}(\Omega_{T})$. We say that $u\in
L^{p}_{\mathrm{loc}}\big{(}0,T;W^{1,p}_{\mathrm{loc}}(\Omega)\big{)}\cap
C_{\mathrm{loc}}\big{(}0,T;L^{2}_{\mathrm{loc}}(\Omega)\big{)}\cap
L^{\infty}_{\mathrm{loc}}\big{(}0,T;L^{p-1}_{ps}(\mathbb{R}^{N})\big{)}$ is a
weak subsolution (or supersolution) of (1.1) if for every
$\Omega^{\prime}\times(t_{1},t_{2})\Subset\Omega_{T}$ and for every
nonnegative $\phi\in
L^{p}_{\mathrm{loc}}\big{(}0,T;W^{1,p}_{0}(\Omega^{\prime})\big{)}\cap
W^{1,2}_{\mathrm{loc}}\big{(}0,T;L^{2}(\Omega^{\prime})\big{)}$, we have

(2.3)
$\begin{split}&\int_{\Omega^{\prime}}u(x,t_{2})\phi(x,t_{2})\,dx-\int_{\Omega^{\prime}}u(x,t_{1})\phi(x,t_{1})\,dx-\int_{t_{1}}^{t_{2}}\int_{\Omega^{\prime}}u(x,t)\partial_{t}\phi(x,t)\,dx\,dt\\\
&+\int_{t_{1}}^{t_{2}}\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\mathcal{A}\big{(}u(x,y,t)\big{)}\big{(}\phi(x,t)-\phi(y,t)\big{)}K(x,y,t)\,dx\,dy\,dt\\\
&+\int_{t_{1}}^{t_{2}}\int_{\Omega^{\prime}}\mathcal{B}(x,t,u,\nabla
u)\nabla\phi(x,t)\,dx\,dt\leq(\text{ or
}\geq)\int_{t_{1}}^{t_{2}}\int_{\Omega^{\prime}}g(x,t,u)\phi(x,t)\,dx\,dt.\end{split}$

Moreover, we say that $u$ is a weak solution of (1.1) if the equality holds in
(2.3) for every $\phi\in
L^{p}_{\mathrm{loc}}\big{(}0,T;W^{1,p}_{0}(\Omega^{\prime})\big{)}\cap
W^{1,2}_{\mathrm{loc}}\big{(}0,T;L^{2}(\Omega^{\prime})\big{)}$ without a sign
restriction.

######  Remark 2.7.

Since $u\in
L^{p}_{\mathrm{loc}}\big{(}0,T;W^{1,p}_{\mathrm{loc}}(\Omega)\big{)}\cap
C_{\mathrm{loc}}\big{(}0,T;L^{2}_{\mathrm{loc}}(\Omega)\big{)}$, by Lemma 2.9
(a) below, we have $u\in
L^{l}_{\mathrm{loc}}(0,T;L^{l}_{\mathrm{loc}}(\Omega))$. Therefore, the
integral involving $g$ in the right-hand side of (2.3) is finite. Noting this
fact along with Lemma 2.2 and Lemma 2.3 imply that Definition 2.6 well stated.

######  Remark 2.8.

When $g\equiv 0$ in $\Omega_{T}\times\mathbb{R}$, it is immediate from
Definition 2.6 that $u$ is a weak subsolution of (1.1) if and only if $-u$ is
a weak supersolution of (1.1).

###  2.3. Auxiliary results

The following result follows from [19, Proposition $3.1$ and Proposition
$3.2$].

######  Lemma 2.9.

Let $p,m\in[1,\infty)$ and $q=p(1+\frac{m}{N})$. Assume that $\Omega$ is a
bounded smooth domain in $\mathbb{R}^{N}$.

  1. (a)

If $u\in L^{p}\big{(}0,T;W^{1,p}(\Omega)\big{)}\cap
L^{\infty}\big{(}0,T;L^{m}(\Omega)\big{)}$, then $u\in
L^{q}\big{(}0,T;L^{q}(\Omega)\big{)}$.

  2. (b)

Moreover, if $u\in L^{p}\big{(}0,T;W^{1,p}_{0}(\Omega)\big{)}\cap
L^{\infty}\big{(}0,T;L^{m}(\Omega)\big{)}$, then there exists a constant
$C=C(p,m,N)$ such that

(2.4) $\int_{0}^{T}\int_{\Omega}|u(x,t)|^{q}\,dx\,dt\leq
C\bigg{(}\int_{0}^{T}\int_{\Omega}|\nabla
u(x,t)|^{p}\,dx\,dt\bigg{)}\bigg{(}\operatorname*{ess\,sup}_{0<t<T}\int_{\Omega}|u(x,t)|^{m}\,dx\bigg{)}^{\frac{p}{N}}.$

For $a,k\in\mathbb{R}$, we define the functions $\zeta_{+}$ and $\zeta_{-}$ by

(2.5)
$\zeta_{+}(a,k)=\int_{k}^{a}(\eta-k)_{+}\,d\eta\quad\text{and}\quad\zeta_{-}(a,k)=-\int_{k}^{a}(\eta-k)_{-}\,d\eta.$

Note that $\zeta_{+}\geq 0$, $\zeta_{-}\geq 0$. The following result follows
from [13, Lemma 2.2].

######  Lemma 2.10.

For every $a,k\in\mathbb{R}$, there exists a constant $\lambda>0$ such that

$\frac{1}{\lambda}(a-k)_{+}^{2}\leq\zeta_{+}(a,k)\leq\lambda(a-k)_{+}^{2}\quad\text{and}\quad\frac{1}{\lambda}(a-k)_{-}^{2}\leq\zeta_{-}(a,k)\leq\lambda(a-k)_{-}^{2}.$

The following iteration lemma is from [31, Lemma $4.3$].

######  Lemma 2.11.

Let $\\{Y_{j}\\}_{j=0}^{\infty}$ be a sequence of positive real numbers such
that

$Y_{j+1}\leq Kb^{j}(Y_{j}^{1+\delta_{1}}+Y_{j}^{1+\delta_{2}}),\quad
j\in\mathbb{N}\cup\\{0\\},$

where $K>0$, $b>1$ and $\delta_{2}\geq\delta_{1}>0$. If

$Y_{0}\leq\min\Big{\\{}1,(2K)^{-\frac{1}{\delta_{1}}}b^{-\frac{1}{\delta_{1}^{2}}}\Big{\\}}\quad\text{or}\quad
Y_{0}\leq\min\Big{\\{}(2K)^{-\frac{1}{\delta_{1}}}b^{-\frac{1}{\delta_{1}^{2}}},(2K)^{-\frac{1}{\delta_{2}}}b^{-\frac{1}{\delta_{1}\delta_{2}}-\frac{\delta_{2}-\delta_{1}}{\delta_{2}^{2}}}\Big{\\}},$

then $Y_{j}\leq 1$ for some $j\in\mathbb{N}\cup\\{0\\}$. Moreover,

$Y_{j}\leq\min\Big{\\{}1,(2K)^{-\frac{1}{\delta_{1}}}b^{-\frac{1}{\delta_{1}^{2}}}b^{-\frac{j}{\delta_{1}}}\Big{\\}}\quad\text{for
every}\quad j\geq j_{0},$

where $j_{0}$ is the smallest $j\in\mathbb{N}\cup\\{0\\}$ such that $Y_{j}\leq
1$. In particular, we have $\lim_{j\to\infty}Y_{j}=0$.

##  3\. Local boundedness result

Our first main result is the following local boundedness estimate for weak
subsolutions. We prove the result by applying Lemma 3.7 below.

######  Theorem 3.1.

(Local boundedness) Let $0<s<1,\,\frac{2N}{N+2}\leq p<\infty$ and $u\in
L^{p}_{\mathrm{loc}}\big{(}0,T;W^{1,p}_{\mathrm{loc}}(\Omega)\big{)}\\\ \cap
C_{\mathrm{loc}}\big{(}0,T;L^{2}_{\mathrm{loc}}(\Omega)\big{)}\cap
L^{\infty}_{\mathrm{loc}}\big{(}0,T;L^{p-1}_{ps}(\mathbb{R}^{N})\big{)}$ be a
weak subsolution of (1.1) in $\Omega_{T}$. Suppose
$(x_{0},t_{0})\in\Omega_{T}$ and $r\in(0,1)$ such that
$\mathcal{Q}_{r}(x_{0},t_{0})=B_{r}(x_{0})\times(t_{0}-r^{p},t_{0}+r^{p})\Subset\Omega_{T}$.
Assume that $g$ satisfies (1.6) for some $\alpha\geq 0$ and $\max\\{p,2\\}\leq
l<p\kappa$ with $\kappa=1+\frac{2}{N}$ such that $h\in L^{\gamma
l^{\prime}}_{\mathrm{loc}}(\Omega_{T})$ for some $\gamma>\frac{N+p}{p}$. Then
there exists a positive constant
$C=C(N,p,s,\Lambda,C_{1},C_{2},C_{3},C_{4},l,\alpha,h)\geq 1$ such that

(3.1)
$\operatorname*{ess\,sup}_{\mathcal{Q}_{\frac{r}{2}}(x_{0},t_{0})}\,u\leq
C\max\bigg{\\{}\bigg{(}\fint_{\mathcal{Q}_{r}(x_{0},t_{0})}u_{+}^{l}\,dx\,dt\bigg{)}^{m},1\bigg{\\}}+\mathrm{Tail}_{\infty}(u_{+};x_{0},\tfrac{r}{2},t_{0}-r^{p},t_{0}+r^{p}),$

where $\mathrm{Tail}_{\infty}$ is defined by (2.2) and
$m=\frac{p}{N(p\kappa-l)}$.

###### Proof.

For $j\in\mathbb{N}\cup\\{0\\}$, let
$B_{j},\hat{B}_{j},\Gamma_{j},\hat{\Gamma}_{j},\hat{k},w_{j},\hat{w}_{j}$ be
given by (3.30)-(3.31) and (3.36)-(3.37) and denote

$Y_{j}=\frac{1}{r^{p}}\fint_{B_{j}}\int_{\Gamma_{j}}w_{j}^{l}\,dx\,dt.$

Setting $\theta=\frac{1}{2}$ and
$\hat{k}\geq\mathrm{Tail}_{\infty}(u_{+};x_{0},\tfrac{r}{2},t_{0}-r^{p},t_{0}+r^{p})+1$
in Lemma 3.7 and using the fact that $r\in(0,1)$, we have

(3.2)
$\begin{split}Y_{j+1}&\leq\frac{C2^{aj}}{\hat{k}^{l(1-\frac{l}{p\kappa})}}\big{(}Y_{j}^{1+\frac{l}{\kappa
N}}+Y_{j}^{1+\frac{l\kappa_{0}}{\kappa N}}\big{)},\end{split}$

where $a=\xi(N+p+l)$ for $\xi=1+\frac{p}{N}$, $\kappa=1+\frac{2}{N}$,
$\kappa_{0}=1-\frac{p+N}{p\gamma}\in(0,1]$, since $\gamma>\frac{N+p}{p}$, and
$C=C(N,p,s,\Lambda,C_{1},C_{2},C_{3},C_{4},l,\alpha,h)\geq 1$. For such a
constant $C$, we choose

$\hat{k}=C\max\bigg{\\{}\bigg{(}\fint_{\mathcal{Q}_{r}(x_{0},t_{0})}u_{+}^{l}\,dx\,dt\bigg{)}^{m},1\bigg{\\}}+\mathrm{Tail}_{\infty}(u_{+};x_{0},\tfrac{r}{2},t_{0}-r^{p},t_{0}+r^{p}).$

Thus setting

$K=\frac{C}{\hat{k}^{l(1-\frac{l}{p\kappa})}},\quad
b=2^{a},\quad\delta_{2}=\frac{l}{N\kappa},\quad\text{and}\quad\delta_{1}=\frac{l\kappa_{0}}{N\kappa}$

in Lemma 2.11, we obtain $\lim_{j\to\infty}Y_{j}=0$. This implies (3.1) and
the result follows. ∎

When $1<p<\frac{2N}{N+2}$, we prove a local boundedness result in Theorem 3.2
below, where we follow the idea of the proof of [21, Theorem $2$]. To this
end, we assume that $m>\frac{N(2-p)}{p}$ and for a given weak subsolution
$u\in L^{m}_{\mathrm{loc}}(\Omega_{T})\cap
L^{p}_{\mathrm{loc}}\big{(}0,T;W^{1,p}_{\mathrm{loc}}(\Omega)\big{)}\cap
C_{\mathrm{loc}}\big{(}0,T;L^{2}_{\mathrm{loc}}(\Omega)\big{)}\cap
L^{\infty}_{\mathrm{loc}}\big{(}0,T;L^{p-1}_{ps}(\mathbb{R}^{N})\big{)}$ of
(1.1) in $\Omega_{T}$, there exists a sequence of bounded weak subsolutions
$\\{u_{k}\\}_{k=1}^{\infty}\subset L^{\infty}_{\mathrm{loc}}(\Omega_{T})\cap
L^{p}_{\mathrm{loc}}\big{(}0,T;W^{1,p}_{\mathrm{loc}}(\Omega)\big{)}\cap
C_{\mathrm{loc}}\big{(}0,T;L^{2}_{\mathrm{loc}}(\Omega)\big{)}\cap
L^{\infty}_{\mathrm{loc}}\big{(}0,T;L^{p-1}_{ps}(\mathbb{R}^{N})\big{)}$ of
(1.1) in $\Omega_{T}$ such that for some constant $C>0$, independent of $k$,
we have

(3.3)
$\|u_{k}\|_{L^{\infty}_{\mathrm{loc}}(0,T;L^{p-1}_{ps}(\mathbb{R}^{N}))}\leq
C,$

and

(3.4) $u_{k}\to u\text{ in }L^{m}_{\mathrm{loc}}(\Omega_{T})\text{ as
}k\to\infty.$

By applying Lemma 3.8 below, we have our second main result.

######  Theorem 3.2.

(Local boundedness) Let $0<s<1,\,1<p<\frac{2N}{N+2},\,m>\frac{N(2-p)}{p}$ and
$u\in L^{m}_{\mathrm{loc}}(\Omega_{T})\cap
L^{p}_{\mathrm{loc}}\big{(}0,T;W^{1,p}_{\mathrm{loc}}(\Omega)\big{)}\cap
C_{\mathrm{loc}}\big{(}0,T;L^{2}_{\mathrm{loc}}(\Omega)\big{)}\cap
L^{\infty}_{\mathrm{loc}}\big{(}0,T;L^{p-1}_{ps}(\mathbb{R}^{N})\big{)}$ be a
weak subsolution of (1.1) in $\Omega_{T}$ for which (3.3) and (3.4) holds.
Suppose $(x_{0},t_{0})\in\Omega_{T}$ and $R\in(0,1)$ such that
$\mathcal{Q}_{R}(x_{0},t_{0})=B_{R}(x_{0})\times(t_{0}-R^{p},t_{0}+R^{p})\Subset\Omega_{T}$.
Assume that $g$ satisfies the hypothesis (1.6) for some $\alpha\geq 0$,
$1<l\leq 2$ and $h\in L^{\infty}_{\mathrm{loc}}(\Omega_{T})$. Then there
exists a positive constant
$C=C(N,p,s,\Lambda,C_{1},C_{2},C_{3},C_{4},l,\alpha,m,h)$ such that

(3.5)
$\begin{split}\operatorname*{ess\,sup}_{\mathcal{Q}_{\frac{R}{2}}(x_{0},t_{0})}\,u&\leq
C\max\bigg{\\{}\bigg{(}\fint_{\mathcal{Q}_{R}(x_{0},t_{0})}u_{+}^{m}\,dx\,dt\bigg{)}^{\frac{p}{(p+N)(m-\mu_{m})}},\bigg{(}\fint_{\mathcal{Q}_{R}(x_{0},t_{0})}u_{+}^{m}\,dx\,dt\bigg{)}^{\frac{p}{(p+N)(m-2-\mu_{m})}}\bigg{\\}}\\\
&\qquad+\mathrm{Tail}_{\infty}(u_{+};x_{0},\tfrac{R}{2},t_{0}-R^{p},t_{0}+R^{p}),\end{split}$

where $\mu_{m}=\frac{(m-p\kappa)N}{(p+N)}$ with $\kappa=1+\frac{2}{N}$ and
$\mathrm{Tail}_{\infty}$ is defined by (2.2).

###### Proof.

By (3.3) and (3.4), we may assume that $u$ is qualitatively locally bounded
and run the arguments below to get the required estimate (3.5). Indeed, since
every $u_{k}$ is a bounded weak subsolution of (1.1), the arguments below
holds for $u_{k}$ in place of $u$. Then the estimate (3.5) holds for every
$u_{k}$, which gives a $k$-independent bound of $u_{k}$ using (3.3) and (3.4).
By the pointwise convergence of $u_{k}$ to $u$, we conclude that $u$ is
locally bounded. To this end, let

$R_{0}=\frac{R}{2},\quad R_{n}=\frac{R}{2}+\sum_{i=1}^{n}2^{-i-1}R,\quad
n\in\mathbb{N}.$

Moreover, let

$\mathcal{U}_{n}=B_{R_{n}}(x_{0})\times(t_{0}-R_{n}^{p},t_{0}+R_{n}^{p}),\quad
S_{n}=\operatorname*{ess\,sup}_{\mathcal{U}_{n}}\,u_{+},\quad
n\in\mathbb{N}\cup\\{0\\}.$

By denoting $r=R_{n+1}$ and $\theta r=R_{n}$, we have

$\theta=\frac{\frac{1}{2}+\sum_{i=1}^{n}2^{-i-1}}{\frac{1}{2}+\sum_{i=1}^{n+1}2^{-i-1}}\in\Big{[}\frac{1}{2},1\Big{)}.$

For $j\in\mathbb{N}\cup\\{0\\}$, let $B_{j},\Gamma_{j},k_{j},\hat{k}$ be
defined as in (3.30), (3.31), (3.35), (3.36) and $m>\frac{N(2-p)}{p}$. By
setting

$X_{j}=\fint_{B_{j}}\int_{\Gamma_{j}}(u-k_{j})_{+}^{m}\,dx\,dt$

in Lemma 3.8, we obtain

(3.6)
$\begin{split}X_{j+1}&\leq\Bigg{[}\frac{C}{r^{\frac{p^{2}}{N}}}\frac{2^{aj}}{(1-\theta)^{\frac{(N+p)^{2}}{N}}}\Big{(}\frac{1}{\hat{k}^{m-2}}+\frac{1}{\hat{k}^{m-p}}\Big{)}^{1+\frac{p}{N}}+C\frac{2^{aj}}{\hat{k}^{m(1+\frac{p}{N})}}\Bigg{]}\|u_{+}\|^{m-p\kappa}_{L^{\infty}(\mathcal{U}_{n+1})}X_{j}^{1+\frac{p}{N}},\end{split}$

for some positive constant
$C=C(N,p,s,\Lambda,C_{1},C_{2},C_{3},C_{4},l,\alpha,m,h)$, where
$a=(N+p+m)(1+\frac{p}{N})$. Now setting $Y_{j}=\frac{X_{j}}{R_{n}^{p}}$ for
$j\in\mathbb{N}\cup\\{0\\}$ from (3.6) we obtain

(3.7) $\begin{split}Y_{j+1}&\leq
C2^{en}\bigg{(}\frac{1}{\hat{k}^{(m-2)(1+\frac{p}{N})}}+\frac{1}{\hat{k}^{m(1+\frac{p}{N})}}\bigg{)}S_{n+1}^{m-p\kappa}2^{aj}Y_{j}^{1+\frac{p}{N}},\end{split}$

for some positive constant
$C=C(N,p,s,\Lambda,C_{1},C_{2},C_{3},C_{4},l,\alpha,m,h)$, where
$e=\frac{(N+p)^{2}}{N}$. By choosing

(3.8)
$\begin{split}\hat{k}&=C2^{\frac{enN}{(m-2)(p+N)}}\bigg{(}\fint_{B_{R_{n+1}}(x_{0})}\fint_{t_{0}-R_{n+1}^{p}}^{t_{0}-R_{n+1}^{p}}u_{+}^{m}\,dx\,dt\bigg{)}^{\frac{p}{(m-2)(p+N)}}S_{n+1}^{\frac{N(m-p\kappa)}{(m-2)(p+N)}}\\\
&\qquad+C2^{\frac{enN}{m(p+N)}}\bigg{(}\fint_{B_{R_{n+1}}(x_{0})}\fint_{t_{0}-R_{n+1}^{p}}^{t_{0}-R_{n+1}^{p}}u_{+}^{m}\,dx\,dt\bigg{)}^{\frac{p}{m(p+N)}}S_{n+1}^{\frac{N(m-p\kappa)}{m(p+N)}}\\\
&\qquad+\frac{1}{2}\mathrm{Tail}_{\infty}(u_{+};x_{0},R_{n},t_{0}-R_{n+1}^{p},t_{0}+R_{n+1}^{p}),\end{split}$

we obtain $Y_{0}\leq(2K)^{-\frac{1}{\delta}}b^{-\frac{1}{\delta^{2}}},$ where

$K=C2^{en-1}\bigg{(}\frac{1}{\hat{k}^{(m-2)(1+\frac{p}{N})}}+\frac{1}{\hat{k}^{m(1+\frac{p}{N})}}\bigg{)}S_{n+1}^{m-p\kappa},\quad
b=2^{a}\quad\text{and}\quad\delta_{2}=\delta_{1}=\delta=\frac{p}{N}.$

Therefore, by Lemma 2.11 we have $\lim_{j\to\infty}Y_{j}=0$ and we get

(3.9) $\begin{split}\operatorname*{ess\,sup}_{\mathcal{U}_{n}}\,u_{+}&\leq
C2^{\frac{enN}{(m-2)(p+N)}}\Big{(}\fint_{B_{R_{n+1}}(x_{0})}\fint_{t_{0}-R_{n+1}^{p}}^{t_{0}-R_{n+1}^{p}}u_{+}^{m}\,dx\,dt\Big{)}^{\frac{p}{(m-2)(p+N)}}S_{n+1}^{\frac{N(m-p\kappa)}{(m-2)(p+N)}}\\\
&\qquad+C2^{\frac{enN}{m(p+N)}}\bigg{(}\fint_{B_{R_{n+1}}(x_{0})}\fint_{t_{0}-R_{n+1}^{p}}^{t_{0}-R_{n+1}^{p}}u_{+}^{m}\,dx\,dt\bigg{)}^{\frac{p}{m(p+N)}}S_{n+1}^{\frac{N(m-p\kappa)}{m(p+N)}}\\\
&\qquad+\frac{1}{2}\mathrm{Tail}_{\infty}(u_{+};x_{0},R_{n},t_{0}-R_{n+1}^{p},t_{0}+R_{n+1}^{p}),\end{split}$

for some positive constant
$C=C(N,p,s,\Lambda,C_{1},C_{2},C_{3},C_{4},l,\alpha,m,h)$. Since
$1<p<\frac{2N}{N+2},\,\kappa=1+\frac{2}{N}$ and $m>\frac{N(2-p)}{p}$, we have

$0<\frac{N(m-p\kappa)}{(m-2)(p+N)},\frac{N(m-p\kappa)}{m(p+N)}<1.$

By Young’s inequality with $\epsilon$ (to be chosen below) in (3.9), we get

(3.10)
$\begin{split}S_{n}=\operatorname*{ess\,sup}_{\mathcal{U}_{n}}\,u_{+}&\leq\epsilon
S_{n+1}+P_{0}+T_{1}^{n}P_{1}+T_{2}^{n}P_{2},\quad
n\in\mathbb{N}\cup\\{0\\},\end{split}$

where

$\displaystyle P_{0}$
$\displaystyle=\frac{1}{2}\mathrm{Tail}_{\infty}(u_{+};x_{0},\frac{R}{2},t_{0}-R^{p},t_{0}+R^{p}),$
$\displaystyle P_{1}$
$\displaystyle=C\epsilon^{-\frac{\mu_{m}}{m-2-\mu_{m}}}\bigg{(}\fint_{B_{R}(x_{0})}\fint_{t_{0}-R^{p}}^{t_{0}+R^{p}}u_{+}^{m}\,dx\,dt\bigg{)}^{\frac{p}{(p+N)(m-2-\mu_{m})}},$
$\displaystyle P_{2}$
$\displaystyle=C\epsilon^{-\frac{\mu_{m}}{m-\mu_{m}}}\bigg{(}\fint_{B_{R}(x_{0})}\fint_{t_{0}-R^{p}}^{t_{0}+R^{p}}u_{+}^{m}\,dx\,dt\bigg{)}^{\frac{p}{(p+N)(m-\mu_{m})}},$
$\displaystyle T_{1}$
$\displaystyle=2^{\frac{eN}{(p+N)(m-2-\mu_{m})}}\quad\text{and}\quad
T_{2}=2^{\frac{eN}{(p+N)(m-\mu_{m})}},$

for $\mu_{m}=\frac{(m-p\kappa)N}{(p+N)}$. We claim that

(3.11)
$S_{0}\leq\epsilon^{n+1}S_{n+1}+P_{0}\sum_{i=0}^{n}\epsilon^{i}+P_{1}\sum_{i=0}^{n}(\epsilon
T_{1})^{i}+P_{2}\sum_{i=0}^{n}(\epsilon T_{2})^{i},\quad
n\in\mathbb{N}\cup\\{0\\}.$

Indeed, by (3.10), the inequality (3.11) holds for $n=0$. We assume that
(3.11) holds for $n=j$ and prove it for $n=j+1$. To this end, assuming (3.11)
for $n=j$, we observe that

(3.12)
$\begin{split}S_{0}&\leq\epsilon^{j+1}S_{j+1}+P_{0}\sum_{i=0}^{j}\epsilon^{i}+P_{1}\sum_{i=0}^{j}(\epsilon
T_{1})^{i}+P_{2}\sum_{i=0}^{j}(\epsilon T_{2})^{i}\\\
&\leq\epsilon^{j+1}(\epsilon
S_{j+2}+P_{0}+T_{1}^{j+1}P_{1}+T_{2}^{j+1}P_{2})+P_{0}\sum_{i=0}^{j}\epsilon^{i}+P_{1}\sum_{i=0}^{j}(\epsilon
T_{1})^{i}+P_{2}\sum_{i=0}^{j}(\epsilon T_{2})^{i}\\\
&=\epsilon^{j+2}S_{j+2}+P_{0}\sum_{i=0}^{j+1}\epsilon^{i}+P_{1}\sum_{i=0}^{j+1}(\epsilon
T_{1})^{i}+P_{2}\sum_{i=0}^{j+1}(\epsilon T_{2})^{i}.\end{split}$

Thus, (3.11) holds for $n=j+1$. By induction (3.11) holds for every
$n\in\mathbb{N}\cup\\{0\\}$. By inserting $P_{0},P_{1},P_{2},T_{1},T_{2}$,
choosing $\epsilon=2^{-\frac{eN}{(p+N)(m-2-\mu_{m})}-1}$ in (3.11) and letting
$n\to\infty$ in the resulting inequality, we conclude that (3.5) holds. This
completes the proof. ∎

We have the following energy estimate for weak subsolutions. For short, we
denote

$d\mu=K(x,y,t)\,dx\,dy,$

where $K(x,y,t)$ is given by (1.4). For $f\in L^{1}(\Omega_{T})$, we define
the mollification in time by

(3.13) $f_{m}(x,t)=\frac{1}{m}\int_{0}^{t}e^{\frac{s-t}{m}}f(x,s)\,ds,\quad
m>0.$

This is useful in our energy estimates below (see for example Lemma 3.3) to
justify the use of test functions depending on the solution itself. For more
details on $f_{m}$, we refer to [33].

######  Lemma 3.3.

(Energy estimate) Let $0<s<1,\,1<p<\infty$ and $u\in
L^{p}_{\mathrm{loc}}\big{(}0,T;W^{1,p}_{\mathrm{loc}}(\Omega)\big{)}\cap
C_{\mathrm{loc}}\big{(}0,T;L^{2}_{\mathrm{loc}}(\Omega)\big{)}\cap
L^{\infty}_{\mathrm{loc}}\big{(}0,T;L^{p-1}_{ps}(\mathbb{R}^{N})\big{)}$ be a
weak subsolution of (1.1) in $\Omega_{T}$. Suppose $x_{0}\in\Omega,\,r>0$ such
that $B_{r}=B_{r}(x_{0})\Subset\Omega$ and $0<\tau_{1}<\tau_{2}$, $\tau>0$
such that $(\tau_{1}-\tau,\tau_{2})\Subset(0,T)$. For $k\in\mathbb{R}$, we
denote by $w_{+}=(u-k)_{+}$. Assume that $g$ satisfies (1.6) for some
$\alpha\geq 0$, $1<l\leq\max\\{2,p(1+\frac{2}{N})\\}$ and $h\in
L^{l^{\prime}}_{\mathrm{loc}}(\Omega_{T})$. Then there exists a positive
constant $C=C(p,\Lambda,C_{1},C_{2},C_{3},C_{4},l,\alpha)$ such that

(3.14)
$\begin{split}&\operatorname*{ess\,sup}_{\tau_{1}-\tau<t<\tau_{2}}\int_{B_{r}}\zeta_{+}(u,k)\xi^{p}\,dx+\int_{\tau_{1}-\tau}^{\tau_{2}}\int_{B_{r}}|\nabla
w_{+}|^{p}\xi^{p}\,dx\,dt\\\ &\leq
C\bigg{(}\int_{\tau_{1}-\tau}^{\tau_{2}}\int_{B_{r}}w_{+}^{p}|\nabla\xi|^{p}\,dx\,dt+\int_{\tau_{1}-\tau}^{\tau_{2}}\int_{B_{r}}\int_{B_{r}}\max\\{w_{+}(x,t),w_{+}(y,t)\\}^{p}|\xi(x,t)-\xi(y,t)|^{p}\,d\mu\,dt\\\
&\qquad+\operatorname*{ess\,sup}_{(x,t)\in\mathrm{supp}\,\xi,\,\tau_{1}-\tau<t<\tau_{2}}\int_{\mathbb{R}^{N}\setminus
B_{r}}\frac{w_{+}(y,t)^{p-1}}{|x-y|^{N+ps}}\,dy\int_{\tau_{1}-\tau}^{\tau_{2}}\int_{B_{r}}w_{+}\xi^{p}\,dx\,dt\\\
&\qquad+\int_{\tau_{1}-\tau}^{\tau_{2}}\int_{B_{r}}\zeta_{+}(u,k)|\partial_{t}\xi^{p}|\,dx\,dt+\int_{B_{r}}\zeta_{+}(u(x,\tau_{1}-\tau),k)\xi(x,\tau_{1}-\tau)^{p}\,dx\\\
&\qquad+\int_{\tau_{1}-\tau}^{\tau_{2}}\int_{B_{r}}\big{(}|u|^{l}+|h|^{l^{\prime}}+w_{+}^{l}\big{)}\chi_{\\{u\geq
k\\}}\xi^{p}\,dx\,dt\bigg{)},\end{split}$

where $\xi(x,t)=\psi(x)\eta(t)$, with $\psi\in C_{c}^{\infty}(B_{r})$ and
$\eta\in C^{\infty}(\tau_{1}-\tau,\tau_{2})$ are nonnegative functions. Here
$\zeta_{+}$ is given by (2.5).

######  Remark 3.4.

If $g(x,t,u)=h(x,t)$ in $\Omega_{T}\times\mathbb{R}$, then the sixth integral
in the right hand side of (3.14) can be replaced by the integral
$\int_{\tau_{1}-\tau}^{\tau_{2}}\int_{B_{r}}|h|w_{+}\xi^{p}\,dx\,dt$, which
vanishes if $h\equiv 0$.

######  Remark 3.5.

Moreover, if $g\equiv 0$ in $\Omega_{T}\times\mathbb{R}$ and $u\in
L^{p}_{\mathrm{loc}}\big{(}0,T;W^{1,p}_{\mathrm{loc}}(\Omega)\big{)}\cap
C_{\mathrm{loc}}\big{(}0,T;L^{2}_{\mathrm{loc}}(\Omega)\big{)}\cap
L^{\infty}_{\mathrm{loc}}\big{(}0,T;L^{p-1}_{ps}(\mathbb{R}^{N})\big{)}$ is a
weak supersolution of (1.1) in $\Omega_{T}$ such that
${B_{r}(x_{0})}\times(\tau_{1}-\tau,\tau_{2})\Subset\Omega_{T}$, then noting
Remark 2.8, Remark 3.4 and the definition of $\zeta_{\pm}$ from (2.5), the
estimate (3.14) will hold by replacing $\zeta_{+},w_{+}$ and the sixth
integral in right hand side of (3.14) with $\zeta_{-},w_{-}$ and zero
respectively. Here $w_{-}=(u-k)_{-}$ for $k\in\mathbb{R}$.

###### Proof.

Let $t_{1}=\tau_{1}-\tau$ and $t_{2}=\tau_{2}$. For small enough $\epsilon>0$
and for fixed $t_{1}<\theta_{1}<\theta_{2}<t_{2}$, we define a Lipschitz
cutoff function $\zeta_{\epsilon}:[t_{1},t_{2}]\to[0,1]$ by

(3.15) $\zeta_{\epsilon}(t)=\begin{cases}0&\text{for }t_{1}\leq
t\leq\theta_{1}-\epsilon,\\\ 1+\frac{t-\theta_{1}}{\epsilon}&\text{for
}\theta_{1}-\epsilon<t\leq\theta_{1},\\\ 1,&\text{for
}\theta_{1}<t\leq\theta_{2},\\\ 1-\frac{t-\theta_{2}}{\epsilon},&\text{for
}\theta_{2}<t\leq\theta_{2}+\epsilon,\\\ 0,&\text{for
}\theta_{2}+\epsilon<t\leq t_{2}.\end{cases}$

Recalling that $\xi(x,t)=\psi(x)\eta(t)$, we choose

$\phi(x,t)=w_{+}(x,t)\xi(x,t)^{p}\zeta_{\epsilon}(t)$

as an admissible test function in (2.3). Indeed, for $(\cdot)_{m}$ as defined
in (3.13) and following [13, 33], the weak subsolution $u$ of (1.1) satisfies
the following mollified inequality

(3.16) $\lim_{\epsilon\to 0}\lim_{m\to
0}(T_{m}^{\epsilon}+L_{m}^{\epsilon}+N_{m}^{\epsilon}-S_{m}^{\epsilon})\leq
0,$

where

$\begin{split}T_{m}^{\epsilon}&=\int_{t_{1}}^{t_{2}}\int_{B_{r}}\partial_{t}{u_{m}}(x,t)\phi(x,t)\,dx\,dt=\int_{t_{1}}^{t_{2}}\int_{B_{r}}\partial_{t}{u_{m}}(x,t)\,w_{+}(x,t)\xi(x,t)^{p}\zeta_{\epsilon}(t)\,dx\,dt,\\\
L_{m}^{\epsilon}&=\int_{t_{1}}^{t_{2}}\int_{B_{r}}(\mathcal{B}(x,t,u,\nabla
u))_{m}\nabla\phi(x,t)\,dx\,dt\\\
&=\int_{t_{1}}^{t_{2}}\int_{B_{r}}(\mathcal{B}(x,t,u,\nabla
u))_{m}\nabla\big{(}w_{+}(x,t)\xi(x,t)^{p}\zeta_{\epsilon}(t)\big{)}\,dx\,dt,\\\
N_{m}^{\epsilon}&=\int_{t_{1}}^{t_{2}}\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}(\mathcal{V}(u(x,y,t)))_{m}(\phi(x,t)-\phi(y,t))\,dx\,dy\,dt\\\
&=\int_{t_{1}}^{t_{2}}\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}(\mathcal{V}(u(x,y,t)))_{m}\big{(}w_{+}(x,t)\xi(x,t)^{p}-w_{+}(y,t)\xi(y,t)^{p}\big{)}\zeta_{\epsilon}(t)\,dx\,dy\,dt,\end{split}$

with $\mathcal{V}(u(x,y,t))=\mathcal{A}(u(x,y,t))K(x,y,t)$, and

$\displaystyle S_{m}^{\epsilon}$
$\displaystyle=\int_{t_{1}}^{t_{2}}\int_{\Omega}(g(x,t,u))_{m}\phi(x,t)\,dx\,dt=\int_{t_{1}}^{t_{2}}\int_{\Omega}(g(x,t,u))_{m}w_{+}(x,t)\xi(x,t)^{p}\zeta_{\epsilon}(t)\,dx\,dt.$

Estimate of $T_{m}^{\epsilon}$: Recalling $\zeta_{+}(u,k)$ defined in (2.5),
from the proof of [13, Proposition $3.1$, p. $9$], we arrive at

(3.17) $\begin{split}\lim_{\epsilon\to 0}\lim_{m\to
0}T_{m}^{\epsilon}&\geq\int_{B_{r}}\zeta_{+}(u(x,\theta_{2}),k)\xi(x,\theta_{2})^{p}\,dx-\int_{B_{r}}\zeta_{+}(u(x,\theta_{1}),k)\xi(x,\theta_{1})^{p}\,dx\\\
&\qquad-\int_{\theta_{1}}^{\theta_{2}}\int_{B_{r}}\zeta_{+}(u(x,t),k)\partial_{t}\xi(x,t)^{p}\,dx\,dt.\end{split}$

Estimate of $L_{m}^{\epsilon}$: From the proof of [13, Proposition $3.1$, p.
$10$], we obtain

(3.18) $\begin{split}\lim_{\epsilon\to 0}\lim_{m\to
0}L_{m}^{\epsilon}&\geq\frac{C_{3}}{p}\int_{\theta_{1}}^{\theta_{2}}\int_{B_{r}}|\nabla
w_{+}|^{p}\xi^{p}\,dx\,dt-C\int_{\theta_{1}}^{\theta_{2}}\int_{B_{r}}w_{+}^{p}|\nabla\xi|^{p}\,dx\,dt,\end{split}$

for some constant $C=C(C_{3},C_{4},p)>0$, where $C_{3},C_{4}$ are given by
(1.5).
Estimate of $N_{m}^{\epsilon}$: From the proof of [1, Lemma $3.1$, p. $9-12$],
we obtain

(3.19) $\begin{split}\lim_{\epsilon\to 0}\lim_{m\to 0}N_{m}^{\epsilon}&\geq
c\int_{\theta_{1}}^{\theta_{2}}\int_{B_{r}}\int_{B_{r}}|w_{+}(x,t)\xi(x,t)-w_{+}(y,t)\xi(y,t)|^{p}\,d\mu\,dt\\\
&\qquad-C\int_{\theta_{1}}^{\theta_{2}}\int_{B_{r}}\int_{B_{r}}\max\\{w_{+}(x,t),w_{+}(y,t)\\}^{p}|\xi(x,t)-\xi(y,t)|^{p}\,d\mu\,dt\\\
&\qquad-C\operatorname*{ess\,sup}_{(x,t)\in\mathrm{supp}\,\zeta,\,\theta_{1}<t<\theta_{2}}\int_{\mathbb{R}^{N}\setminus
B_{r}}\frac{w_{+}(y,t)^{p-1}}{|x-y|^{N+ps}}\,dy\int_{\theta_{1}}^{\theta_{2}}\int_{B_{r}}w_{+}(x,t)\xi(x,t)^{p}\,dx\,dt,\end{split}$

for some positive constants $c=c(C_{1},C_{2},\Lambda,p)$ and
$C=C(C_{1},C_{2},\Lambda,p)$, where $C_{1},C_{2}$ are given by (1.3).
Estimate of $S_{m}^{\epsilon}$: Since $u\in
L^{p}(t_{1},t_{2};W^{1,p}(B_{r}))\cap C(t_{1},t_{2};L^{2}(B_{r}))$, by Lemma
2.9 (a), we have $u\in L^{l}(t_{1},t_{2};L^{l}(B_{r}))$. Using this fact along
with the given hypothesis (1.6) on $g$, we obtain $g\in
L^{l^{\prime}}(t_{1},t_{2};L^{l^{\prime}}(B_{r}))$ and
$\phi=w_{+}(x,t)\xi(x,t)^{p}\zeta_{\epsilon}(t)\in
L^{l}(t_{1},t_{2};L^{l}(B_{r}))$. Setting

$S^{\epsilon}=\int_{t_{1}}^{t_{2}}\int_{B_{r}}g(x,t,u)w_{+}(x,t)\xi(x,t)^{p}\zeta_{\epsilon}(t)\,dx\,dt,$

by Hölder’s inequality, we obtain

(3.20)
$\begin{split}|S_{m}^{\epsilon}-S^{\epsilon}|&\leq\left(\int_{t_{1}}^{t_{2}}\int_{B_{r}}|g_{m}-g|^{l^{\prime}}\,dx\,dt\right)^{\frac{1}{l^{\prime}}}\left(\int_{t_{1}}^{t_{2}}\int_{B_{r}}|\phi|^{l}\,dx\,dt\right)^{\frac{1}{l}},\end{split}$

where $\phi(x,t)=w_{+}(x,t)\xi(x,t)^{p}\zeta_{\epsilon}(t)$. Using [33, Lemma
$2.9$], the first integral in the above estimate (3.20) goes to zero as $m\to
0$. This implies $\lim_{m\to 0}S_{m}^{\epsilon}=S^{\epsilon}$. Letting
$\epsilon\to 0$, by the Lebesgue dominated convergence theorem, we obtain

(3.21) $\lim_{\epsilon\to 0}\lim_{m\to 0}S_{m}^{\epsilon}=S,$

where

$S=\int_{\theta_{1}}^{\theta_{2}}\int_{B_{r}}g(x,t,u)w_{+}(x,t)\xi(x,t)^{p}\,dx\,dt.$

Using Young’s inequality with exponents $l$ and $l^{\prime}$ in (1.6), for
some positive constant $C=C(l,\alpha)$, we obtain

$g(x,t,u)w_{+}(x,t)\leq C\big{(}w_{+}(x,t)^{l}+|u|^{l}\chi_{\\{u\geq
k\\}}(x,t)+|h(x,t)|^{l^{{}^{\prime}}}\chi_{\\{u\geq k\\}}(x,t)\big{)}.$

Therefore, we have

(3.22) $S\leq
C\int_{\theta_{1}}^{\theta_{2}}\int_{B_{r}}\big{(}|u|^{l}+|h|^{l^{\prime}}+w_{+}^{l}\big{)}\chi_{\\{u\geq
k\\}}\xi^{p}\,dx\,dt,$

for some positive constant $C=C(l,\alpha)$. From (3.21) and (3.22), we have

(3.23) $\lim_{\epsilon\to 0}\lim_{m\to 0}S_{m}^{\epsilon}\leq
C\int_{\theta_{1}}^{\theta_{2}}\int_{B_{r}}\big{(}|u|^{l}+|h|^{l^{\prime}}+w_{+}^{l}\big{)}\chi_{\\{u\geq
k\\}}\xi^{p}\,dx\,dt,$

for some positive constant $C=C(l,\alpha)$. Combining the estimates (3.17),
(3.18), (3.19) and (3.23) in (3.16), we obtain

(3.24)
$\begin{split}&\int_{B_{r}}\zeta_{+}(u(x,\theta_{2}),k)\xi(x,\theta_{2})^{p}\,dx+\int_{\theta_{1}}^{\theta_{2}}\int_{B_{r}}|\nabla
w_{+}|^{p}\xi^{p}\,dx\,dt\\\ &\leq
C\bigg{(}\int_{\theta_{1}}^{\theta_{2}}\int_{B_{r}}w_{+}^{p}|\nabla\xi|^{p}\,dx\,dt+\int_{\theta_{1}}^{\theta_{2}}\int_{B_{r}}\int_{B_{r}}\max\\{w_{+}(x,t),w_{+}(y,t)\\}^{p}|\xi(x,t)-\xi(y,t)|^{p}\,d\mu\,dt\\\
&\qquad+\operatorname*{ess\,sup}_{(x,t)\in\mathrm{supp}\zeta,\,\theta_{1}<t<\theta_{2}}\int_{\mathbb{R}^{N}\setminus
B_{r}}\frac{w_{+}(y,t)^{p-1}}{|x-y|^{N+ps}}\,dy\int_{\theta_{1}}^{\theta_{2}}\int_{B_{r}}w_{+}\xi^{p}\,dx\,dt+\int_{\theta_{1}}^{\theta_{2}}\int_{B_{r}}\zeta_{+}(u,k)\partial_{t}\xi^{p}\,dx\,dt\\\
&\qquad+\int_{B_{r}}\zeta_{+}(u(x,\theta_{1}),k)\xi(x,\theta_{1})^{p}\,dx+\int_{\theta_{1}}^{\theta_{2}}\int_{B_{r}}\big{(}|u|^{l}+|h|^{l^{\prime}}+w_{+}^{l}\big{)}\chi_{\\{u\geq
k\\}}\xi^{p}\,dx\,dt\bigg{)},\end{split}$

whenever $t_{1}<\theta_{1}<\theta_{2}<t_{2}$, for some constant
$C=C(p,\Lambda,C_{1},C_{2},C_{3},C_{4},l,\alpha)>0$. Letting $\theta_{1}\to
t_{1}$ in (3.24) gives

(3.25)
$\begin{split}&\int_{B_{r}}\zeta_{+}(u(x,\theta_{2}),k)\xi(x,\theta_{2})^{p}\,dx+\int_{t_{1}}^{\theta_{2}}\int_{B_{r}}|\nabla
w_{+}|^{p}\xi^{p}\,dx\,dt\\\ &\leq
C\bigg{(}\int_{t_{1}}^{t_{2}}\int_{B_{r}}w_{+}^{p}|\nabla\xi|^{p}\,dx\,dt+\int_{t_{1}}^{t_{2}}\int_{B_{r}}\int_{B_{r}}\max\\{w_{+}(x,t),w_{+}(y,t)\\}^{p}|\xi(x,t)-\xi(y,t)|^{p}\,d\mu\,dt\\\
&\qquad+\operatorname*{ess\,sup}_{(x,t)\in\mathrm{supp}\zeta,\,t_{1}<t<t_{2}}\int_{\mathbb{R}^{N}\setminus
B_{r}}\frac{w_{+}(y,t)^{p-1}}{|x-y|^{N+ps}}\,dy\int_{\theta_{1}}^{\theta_{2}}\int_{B_{r}}w_{+}\xi^{p}\,dx\,dt+\int_{t_{1}}^{t_{2}}\int_{B_{r}}\zeta_{+}(u,k)|\partial_{t}\xi^{p}|\,dx\,dt\\\
&\qquad+\int_{B_{r}}\zeta_{+}(u(x,t_{1}),k)\xi(x,t_{1})^{p}\,dx+\int_{t_{1}}^{t_{2}}\int_{B_{r}}\big{(}|u|^{l}+|h|^{l^{\prime}}+w_{+}^{l}\big{)}\chi_{\\{u\geq
k\\}}\xi^{p}\,dx\,dt\bigg{)},\end{split}$

for some constant $C=C(p,\Lambda,C_{1},C_{2},C_{3},C_{4},l,\alpha)>0$.

Since $\zeta_{+}$ and $\xi$ are nonnegative, ignoring the second integral on
the left hand side of (3.25) and then taking essential supremum with respect
to $\theta_{2}\in(t_{1},t_{2})$, we arrive at

(3.26)
$\begin{split}&\operatorname*{ess\,sup}_{\tau_{1}-\tau<t<\tau_{2}}\int_{B_{r}}\zeta_{+}(u,k)\xi^{p}\,dx\leq
C\bigg{(}\int_{t_{1}}^{t_{2}}\int_{B_{r}}w_{+}^{p}|\nabla\xi|^{p}\,dx\,dt\\\
&\qquad+\int_{t_{1}}^{t_{2}}\int_{B_{r}}\int_{B_{r}}\max\\{w_{+}(x,t),w_{+}(y,t)\\}^{p}|\xi(x,t)-\xi(y,t)|^{p}\,d\mu\,dt\\\
&\qquad+\operatorname*{ess\,sup}_{(x,t)\in\mathrm{supp}\zeta,\,t_{1}<t<t_{2}}\int_{\mathbb{R}^{N}\setminus
B_{r}}\frac{w_{+}(y,t)^{p-1}}{|x-y|^{N+ps}}\,dy\int_{\theta_{1}}^{\theta_{2}}\int_{B_{r}}w_{+}\xi^{p}\,dx\,dt+\int_{t_{1}}^{t_{2}}\int_{B_{r}}\zeta_{+}(u,k)|\partial_{t}\xi^{p}|\,dx\,dt\\\
&\qquad+\int_{B_{r}}\zeta_{+}(u(x,t_{1}),k)\xi(x,t_{1})^{p}\,dx+\int_{t_{1}}^{t_{2}}\int_{B_{r}}\big{(}|u|^{l}+|h|^{l^{\prime}}+w_{+}^{l}\big{)}\chi_{\\{u\geq
k\\}}\xi^{p}\,dx\,dt\bigg{)},\end{split}$

for some constant $C=C(p,\Lambda,C_{1},C_{2},C_{3},C_{4},l,\alpha)>0$. Next,
discarding the first integral in (3.25) and then we let $\theta_{2}\to t_{2}$
in (3.25) to obtain

(3.27) $\begin{split}&\int_{t_{1}}^{t_{2}}\int_{B_{r}}|\nabla
w_{+}|^{p}\xi^{p}\,dx\,dt\leq
C\bigg{(}\int_{t_{1}}^{t_{2}}\int_{B_{r}}w_{+}^{p}|\nabla\xi|^{p}\,dx\,dt\\\
&\qquad+\int_{t_{1}}^{t_{2}}\int_{B_{r}}\int_{B_{r}}\max\\{w_{+}(x,t),w_{+}(y,t)\\}^{p}|\xi(x,t)-\xi(y,t)|^{p}\,d\mu\,dt\\\
&\qquad+\operatorname*{ess\,sup}_{(x,t)\in\mathrm{supp}\zeta,\,t_{1}<t<t_{2}}\int_{\mathbb{R}^{N}\setminus
B_{r}}\frac{w_{+}(y,t)^{p-1}}{|x-y|^{N+ps}}\,dy\int_{\theta_{1}}^{\theta_{2}}\int_{B_{r}}w_{+}\xi^{p}\,dx\,dt+\int_{t_{1}}^{t_{2}}\int_{B_{r}}\zeta_{+}(u,k)|\partial_{t}\xi^{p}|\,dx\,dt\\\
&\qquad+\int_{B_{r}}\zeta_{+}(u(x,t_{1}),k)\xi(x,t_{1})^{p}\,dx+\int_{t_{1}}^{t_{2}}\int_{B_{r}}\big{(}|u|^{l}+|h|^{l^{\prime}}+w_{+}^{l}\big{)}\chi_{\\{u\geq
k\\}}\xi^{p}\,dx\,dt\bigg{)},\end{split}$

for some constant $C=C(p,\Lambda,C_{1},C_{2},C_{3},C_{4},l,\alpha)>0$. From
(3.26) and (3.27) we conclude that the estimate (3.14) holds. Hence the result
follows. ∎

Next, we obtain some auxiliary results using the energy estimate in Lemma 3.3.
Before stating them, let us define the following parameters. Let
$(x_{0},t_{0})\in\Omega_{T}$, $r\in(0,1)$ be such that the parabolic cylinder
$\mathcal{Q}_{r}=\mathcal{Q}_{r}(x_{0},t_{0})=B_{r}(x_{0})\times(t_{0}-r^{p},t_{0}+r^{p})\Subset\Omega_{T}$.
For $\frac{1}{2}\leq\theta<1$ we define

(3.28) $r_{j}=\theta
r+(1-\theta)2^{-j}r\quad\text{and}\quad\hat{r}_{j}=\frac{r_{j}+r_{j+1}}{2},\quad
j\in\mathbb{N}\cup\\{0\\}.$

Note that

(3.29) $r_{j+1}<\hat{r}_{j}<r_{j},\quad j\in\mathbb{N}\cup\\{0\\}.$

We denote

(3.30) $B_{j}=B_{r_{j}}(x_{0}),\quad\hat{B}_{j}=B_{\hat{r}_{j}}(x_{0}),$

(3.31)
$\Gamma_{j}=(t_{0}-r_{j}^{p},t_{0}+r_{j}^{p}),\quad\hat{\Gamma}_{j}=(t_{0}-\hat{r}_{j}^{p},t_{0}+\hat{r}_{j}^{p})$

and

(3.32)
$\mathcal{Q}_{j}=B_{j}\times\Gamma_{j},\quad\hat{\mathcal{Q}}_{j}=\hat{B}_{j}\times\hat{\Gamma}_{j},\quad
j\in\mathbb{N}\cup\\{0\\}.$

By (3.29) we obtain

(3.33) $B_{j+1}\subset\hat{B}_{j}\subset
B_{j},\quad\Gamma_{j+1}\subset\hat{\Gamma}_{j}\subset\Gamma_{j},\quad
j\in\mathbb{N}\cup\\{0\\}$

and therefore, we have

(3.34)
$\mathcal{Q}_{j+1}\subset\hat{\mathcal{Q}}_{j}\subset\mathcal{Q}_{j},\quad
j\in\mathbb{N}\cup\\{0\\}.$

Next, we set

(3.35) $k_{j}=(1-2^{-j})\hat{k},\quad\hat{k}_{j}=\frac{k_{j}+k_{j+1}}{2},\quad
j\in\mathbb{N}\cup\\{0\\},$

for

(3.36) $\hat{k}\geq\frac{1}{2}\mathrm{Tail}_{\infty}(u_{+};x_{0},\theta
r,t_{0}-r^{p},t_{0}+r^{p}),$

where $\mathrm{Tail}_{\infty}$ is defined in (2.2). Finally, we define the
functions

(3.37) $w_{j}=(u-k_{j})_{+},\quad\hat{w}_{j}=(u-\hat{k}_{j})_{+},\quad
j\in\mathbb{N}\cup\\{0\\}.$

We apply Lemma 3.3 to obtain the following result.

######  Lemma 3.6.

Let $0<s<1,\,1<p<\infty$ and $u\in
L^{p}_{\mathrm{loc}}\big{(}0,T;W^{1,p}_{\mathrm{loc}}(\Omega)\big{)}\cap
C_{\mathrm{loc}}\big{(}0,T;L^{2}_{\mathrm{loc}}(\Omega)\big{)}\cap
L^{\infty}_{\mathrm{loc}}\big{(}0,T;L^{p-1}_{ps}(\mathbb{R}^{N})\big{)}$ be a
weak subsolution of (1.1) in $\Omega_{T}$. Assume that
$(x_{0},t_{0})\in\Omega_{T}$ and $r\in(0,1)$ such that
$\mathcal{Q}_{r}=\mathcal{Q}_{r}(x_{0},t_{0})=B_{r}(x_{0})\times(t_{0}-r^{p},t_{0}+r^{p})\Subset\Omega_{T}$.
Suppose that $g$ satisfies (1.6) for some $\alpha\geq 0$,
$1<l\leq\max\\{2,p(1+\frac{2}{N})\\}$ and $h\in L^{\gamma
l^{\prime}}_{\mathrm{loc}}(\Omega_{T})$ for some $\gamma>\frac{N+p}{p}$. For
$j\in\mathbb{N}\cup\\{0\\}$, let
$B_{j},\hat{B}_{j},\Gamma_{j},\hat{\Gamma}_{j},\mathcal{Q}_{j},\hat{\mathcal{Q}}_{j}$
be given by (3.30)-(3.32) and $k_{j},\hat{k}_{j},\hat{k},w_{j},\hat{w}_{j}$
are given by (3.35)-(3.37). Assume that $\frac{1}{2}\leq\theta<1$. Then for
any $q\geq\max\\{p,2,l\\}$ and for any $j\in\mathbb{N}\cup\\{0\\}$, there
exists a positive constant
$C=C(N,p,q,s,\Lambda,C_{1},C_{2},C_{3},C_{4},l,\alpha,h)$ such that

(3.38)
$\begin{split}&\operatorname*{ess\,sup}_{t\in\hat{\Gamma}_{j}}\int_{\hat{B}_{j}}\hat{w}_{j}^{2}\,dx+\int_{\hat{\Gamma}_{j}}\int_{\hat{B}_{j}}|\nabla\hat{w}_{j}|^{p}\,dx\,dt\\\
&\leq\frac{C}{r^{p}}\Bigg{[}\frac{1}{\theta^{p}(1-\theta)^{N+p}}+\frac{1}{(1-\theta)^{p}}\Bigg{]}\Bigg{[}\frac{2^{(p+q-2)j}}{\hat{k}^{q-2}}+\frac{2^{(N+p+q-1)j}}{\hat{k}^{q-p}}\Bigg{]}\int_{\Gamma_{j}}\int_{B_{j}}w_{j}^{q}\,dx\,dt\\\
&\qquad+C\frac{2^{qj}}{\hat{k}^{q-l}}\int_{\Gamma_{j}}\int_{B_{j}}w_{j}^{q}\,dx\,dt+C\frac{2^{\frac{q(N+p\kappa_{0})j}{N+p}}}{\hat{k}^{\frac{q(N+p\kappa_{0})}{N+p}}}\bigg{(}\int_{\Gamma_{j}}\int_{B_{j}}w_{j}^{q}\,dx\,dt\bigg{)}^{\frac{N+p\kappa_{0}}{N+p}},\end{split}$

where $\kappa_{0}=1-\frac{N+p}{p\gamma}\in(0,1]$.

###### Proof.

Noting that $\hat{k}_{j}>k_{j}$ and as in [21, Lemma $4.1$, p. $25$], for any
$0\leq\lambda<q$ where $q\geq\max\\{p,2,l\\}$, we get

(3.39) $\hat{w}_{j}\leq w_{j}\quad\text{and}\quad\hat{w}_{j}^{\lambda}\leq
C\hat{k}^{\lambda-q}2^{(q-\lambda)j}w_{j}^{q}\text{ in }\Omega_{T},$

for some constant $C=C(\lambda,q)>0$. Let
$\xi_{j}(x,t)=\psi_{j}(x)\eta_{j}(t)$, where $\psi_{j}\in
C_{c}^{\infty}(B_{j})$ and $\eta_{j}\in C_{c}^{\infty}(\Gamma_{j})$ such that

(3.40) $\begin{split}&0\leq\psi_{j}\leq 1\text{ in
}B_{j},\quad|\nabla\psi_{j}|\leq C\frac{2^{j}}{(1-\theta)r}\text{ in
}B_{j},\\\ &\psi_{j}\equiv 1\text{ in
}\hat{B}_{j},\quad\mathrm{dist}(\mathrm{supp}\,\psi_{j},\mathbb{R}^{N}\setminus
B_{j})\geq C\frac{(1-\theta)r}{2^{j}}\end{split}$

and

(3.41) $\begin{split}&0\leq\eta_{j}\leq 1\text{ in
}\Gamma_{j},\quad|\partial_{t}\eta_{j}|\leq
C\frac{2^{pj}}{(1-\theta)^{p}r^{p}}\text{ in }\Gamma_{j},\quad\eta_{j}\equiv
1\text{ in }\hat{\Gamma}_{j},\end{split}$

for some positive constant $C=C(N,p)$. Noting Lemma 2.10 and setting
$r=r_{j},\tau_{1}=t_{0}-\hat{r}_{j}^{p},\tau_{2}=t_{0}+r_{j}^{p}$,
$\tau=r_{j}^{p}-\hat{r}_{j}^{p}$ in Lemma 3.3, we obtain

(3.42)
$\begin{split}&\operatorname*{ess\,sup}_{t\in\hat{\Gamma}_{j}}\int_{\hat{B}_{j}}\hat{w}_{j}^{2}\,dx+\int_{\hat{\Gamma}_{j}}\int_{\hat{B}_{j}}|\nabla\hat{w}_{j}|^{p}\,dx\,dt\leq
I_{0}+I_{1}+I_{2}+I_{3}+I_{4}+I_{5},\end{split}$

where

$\begin{split}I_{0}&=C\int_{\Gamma_{j}}\int_{B_{j}}\hat{w}_{j}^{p}|\nabla\psi_{j}|^{p}\eta_{j}^{p}\,dx\,dt,\\\
I_{1}&=C\int_{\Gamma_{j}}\int_{B_{j}}\int_{B_{j}}\max\\{\hat{w}_{j}(x,t),\hat{w}_{j}(y,t)\\}^{p}|\psi_{j}(x)-\psi_{j}(y)|^{p}\eta_{j}^{p}\,d\mu\,dt,\\\
I_{2}&=C\operatorname*{ess\,sup}_{x\in\mathrm{supp}\psi_{j},\,t\in\Gamma_{j}}\int_{\mathbb{R}^{N}\setminus
B_{j}}\frac{\hat{w}_{j}(y,t)^{p-1}}{|x-y|^{N+ps}}\,dy\int_{\Gamma_{j}}\int_{B_{j}}\hat{w}_{j}\psi_{j}^{p}\eta_{j}^{p}\,dx\,dt,\\\
I_{3}&=C\int_{\Gamma_{j}}\int_{B_{j}}\hat{w}_{j}^{2}\psi_{j}^{p}|\partial_{t}\eta_{j}^{p}|\,dx\,dt,\\\
I_{4}&=C\int_{\Gamma_{j}}\int_{B_{j}}\big{(}|u|^{l}\chi_{\\{u\geq\hat{k}_{j}\\}}+\hat{w}_{j}^{l}\big{)}\psi_{j}^{p}\eta_{j}^{p}\,dx\,dt\quad\text{and}\\\
I_{5}&=C\int_{\Gamma_{j}}\int_{B_{j}}|h|^{l^{\prime}}\chi_{\\{u\geq\hat{k}_{j}\\}}\psi_{j}^{p}\eta_{j}^{p}\,dx\,dt\end{split}$

for some constant $C=C(p,\Lambda,C_{1},C_{2},C_{3},C_{4},l,\alpha)>0$.
Estimate of $I_{0}$: Using (3.39) with $\lambda=p$ we obtain
$\hat{w}_{j}^{p}\leq C\hat{k}^{p-q}2^{(q-p)j}w_{j}^{q}$ in $\mathcal{Q}_{j}$
for some constant $C=C(p,q)>0$. Using this fact along with the properties of
$\psi_{j}$ and $\eta_{j}$ from (3.40) and (3.41), for some constant
$C=C(N,p,q,s,\Lambda,C_{1},C_{2},C_{3},C_{4},l,\alpha,h)>0$, we have

(3.43)
$\begin{split}I_{0}&=C\int_{\Gamma_{j}}\int_{B_{j}}\hat{w}_{j}^{p}|\nabla\psi_{j}|^{p}\eta_{j}^{p}\,dx\,dt\leq
C\frac{2^{pj}}{(1-\theta)^{p}r^{p}}\int_{\Gamma_{j}}\int_{B_{j}}\hat{w}_{j}^{p}\,dx\,dt\\\
&\leq
C\frac{2^{qj}}{\hat{k}^{q-p}(1-\theta)^{p}r^{p}}\int_{\Gamma_{j}}\int_{B_{j}}w_{j}^{q}\,dx\,dt.\end{split}$

Estimate of $I_{1}$: Choosing $\lambda=p$ in (3.39), we obtain
$\hat{w}_{j}^{p}\leq C\hat{k}^{p-q}2^{(q-p)j}w_{j}^{q}$ in $\mathcal{Q}_{j}$
for some constant $C(p,q)>0$. Using this estimate along with the properties of
$\psi_{j}$ and $\eta_{j}$ from (3.40) and (3.41), for some constant
$C=C(N,p,q,s,\Lambda,C_{1},C_{2},C_{3},C_{4},l,\alpha,h)>0$, we have

(3.44)
$\begin{split}I_{1}&=C\int_{\Gamma_{j}}\int_{B_{j}}\int_{B_{j}}\max\\{\hat{w}_{j}(x,t),\hat{w}_{j}(y,t)\\}^{p}|\psi_{j}(x)-\psi_{j}(y)|^{p}\eta_{j}^{p}\,d\mu\,dt\\\
&\leq C\frac{2^{pj}}{(1-\theta)^{p}r^{p}}\operatorname*{ess\,sup}_{x\in
B_{j}}\int_{B_{j}}\frac{dy}{|x-y|^{N+ps-p}}\int_{\Gamma_{j}}\int_{B_{j}}\hat{w}_{j}^{p}\,dx\,dt\\\
&\leq
C\frac{2^{qj}}{\hat{k}^{q-p}(1-\theta)^{p}r^{ps}}\int_{\Gamma_{j}}\int_{B_{j}}w_{j}^{q}\,dx\,dt\leq
C\frac{2^{qj}}{\hat{k}^{q-p}(1-\theta)^{p}r^{p}}\int_{\Gamma_{j}}\int_{B_{j}}w_{j}^{q}\,dx\,dt,\end{split}$

where to deduce the last line above, we have also used the fact that
$r\in(0,1)$.
Estimate of $I_{2}$: Without loss of generality, we assume that $x_{0}=0$. For
$x\in\mathrm{supp}\,\psi_{j}$ and $y\in\mathbb{R}^{N}\setminus B_{j}$, we have

$\frac{1}{|x-y|}=\frac{1}{|y|}\frac{|y|}{|x-y|}\leq\frac{1}{|y|}\frac{|x|+|x-y|}{|x-y|}\leq\frac{1}{|y|}\frac{2^{j+4}}{(1-\theta)}.$

This implies

(3.45)
$\begin{split}\operatorname*{ess\,sup}_{x\in\mathrm{supp}\,\psi_{j},\,t\in\Gamma_{j}}\int_{\mathbb{R}^{N}\setminus
B_{j}}\frac{\hat{w}_{j}(y,t)^{p-1}}{|x-y|^{N+ps}}\,dy&\leq
C\frac{2^{j(N+ps)}}{(1-\theta)^{N+ps}}\operatorname*{ess\,sup}_{t\in\Gamma_{j}}\int_{\mathbb{R}^{N}\setminus
B_{j}}\frac{\hat{w}_{j}(y,t)^{p-1}}{|y|^{N+ps}}\,dy\\\ &\leq
C\frac{2^{j(N+ps)}}{(1-\theta)^{N+ps}}\operatorname*{ess\,sup}_{t\in\Gamma_{j}}\int_{\mathbb{R}^{N}\setminus
B_{\theta r}}\frac{\hat{w}_{j}(y,t)^{p-1}}{|y|^{N+ps}}\,dy\\\ &\leq
C\frac{2^{j(N+p)}}{r^{p}\theta^{p}(1-\theta)^{N+p}}\mathrm{Tail}_{\infty}(u_{+};x_{0},\theta
r,t_{0}-r^{p},t_{0}+r^{p})^{p-1},\end{split}$

for some constant $C=C(N,p,s)>0$. Again using (3.39) with $\lambda=1$, we
obtain $\hat{w}_{j}\leq C\hat{k}^{1-q}2^{(q-1)j}w_{j}^{q}$ in
$\mathcal{Q}_{j}$ for some constant $C=C(q)>0$. This fact along with (3.45)
and the choice of $\hat{k}$ in (3.36), for some constant
$C=C(N,p,q,s,\Lambda,C_{1},C_{2},C_{3},C_{4},l,\alpha,h)>0$ gives us

(3.46)
$\begin{split}I_{2}&=\operatorname*{ess\,sup}_{x\in\mathrm{supp}\,\psi_{j},\,t\in\hat{\Gamma}_{j}}\int_{\mathbb{R}^{N}\setminus
B_{j}}\frac{\hat{w}_{j}(y,t)^{p-1}}{|x-y|^{N+ps}}\,dy\int_{\Gamma_{j}}\int_{B_{j}}\hat{w}_{j}\psi_{j}^{p}\eta_{j}^{p}\,dx\,dt\\\
&\leq
C\frac{2^{j(N+p+q-1)}}{r^{p}\theta^{p}(1-\theta)^{N+p}\hat{k}^{q-p}}\int_{\Gamma_{j}}\int_{B_{j}}w_{j}^{q}\,dx\,dt.\end{split}$

Estimate of $I_{3}$: Again, using (3.39) for $\lambda=2$, we get
$\hat{w}_{j}^{2}\leq C\hat{k}^{2-q}2^{(q-2)j}w_{j}^{q}$ in $\mathcal{Q}_{j}$
for some constant $C=C(q)>0$. Using the properties of $\psi_{j},\eta_{j}$ for
a constant $C=C(N,p,q,s,\Lambda,C_{1},C_{2},C_{3},\\\ C_{4},l,\alpha,h)>0$, we
have

(3.47)
$\begin{split}I_{3}&=\int_{\Gamma_{j}}\int_{B_{j}}\hat{w}_{j}^{2}\psi_{j}^{p}|\partial_{t}\eta_{j}^{p}|\,dx\,dt\leq
C\frac{2^{pj}}{(1-\theta)^{p}r^{p}}\int_{\Gamma_{j}}\int_{B_{j}}\hat{w}_{j}^{2}\,dx\,dt\\\
&\leq
C\frac{2^{(p+q-2)j}}{(1-\theta)^{p}r^{p}\hat{k}^{q-2}}\int_{\Gamma_{j}}\int_{B_{j}}w_{j}^{q}\,dx\,dt.\end{split}$

Estimate of $I_{4}$: From the proof of [21, Lemma $4.1$, p. $27$], we have

(3.48)
$\begin{split}I_{4}&=\int_{\Gamma_{j}}\int_{B_{j}}\big{(}|u|^{l}\chi_{\\{u\geq\hat{k}_{j}\\}}+\hat{w}_{j}^{l}\big{)}\psi_{j}^{p}\eta_{j}^{p}\,dx\,dt\leq
C\frac{2^{qj}}{\hat{k}^{q-l}}\int_{\Gamma_{j}}\int_{B_{j}}w_{j}^{q}\,dx\,dt,\end{split}$

for a positive constant
$C=C(N,p,q,s,\Lambda,C_{1},C_{2},C_{3},C_{4},l,\alpha,h)$.
Estimate of $I_{5}$: Using Hölder’s inequality with exponents $\gamma$ and
$\gamma^{\prime}$ along with (3.39), for a constant
$C=C(N,p,q,s,\Lambda,C_{1},C_{2},C_{3},C_{4},l,\alpha,h)>0$, we have

(3.49)
$\begin{split}I_{5}&=\int_{\Gamma_{j}}\int_{B_{j}}|h|^{l^{\prime}}\chi_{\\{u\geq\hat{k}_{j}\\}}\psi_{j}^{p}\eta_{j}^{p}\,dx\,dt\leq
C\||h|^{l^{{}^{\prime}}}\|_{L^{\gamma}(\Omega_{T})}\frac{2^{\frac{qj}{\gamma^{\prime}}}}{\hat{k}^{\frac{q}{\gamma^{\prime}}}}\bigg{(}\int_{\Gamma_{j}}\int_{B_{j}}w_{j}^{q}\,dx\,dt\bigg{)}^{\frac{1}{\gamma^{\prime}}}\\\
&\leq
C\frac{2^{\frac{q(N+p\kappa_{0})j}{N+p}}}{\hat{k}^{\frac{q(N+p\kappa_{0})}{N+p}}}\Big{(}\int_{\Gamma_{j}}\int_{B_{j}}w_{j}^{q}\,dx\,dt\Big{)}^{\frac{N+p\kappa_{0}}{N+p}},\end{split}$

where $\kappa_{0}=1-\frac{p+N}{p\gamma}\in(0,1]$, since
$\gamma>\frac{N+p}{p}$. The estimate (3.38) follows by combining (3.43),
(3.44), (3.46), (3.47), (3.48) and (3.49) in (3.42), ∎

######  Lemma 3.7.

Let $0<s<1,\,\frac{2N}{N+2}\leq p<\infty$ and $u\in
L^{p}_{\mathrm{loc}}\big{(}0,T;W^{1,p}_{\mathrm{loc}}(\Omega)\big{)}\cap
C_{\mathrm{loc}}\big{(}0,T;L^{2}_{\mathrm{loc}}(\Omega)\big{)}\cap
L^{\infty}_{\mathrm{loc}}\big{(}0,T;L^{p-1}_{ps}(\mathbb{R}^{N})\big{)}$ be a
weak subsolution of (1.1) in $\Omega_{T}$. Assume that
$(x_{0},t_{0})\in\Omega_{T}$ and $r\in(0,1)$ such that
$\mathcal{Q}_{r}=\mathcal{Q}_{r}(x_{0},t_{0})=B_{r}(x_{0})\times(t_{0}-r^{p},t_{0}+r^{p})\Subset\Omega_{T}$.
Let $\max\\{p,2\\}\leq l<p\kappa$ where $\kappa=\frac{N+2}{N}$ and $g$
satisfies (1.6), where $h\in L^{\gamma l^{\prime}}_{\mathrm{loc}}(\Omega_{T})$
for some $\gamma>\frac{N+p}{p}$. For $j\in\mathbb{N}\cup\\{0\\}$, let
$B_{j},\hat{B}_{j},\Gamma_{j},\hat{\Gamma}_{j},\mathcal{Q}_{j},\hat{\mathcal{Q}}_{j}$
be given by (3.30)-(3.32) and $k_{j},\hat{k}_{j},\hat{k},w_{j},\hat{w}_{j}$
are given by (3.35)-(3.37). Assume that $\frac{1}{2}\leq\theta<1$. Then for
any $j\in\mathbb{N}\cup\\{0\\}$, there exists a positive constant
$C=C(N,p,s,\Lambda,C_{1},C_{2},C_{3},C_{4},l,\alpha,h)$ such that

(3.50) $\begin{split}&\int_{\Gamma_{j+1}}\int_{B_{j+1}}w_{j+1}^{l}\,dx\,dt\\\
&\leq C\frac{2^{aj}}{r^{\frac{\xi
l}{\kappa}}}\Bigg{[}\frac{1}{\theta^{\frac{\xi
l}{\kappa}}(1-\theta)^{\frac{(N+p)\xi
l}{p\kappa}}}+\frac{1}{(1-\theta)^{\frac{\xi
l}{\kappa}}}\Bigg{]}\frac{1}{\hat{k}^{l(1-\frac{l}{p\kappa})}}\Bigg{[}\frac{1}{\hat{k}^{(l-2)\xi}}+\frac{1}{\hat{k}^{(l-p)\xi}}\Bigg{]}^{\frac{l}{p\kappa}}\bigg{(}\int_{\Gamma_{j}}\int_{B_{j}}w_{j}^{l}\,dxdt\bigg{)}^{1+\frac{l}{\kappa
N}}\\\
&\qquad+C\frac{2^{aj}}{\hat{k}^{l(1-\frac{l}{p\kappa})}}\bigg{(}\int_{\Gamma_{j}}\int_{B_{j}}w_{j}^{l}\,dx\,dt\bigg{)}^{1+\frac{l}{\kappa
N}}+C\frac{2^{aj}}{\hat{k}^{{l(1-\frac{l}{p\kappa})}+\frac{l^{2}}{p\kappa}(1+\frac{\kappa_{0}p}{N})}}\bigg{(}\int_{\Gamma_{j}}\int_{B_{j}}w_{j}^{l}\,dx\,dt\bigg{)}^{1+\frac{l\kappa_{0}}{\kappa
N}},\end{split}$

where $\xi=1+\frac{p}{N}$, $a=\xi(N+p+l)$ and
$\kappa_{0}=1-\frac{p+N}{p\gamma}\in(0,1]$.

###### Proof.

Let $\Phi_{j}\in C_{c}^{\infty}(\hat{\mathcal{Q}}_{j})$ be such that

(3.51) $0\leq\Phi_{j}\leq 1\text{ in
}\hat{\mathcal{Q}}_{j},\quad|\nabla\Phi_{j}|\leq
C\frac{2^{j}}{(1-\theta)r},\quad\Phi_{j}\equiv 1\text{ in }\mathcal{Q}_{j+1},$

for some constant $C=C(N,p,s)>0$. Note that $k_{j+1}\geq\hat{k}_{j}$, thus, we
have $w_{j+1}\leq\hat{w}_{j}$. Since $l<p\kappa$, where $\kappa=\frac{N+2}{N}$
and $\mathcal{Q}_{j+1}\subset\hat{\mathcal{Q}}_{j}$, using Hölder’s
inequality, we obtain

(3.52)
$\begin{split}\int_{\Gamma_{j+1}}\int_{B_{j+1}}w_{j+1}^{l}\,dx\,dt&\leq\int_{\Gamma_{j+1}}\int_{B_{j+1}}\hat{w}_{j}^{l}\,dx\,dt\\\
&\leq\bigg{(}\int_{\Gamma_{j+1}}\int_{B_{j+1}}\hat{w}_{j}^{p\kappa}\,dx\,dt\bigg{)}^{\frac{l}{p\kappa}}\bigg{(}\int_{\Gamma_{j+1}}\int_{B_{j+1}}\chi_{\\{u\geq\hat{k}_{j}\\}}\,dx\,dt\bigg{)}^{1-\frac{l}{p\kappa}}\\\
&\leq\bigg{(}\int_{\hat{\Gamma}_{j}}\int_{\hat{B}_{j}}\big{(}\hat{w}_{j}\Phi_{j}\big{)}^{p\kappa}\,dx\,dt\bigg{)}^{\frac{l}{p\kappa}}\bigg{(}C\frac{2^{aj}}{\hat{k}^{l}}\int_{\Gamma_{j}}\int_{B_{j}}w_{j}^{l}\,dx\,dt\bigg{)}^{1-\frac{l}{p\kappa}},\end{split}$

for a constant $C=C(l)>0$, where $a=\xi(N+p+l)$ with $\xi=1+\frac{p}{N}$. To
obtain the last line above in (3.52), we have also used the estimate

$\int_{\Gamma_{j+1}}\int_{B_{j+1}}\chi_{\\{u\geq\hat{k}_{j}\\}}\,dx\,dt\leq
C\frac{2^{lj}}{\hat{k}^{l}}\int_{\Gamma_{j}}\int_{B_{j}}w_{j}^{l}\,dx\,dt,$

for some constant $C=C(l)>0$. This follows by choosing $\lambda=0$ and $q=l$
in (3.39), which is possible, since $l\geq\max\\{p,2\\}$. Since
$\kappa=1+\frac{2}{N}$, by Lemma 2.9 (b), for some constant $C=C(p,N)>0$, we
have

(3.53)
$\int_{\hat{\Gamma}_{j}}\int_{\hat{B}_{j}}\big{(}\hat{w}_{j}\Phi_{j}\big{)}^{p\kappa}\,dx\,dt\leq
C\bigg{(}\int_{\hat{\Gamma}_{j}}\int_{\hat{B}_{j}}|\nabla(\hat{w}_{j}\Phi_{j})|^{p}\,dx\,dt\bigg{)}\bigg{(}\operatorname*{ess\,sup}_{\hat{\Gamma}_{j}}\int_{\hat{B}_{j}}|\hat{w}_{j}\Phi_{j}|^{2}\,dx\bigg{)}^{\frac{p}{N}}.$

Using the properties of $\Phi_{j}$ and combining the estimates (3.52) and
(3.53), for some constant $C=C(N,p,s,l)>0$, we have

(3.54) $\begin{split}\int_{\Gamma_{j+1}}\int_{B_{j+1}}w_{j+1}^{l}\,dx\,dt&\leq
C\big{(}(I+\hat{I})J^{\frac{p}{N}}\big{)}^{\frac{l}{p\kappa}}\bigg{(}\frac{2^{aj}}{\hat{k}^{l}}\int_{\Gamma_{j}}\int_{B_{j}}w_{j}^{l}\,dx\,dt\bigg{)}^{1-\frac{l}{p\kappa}},\end{split}$

where

$I=\int_{\hat{\Gamma}_{j}}\int_{\hat{B}_{j}}|\nabla\hat{w}_{j}|^{p}\,dx\,dt,\quad\hat{I}=\int_{\hat{\Gamma}_{j}}\int_{\hat{B}_{j}}\hat{w}_{j}^{p}|\nabla\Phi_{j}|^{p}\,dx\,dt\quad\text{and}\quad
J=\operatorname*{ess\,sup}_{\hat{\Gamma}_{j}}\int_{\hat{B}_{j}}|\hat{w}_{j}|^{2}\,dx\,dt.$

Estimate of $\hat{I}$: Using (3.39) with $\lambda=p$ we obtain
$\hat{w}_{j}^{p}\leq C\hat{k}^{p-q}2^{(q-p)j}w_{j}^{q}$ in $\mathcal{Q}_{j}$
for some constant $C=C(p,q)>0$. Using this fact along with the property of
$\Phi_{j}$, for some constant
$C=C(N,p,s,\Lambda,C_{1},C_{2},C_{3},C_{4},l,\alpha,h)>0$, we have

(3.55)
$\begin{split}\hat{I}&=\int_{\Gamma_{j}}\int_{B_{j}}\hat{w}_{j}^{p}|\nabla\Phi_{j}|^{p}\,dx\,dt\leq
C\frac{2^{pj}}{(1-\theta)^{p}r^{p}}\int_{\Gamma_{j}}\int_{B_{j}}\hat{w}_{j}^{p}\,dx\,dt\\\
&\leq
C\frac{2^{qj}}{\hat{k}^{q-p}(1-\theta)^{p}r^{p}}\int_{\Gamma_{j}}\int_{B_{j}}w_{j}^{q}\,dx\,dt.\end{split}$

Using the above estimate of $\hat{I}$ from (3.55) and choosing $q=l$ in Lemma
3.6, for some constant
$C=C(N,p,s,\Lambda,C_{1},C_{2},C_{3},C_{4},l,\alpha,h)>0$, we have

(3.56)
$\begin{split}I+\hat{I},J&\leq\frac{C}{r^{p}}\Bigg{[}\frac{1}{\theta^{p}(1-\theta)^{N+p}}+\frac{1}{(1-\theta)^{p}}\Bigg{]}\Bigg{[}\frac{2^{(p+l-2)j}}{\hat{k}^{l-2}}+\frac{2^{(N+p+l-1)j}}{\hat{k}^{l-p}}\Bigg{]}\int_{\Gamma_{j}}\int_{B_{j}}w_{j}^{l}\,dx\,dt\\\
&\qquad+C{2^{lj}}\int_{\Gamma_{j}}\int_{B_{j}}w_{j}^{l}\,dx\,dt+C\frac{2^{\frac{l(N+p\kappa_{0})j}{N+p}}}{\hat{k}^{\frac{l(N+p\kappa_{0})}{N+p}}}\bigg{(}\int_{\Gamma_{j}}\int_{B_{j}}w_{j}^{l}\,dx\,dt\bigg{)}^{\frac{N+p\kappa_{0}}{N+p}}.\end{split}$

By applying (3.56) in (3.54), we obtain (3.50). Hence the result follows. ∎

The Lemma below helps us to conclude the local boundedness result in Theorem
3.2 for $1<p<\frac{2N}{N+2}$.

######  Lemma 3.8.

Let $0<s<1,\,1<p<\frac{2N}{N+2}$ and $u\in
L^{\infty}_{\mathrm{loc}}(\Omega_{T})\cap
L^{p}_{\mathrm{loc}}\big{(}0,T;W^{1,p}_{\mathrm{loc}}(\Omega)\big{)}\cap
C_{\mathrm{loc}}\big{(}0,T;L^{2}_{\mathrm{loc}}(\Omega)\big{)}\cap
L^{\infty}_{\mathrm{loc}}\big{(}0,T;L^{p-1}_{ps}(\mathbb{R}^{N})\big{)}$ be a
weak subsolution of (1.1) in $\Omega_{T}$. Suppose
$(x_{0},t_{0})\in\Omega_{T}$ and $r\in(0,1)$ such that
$\mathcal{Q}_{r}=\mathcal{Q}_{r}(x_{0},t_{0})=B_{r}(x_{0})\times(t_{0}-r^{p},t_{0}+r^{p})\Subset\Omega_{T}$.
Assume that $m>\frac{N(2-p)}{p}$, $g$ satisfies (1.6) for some $\alpha\geq 0$
with $1<l\leq 2$ and $h\in L^{\infty}_{\mathrm{loc}}(\Omega_{T})$. For
$j\in\mathbb{N}\cup\\{0\\}$, let
$B_{j},\hat{B}_{j},\Gamma_{j},\hat{\Gamma}_{j},\mathcal{Q}_{j},\mathcal{\hat{Q}}_{j}$
be given by (3.30)-(3.32) and $k_{j},\hat{k}_{j},\hat{k},w_{j},\hat{w}_{j}$
are given by (3.35)-(3.37). Assume that $\frac{1}{2}\leq\theta<1$. Then for
any $j\in\mathbb{N}\cup\\{0\\}$ there exists a positive constant
$C=C(N,p,s,\Lambda,C_{1},C_{2},C_{3},C_{4},l,\alpha,m,h)$ such that

(3.57)
$\begin{split}\int_{B_{j+1}}\int_{\Gamma_{j+1}}w_{j+1}^{m}\,dx\,dt&\leq\Bigg{[}\frac{C}{r^{p(1+\frac{p}{N})}}\frac{2^{aj}}{(1-\theta)^{\frac{(N+p)^{2}}{N}}}\Big{(}\frac{1}{\hat{k}^{m-2}}+\frac{1}{\hat{k}^{m-p}}\Big{)}^{1+\frac{p}{N}}+\frac{C}{r^{p}}\frac{2^{aj}}{\hat{k}^{m(1+\frac{p}{N})}}\Bigg{]}\\\
&\qquad\cdot\|\hat{w}_{j}\|^{m-p\kappa}_{L^{\infty}(\mathcal{Q}_{j+1})}\bigg{(}\int_{B_{j}}\int_{\Gamma_{j}}w_{j}^{m}\,dx\,dt\bigg{)}^{1+\frac{p}{N}},\end{split}$

where $a=(N+p+m)(1+\frac{p}{N})$.

###### Proof.

Recall that $\mathcal{Q}_{j}=B_{j}\times\Gamma_{j}$ and
$\hat{\mathcal{Q}}_{j}=\hat{B}_{j}\times\hat{\Gamma}_{j}$ for
$j\in\mathbb{N}\cup\\{0\\}$. Then we observe that

(3.58)
$\int_{B_{j+1}}\int_{\Gamma_{j+1}}w_{j+1}^{m}\,dx\,dt\leq\int_{B_{j+1}}\int_{\Gamma_{j+1}}\hat{w}_{j}^{m}\,dx\,dt\leq\|\hat{w}_{j}\|^{m-p\kappa}_{L^{\infty}(\mathcal{Q}_{j+1})}\int_{B_{j+1}}\int_{\Gamma_{j+1}}\hat{w}_{j}^{p\kappa}\,dx\,dt.$

Below, we estimate the integral

$I=\int_{B_{j+1}}\int_{\Gamma_{j+1}}\hat{w}_{j}^{p\kappa}\,dx\,dt.$

Recall that $\kappa=1+\frac{2}{N}$. By Lemma 2.9 (b), for some constant
$C=C(p,N)>0$, we have

(3.59)
$\int_{\hat{B}_{j}}\int_{\hat{\Gamma}_{j}}\big{(}\hat{w}_{j}\Phi_{j}\big{)}^{p\kappa}\,dx\,dt\leq
C\int_{\hat{B}_{j}}\int_{\hat{\Gamma}_{j}}|\nabla(\hat{w}_{j}\Phi_{j})|^{p}\,dx\,dt\bigg{(}\operatorname*{ess\,sup}_{\hat{\Gamma}_{j}}\int_{\hat{B}_{j}}|\hat{w}_{j}\Phi_{j}|^{2}\,dx\bigg{)}^{\frac{p}{N}},$

where $\Phi_{j}\in C_{c}^{\infty}(\hat{\mathcal{Q}}_{j})$ is such that

(3.60) $0\leq\Phi_{j}\leq 1\text{ in
}\hat{\mathcal{Q}}_{j},\quad|\nabla\Phi_{j}|\leq
C\frac{2^{j}}{(1-\theta)r},\quad\Phi_{j}\equiv 1\text{ in }\mathcal{Q}_{j+1},$

for some constant $C=C(N,p)>0$. Since
$\mathcal{Q}_{j+1}\subset\hat{\mathcal{Q}}_{j}$, from (3.59) we deduce that

(3.61) $\int_{B_{j+1}}\int_{\Gamma_{j+1}}\hat{w}_{j}^{p\kappa}\,dxdt\leq
C(I_{1}+I_{2})I_{3}^{\frac{p}{N}},$

for some constant $C=C(p,N)>0$, where

$I_{1}=\int_{\hat{B}_{j}}\int_{\hat{\Gamma}_{j}}|\nabla\hat{w}_{j}|^{p}\,dx\,dt,\quad
I_{2}=\int_{\hat{B}_{j}}\int_{\hat{\Gamma}_{j}}\hat{w}_{j}^{p}|\nabla\Phi_{j}|^{p}\,dx\,dt\quad\text{and}\quad
I_{3}=\operatorname*{ess\,sup}_{\hat{\Gamma}_{j}}\int_{\hat{B}_{j}}|\hat{w}_{j}|^{2}\,dx.$

Estimate of $I_{1}$ and $I_{3}$: Since $1<l\leq 2$, $m>\frac{N(2-p)}{p}$ and
$h\in L^{\infty}_{\mathrm{loc}}(\Omega_{T})$, choosing $q=m$ and
$\kappa_{0}=1$ in Lemma 3.6, for a positive constant
$C=C(N,p,s,\Lambda,C_{1},C_{2},C_{3},C_{4},l,\alpha,m,h)$, we have

(3.62)
$\begin{split}I_{i}&\leq\frac{C}{r^{p}}\frac{2^{bj}}{(1-\theta)^{N+p}}\bigg{(}\frac{1}{\hat{k}^{m-2}}+\frac{1}{\hat{k}^{m-p}}\bigg{)}\int_{B_{j}}\int_{\Gamma_{j}}w_{j}^{m}\,dxdt+C\frac{2^{bj}}{\hat{k}^{m}}\int_{B_{j}}\int_{\Gamma_{j}}w_{j}^{m}\,dxdt,\end{split}$

for $i=1,3$ where we have also used the fact that $\frac{1}{2}\leq\theta<1$
and $r\in(0,1)$.
Estimate of $I_{2}$: Using the properties of $\Phi_{j}$ from (3.60), we have

(3.63) $I_{2}\leq
C\frac{2^{pj}}{(1-\theta)^{p}r^{p}}\int_{B_{j}}\int_{\Gamma_{j}}\hat{w}_{j}^{p}\,dx\,dt\leq\frac{C}{r^{p}}\frac{2^{mj}}{(1-\theta)^{p}}\frac{1}{\hat{k}^{m-p}}\int_{B_{j}}\int_{\Gamma_{j}}w_{j}^{m}\,dx\,dt,$

for some positive constant $C=C(N,p,m)$, where we have also applied

$\hat{w}_{j}^{p}\leq C\frac{2^{(m-p)j}}{\hat{k}^{m-p}}w_{j}^{m}\text{ in
}\mathcal{Q}_{j},$

for some $C=C(m,p)>0$, which follows by choosing $\lambda=p$ and $q=m$ in
(3.39). Combining (3.62) and (3.63) in (3.61), for some positive constant
$C=C(N,p,s,\Lambda,C_{1},C_{2},C_{3},C_{4},l,\alpha,m,h)$, we obtain

(3.64)
$\begin{split}&\int_{B_{j+1}}\int_{\Gamma_{j+1}}\hat{w}_{j}^{p\kappa}\,dx\,dt\\\
&\leq\Bigg{[}\frac{C}{r^{p(1+\frac{p}{N})}}\frac{2^{aj}}{(1-\theta)^{\frac{(N+p)^{2}}{N}}}\Big{(}\frac{1}{\hat{k}^{m-2}}+\frac{1}{\hat{k}^{m-p}}\Big{)}^{1+\frac{p}{N}}+\frac{C}{r^{p}}\frac{2^{aj}}{\hat{k}^{m(1+\frac{p}{N})}}\Bigg{]}\bigg{(}\int_{B_{j}}\int_{\Gamma_{j}}w_{j}^{m}\,dx\,dt\bigg{)}^{1+\frac{p}{N}},\end{split}$

where $a=(N+p+m)(1+\frac{p}{N})$. Therefore, using (3.64) in (3.58), the
estimate (3.57) follows. ∎

##  4\. Hölder continuity result

Our main result in this section is the following local Hölder continuity
result for weak solutions of (1.1). We prove it by assuming the oscillation
Lemma 4.6, where the logarithmic energy estimate in Corollary 4.4 plays a
crucial role. The essential oscillation of a function $u$ over a set $S$ is
denoted by
$\text{osc}_{S}\,u=\operatorname*{ess\,sup}_{S}\,u-\operatorname*{ess\,inf}_{S}\,u$.

######  Theorem 4.1.

(Hölder continuity) Let $0<s<1,\,2<p<\infty$ and $u\in
L^{p}_{\mathrm{loc}}\big{(}0,T;W^{1,p}_{\mathrm{loc}}(\Omega)\big{)}\\\ \cap
C_{\mathrm{loc}}\big{(}0,T;L^{2}_{\mathrm{loc}}(\Omega)\big{)}\cap
L^{\infty}_{\mathrm{loc}}\big{(}0,T;L^{p-1}_{ps}(\mathbb{R}^{N})\big{)}$ be a
weak solution of (1.1) in $\Omega_{T}$. Suppose that
$(x_{0},t_{0})\in\Omega_{T}$, $R\in(0,1)$,
$\mathcal{Q}_{R}=\mathcal{Q}_{R}(x_{0},t_{0})=B_{R}(x_{0})\times(t_{0}-R^{p},t_{0}+R^{p})\Subset\Omega_{T}$
and $g(x,t,u)=h(x,t)$ in $\Omega_{T}\times\mathbb{R}$ where $h\in
L^{\infty}_{\mathrm{loc}}(\Omega_{T})$. Then there exist constants $b\in(0,1)$
and $\gamma\in(0,\frac{p}{p-1})$ such that for all $\rho\in(0,\frac{R}{2}]$,
we have

(4.1) $\mathrm{osc}_{\mathcal{Q}_{\rho,b}(x_{0},t_{0})}\,u\leq
C\Big{(}\frac{\rho}{R}\Big{)}^{\gamma}\Bigg{[}\max\Bigg{\\{}\bigg{(}\fint_{\mathcal{Q}_{R}}|u|^{p}\,dx\,dt\bigg{)}^{\frac{1}{2}},1\bigg{\\}}+\mathrm{Tail}_{\infty}\big{(}u;x_{0},\frac{R}{2},t_{0}-R^{p},t_{0}+R^{p}\big{)}\Bigg{]},$

for some positive constant $C=C(N,p,s,\Lambda,C_{1},C_{2},C_{3},C_{4},h)\geq
1$, where
$\mathcal{Q}_{\rho,b}(x_{0},t_{0})=B_{\rho}(x_{0})\times\big{(}t_{0}-\rho^{p},t_{0}+\rho^{p}\big{)}$.

###### Proof.

Choosing $r=R$ in Lemma 4.6 there exists $\eta\in(0,\frac{1}{4})$,
$\gamma\in(0,\frac{p}{p-1})$ a constant
$C=C(N,p,s,\Lambda,C_{1},C_{2},C_{3},C_{4},h)\geq 1$ such that

(4.2) $\text{osc}_{\mathcal{Q}_{j}}\,u\leq
C\Big{(}\frac{r_{j}}{R}\Big{)}^{\gamma}M\quad\text{for every}\quad
j\in\mathbb{N}\cup\\{0\\},$

where $r_{j}$ and $\mathcal{Q}_{j}$ be given by (4.18) and (4.22) respectively
and

(4.3)
$M=\max\bigg{\\{}\bigg{(}\fint_{\mathcal{Q}_{R}}|u|^{p}\,dx\,dt\bigg{)}^{\frac{1}{2}},1\bigg{\\}}+\mathrm{Tail}_{\infty}\big{(}u;x_{0},\tfrac{R}{2},t_{0}-R^{p},t_{0}+R^{p}\big{)}.$

Let $\rho\in(0,\frac{R}{2}]$ then we choose $i_{0}\in\mathbb{N}$ such that
$\rho\in(r_{i_{0}+1},r_{i_{0}}]$. We set $b=(CM)^{2-p}\in(0,1)$, then we
obtain $\mathcal{Q}_{\rho,b}(x_{0},t_{0})\subset\mathcal{Q}_{i_{0}}$.
Therefore, using the estimate (4.2) we get

(4.4)
$\text{osc}_{\mathcal{Q}_{\rho,b}(x_{0},t_{0})}\,u\leq\text{osc}_{\mathcal{Q}_{i_{0}}}\,u\leq
C\eta^{-\gamma}\Big{(}\frac{r_{i_{0}+1}}{R}\Big{)}^{\gamma}M\leq
C\eta^{-\gamma}\Big{(}\frac{\rho}{R}\Big{)}^{\gamma}M.$

Combining (4.3) and (4.4) the result follows. ∎

As a direct consequence Theorem 3.1 and Theorem 4.1 we have the following
result.

######  Proposition 4.2.

(Hölder continuity) Let $0<s<1,\,2<p<\infty$ and $u\in
L^{p}_{\mathrm{loc}}\big{(}0,T;W^{1,p}_{\mathrm{loc}}(\Omega)\big{)}\\\ \cap
C_{\mathrm{loc}}\big{(}0,T;L^{2}_{\mathrm{loc}}(\Omega)\big{)}\cap
L^{\infty}_{\mathrm{loc}}\big{(}0,T;L^{p-1}_{ps}(\mathbb{R}^{N})\big{)}$ be a
weak solution of (1.1) in $\Omega_{T}$. Suppose that
$(x_{0},t_{0})\in\Omega_{T}$, $R\in(0,1)$,
$\mathcal{Q}_{R}=\mathcal{Q}_{R}(x_{0},t_{0})=B_{R}(x_{0})\times(t_{0}-R^{p},t_{0}+R^{p})\Subset\Omega_{T}$
and $g$ satisfies (1.6) with $1<l<p\kappa\text{ and }h\in
L^{\infty}_{\mathrm{loc}}(\Omega_{T})$, where $\kappa=1+\frac{2}{N}$. Then $u$
is Hölder continuous in $\mathcal{Q}_{\frac{R}{2}}$.

###  4.1. Preliminaries

First, we establish the following logarithmic estimate, which is very crucial
to prove our Hölder continuity result.

######  Lemma 4.3.

(Logarithmic energy estimate) Let $0<s<1,\,2<p<\infty$ and $u\in
L^{p}_{\mathrm{loc}}\big{(}0,T;W^{1,p}_{\mathrm{loc}}(\Omega)\big{)}\cap
C_{\mathrm{loc}}\big{(}0,T;L^{2}_{\mathrm{loc}}(\Omega)\big{)}\cap
L^{\infty}_{\mathrm{loc}}\big{(}0,T;L^{p-1}_{ps}(\mathbb{R}^{N})\big{)}$ be a
weak solution of (1.1) in $\Omega_{T}$. Assume that
$(x_{0},t_{0})\in\Omega_{T}$,
$\hat{\mathcal{Q}}=B_{R}(x_{0})\times(t_{0}-2\hat{T},t_{0}+2\hat{T})\Subset\Omega_{T}$,
where $r\in(0,1)$ such that $0<r\leq\frac{R}{2}$, $\hat{T}>0$. Suppose $u\in
L^{\infty}(\hat{\mathcal{Q}})$ such that $u\geq 0$ in $\hat{\mathcal{Q}}$ and
$g(x,t,u)=h(x,t)$ in $\Omega_{T}\times\mathbb{R}$ where $h\in
L^{\infty}_{\mathrm{loc}}(\Omega_{T})$. Then, for any $d>0$, there exists a
constant $C=C(N,p,s,\Lambda,C_{1},C_{2},C_{3},C_{4},h)>0$ such that

(4.5)
$\begin{split}&\int_{t_{0}-\hat{T}}^{t_{0}+\hat{T}}\int_{B_{r}}|\nabla\log(u+d)|^{p}\,dx\,dt\\\
&\leq
C\left(\hat{T}r^{N-p}d^{1-p}\Big{(}\frac{r}{R}\Big{)}^{p}\mathrm{Tail}_{\infty}(u;x_{0},R,t_{0}-2\hat{T},t_{0}+\hat{T})^{p-1}+\hat{T}r^{N-p}+r^{N}d^{2-p}+\hat{T}r^{N}d^{1-p}\right),\end{split}$

where $B_{r}=B_{r}(x_{0})$ and $\mathrm{Tail}_{\infty}$ is defined in (2.2).

###### Proof.

Let $\psi\in C_{c}^{\infty}(B_{\frac{3r}{2}})$ and $\eta\in
C_{c}^{\infty}(t_{0}-2\hat{T},t_{0}+2\hat{T})$ be such that

(4.6) $\begin{split}&0\leq\psi\leq 1,\quad|\nabla\psi|\leq\frac{C}{r}\text{ in
}B_{2r}\text{ and }\psi\equiv 1\text{ in }B_{r},\\\ &0\leq\eta\leq
1,\quad|\partial_{t}\eta|\leq\frac{C}{\hat{T}}\text{ in
}(t_{0}-2\hat{T},t_{0}+2\hat{T})\quad\text{and}\quad\eta\equiv 1\text{ in
}(t_{0}-\hat{T},t_{0}+\hat{T}),\end{split}$

for some positive constant $C=C(N,p)$. For $d>0$, we define $\hat{u}=u+d$ and
choose $\phi=\hat{u}^{1-p}\psi^{p}\eta^{p}$ as a test function in (2.3) (whose
admissibility can be justified analogous to the proof of Lemma 3.3), we obtain

(4.7) $\begin{split}0&=I_{1}+I_{2}+I_{3}+I_{4}+I_{5},\end{split}$

where

$\begin{split}I_{1}&=-\int_{t_{0}-2\hat{T}}^{t_{0}-2\hat{T}}\int_{B_{2r}}u\partial_{t}(\hat{u}^{1-p}\psi^{p}\eta^{p})\,dx\,dt,\\\
I_{2}&=\int_{t_{0}-2\hat{T}}^{t_{0}-2\hat{T}}\int_{B_{2r}}\int_{B_{2r}}\mathcal{A}\big{(}u(x,y,t)\big{)}\big{(}\hat{u}^{1-p}(x,t)\psi^{p}(x)-\hat{u}^{1-p}(y,t)\psi^{p}(y)\big{)}\eta(t)^{p}\,d\mu\,dt,\\\
I_{3}&=2\int_{t_{0}-2\hat{T}}^{t_{0}-2\hat{T}}\int_{\mathbb{R}^{N}\setminus
B_{2r}}\int_{B_{2r}}\mathcal{A}\big{(}u(x,y,t)\big{)}\hat{u}(x,t)^{1-p}\psi(x)^{p}\eta(t)^{p}\,d\mu\,dt,\\\
I_{4}&=\int_{t_{0}-2\hat{T}}^{t_{0}-2\hat{T}}\int_{B_{2r}}\mathcal{B}(x,t,u,\nabla
u)\nabla(\hat{u}^{1-p}\psi^{p}\eta^{p})\,dx\,dt\quad\text{and}\\\
I_{5}&=-\int_{t_{0}-2\hat{T}}^{t_{0}-2\hat{T}}\int_{B_{2r}}h(x,t)\hat{u}(x,t)^{1-p}\psi(x)^{p}\eta(t)^{p}\,dx\,dt.\end{split}$

Estimate of $I_{1}$: Since $p>2$, integrating by parts, we obtain

(4.8)
$\begin{split}I_{1}&=-\int_{t_{0}-2\hat{T}}^{t_{0}-2\hat{T}}\int_{B_{2r}}u\partial_{t}(\hat{u}^{1-p}\psi^{p}\eta^{p})\,dx\,dt\\\
&\leq\frac{1}{p-2}\int_{t_{0}-2\hat{T}}^{t_{0}-2\hat{T}}\int_{B_{2r}}\hat{u}^{2-p}\psi^{p}|\partial_{t}\eta^{p}|\,dx\,dt\leq
Cr^{N}d^{2-p},\end{split}$

for some positive constant $C=C(N,p)$.
Estimate of $I_{2}$: Using the structural hypothesis (1.3) on
$\mathcal{A}(u(x,y,t))$, we obtain

$I_{2}\leq
C\int_{t_{0}-2\hat{T}}^{t_{0}-2\hat{T}}\int_{B_{2r}}\int_{B_{2r}}|u(x,t)-u(y,t)|^{p-2}\big{(}u(x,t)-u(y,t)\big{)}\big{(}\phi(x,t)-\phi(y,t)\big{)}\,d\mu\,dt,$

for some positive constant $C$ depending on $C_{1}$, $C_{2}$ given by (1.3),
where $\phi=\hat{u}^{1-p}\psi^{p}\eta^{p}$. From the proof of [21, Lemma
$3.5$, p. 20–22], we get

(4.9)
$\begin{split}I_{2}&\leq-C\int_{t_{0}-\hat{T}}^{t_{0}-\hat{T}}\int_{B_{r}}\int_{B_{r}}\left|\log\Big{(}\frac{\hat{u}(x,t)}{\hat{u}(y,t)}\Big{)}\right|^{p}\,d\mu\,dt+C\hat{T}r^{N-ps}\\\
&\leq-C\int_{t_{0}-\hat{T}}^{t_{0}-\hat{T}}\int_{B_{r}}\int_{B_{r}}\left|\log\Big{(}\frac{\hat{u}(x,t)}{\hat{u}(y,t)}\Big{)}\right|^{p}\,d\mu\,dt+C\hat{T}r^{N-p},\end{split}$

for some positive constant $C=C(N,p,s,\Lambda,C_{1},C_{2})$, where to obtain
the final step above, we have used that $r^{-ps}\leq r^{-p}$, since
$r\in(0,1)$.
Estimate of $I_{3}$: Again, using the structural hypothesis (1.3) on
$\mathcal{A}(u(x,y,t))$, we have

$I_{3}\leq
C\int_{t_{0}-2\hat{T}}^{t_{0}-2\hat{T}}\int_{\mathbb{R}^{N}\setminus
B_{2r}}\int_{B_{2r}}|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t))\psi(x)^{p}\eta(t)^{p}\,d\mu
dt,$

for some positive constant $C$ depending on $C_{1}$, $C_{2}$ given by (1.3).
From the proof of [21, Lemma $3.5$, p. 22–23], we arrive at

(4.10) $\begin{split}I_{3}&\leq
C\hat{T}r^{N-ps}+C\hat{T}r^{N}d^{1-p}\operatorname*{ess\,sup}_{t_{0}-2\hat{T}<t<t_{0}+2\hat{T}}\int_{\mathbb{R}^{N}\setminus
B_{R}}\frac{|u(y,t)|^{p-1}}{|y-x_{0}|^{N+ps}}\,dy\\\ &\leq
C\hat{T}r^{N-p}+\frac{C}{R^{p}}\hat{T}r^{N}d^{1-p}\mathrm{Tail}_{\infty}(u;x_{0},R,t_{0}-2\hat{T},t_{0}+2\hat{T})^{p-1},\end{split}$

for some positive constant $C=C(N,p,s,\Lambda,C_{1},C_{2})$, where in the
final step above, we have used that $r^{-ps}\leq r^{-p}$, since $r\in(0,1)$
and the definition of $\mathrm{Tail}_{\infty}$ given by (2.2).
Estimate of $I_{4}$: Since $\hat{u}\geq 0$ in $\hat{\mathcal{Q}}$, using the
hypothesis (1.5) on $\mathcal{B}$ and Young’s inequality, for any
$\epsilon>0$, we obtain

(4.11)
$\begin{split}I_{4}&=\int_{t_{0}-2\hat{T}}^{t_{0}-2\hat{T}}\int_{B_{2r}}\mathcal{B}(x,t,u,\nabla
u)\nabla(\hat{u}^{1-p}\psi^{p}\eta^{p})\,dx\,dt\\\
&=\int_{t_{0}-2\hat{T}}^{t_{0}-2\hat{T}}\int_{B_{2r}}\mathcal{B}(x,t,u,\nabla
u)\big{(}(1-p)\hat{u}^{-p}\nabla\hat{u}\psi^{p}+p\psi^{p-1}\hat{u}^{1-p}\nabla\psi\big{)}\eta^{p}\,dx\,t\\\
&=-(p-1)\int_{t_{0}-2\hat{T}}^{t_{0}-2\hat{T}}\int_{B_{2r}}\hat{u}^{-p}\psi^{p}\mathcal{B}(x,t,u,\nabla\hat{u})\nabla\hat{u}\eta^{p}\,dx\,dt\\\
&\qquad+p\int_{t_{0}-2\hat{T}}^{t_{0}-2\hat{T}}\int_{B_{2r}}\psi^{p-1}\hat{u}^{1-p}\nabla\psi
B(x,t,u,\nabla\hat{u})\eta^{p}\,dx\,dt\\\
&\leq-(p-1)C_{3}\int_{t_{0}-2\hat{T}}^{t_{0}-2\hat{T}}\int_{B_{2r}}|\nabla\log{\hat{u}}|^{p}\psi^{p}\eta^{p}\,dx\,dt+pC_{4}\epsilon\int_{t_{0}-2\hat{T}}^{t_{0}-2\hat{T}}\int_{B_{2r}}|\nabla\log{\hat{u}}|^{p}\psi^{p}\eta^{p}\,dx\,dt\\\
&\qquad+(p^{\prime})^{1-p}\epsilon^{1-p}C_{4}\int_{t_{0}-2\hat{T}}^{t_{0}-2\hat{T}}\int_{B_{2r}}|\nabla\psi|^{p}\eta^{p}\,dx\,dt,\end{split}$

where $C_{3}$ and $C_{4}$ are given by (1.5). Now choosing
$\epsilon=\frac{C_{3}(p-1)}{2C_{4}p}$ and using the properties of $\psi$ and
$\eta$ from (4.6), we obtain

(4.12)
$\begin{split}I_{4}&\leq-\frac{C_{3}(p-1)}{2}\int_{t_{0}-\hat{T}}^{t_{0}-\hat{T}}\int_{B_{r}}|\nabla\log{\hat{u}}|^{p}\,dx\,dt+C\hat{T}r^{N-p},\end{split}$

for some constant $C=C(N,p,s,\Lambda,C_{3},C_{4})>0$.
Estimate of $I_{5}$: Since $h\in L^{\infty}_{\mathrm{loc}}(\Omega_{T})$ and
$\hat{u}\geq d$ in $\hat{\mathcal{Q}}$, we have

(4.13)
$\begin{split}I_{5}&=-\int_{t_{0}-2\hat{T}}^{t_{0}-2\hat{T}}\int_{B_{2r}}h(x,t)\hat{u}(x,t)^{1-p}\psi(x)^{p}\eta(t)^{p}\,dx\,dt\\\
&\leq\|h\|_{L^{\infty}(\Omega_{T})}d^{1-p}\int_{t_{0}-2\hat{T}}^{t_{0}-2\hat{T}}\int_{B_{2r}}\psi(x)^{p}\eta(t)^{p}\,dx\,dt\leq
C\hat{T}r^{N}d^{1-p},\end{split}$

for some constant $C=C(N,p,s,\Lambda,h)>0$. Combining the estimates (4.8),
(4.9), (4.10), (4.12) and (4.13) in (4.7), the required estimate (4.5) holds.
Hence the result follows. ∎

As a consequence of Lemma 4.3 we have the following result.

######  Corollary 4.4.

Let $0<s<1,\,2<p<\infty$ and $u\in
L^{p}_{\mathrm{loc}}\big{(}0,T;W^{1,p}_{\mathrm{loc}}(\Omega)\big{)}\cap
C_{\mathrm{loc}}\big{(}0,T;L^{2}_{\mathrm{loc}}(\Omega)\big{)}\cap
L^{\infty}_{\mathrm{loc}}\big{(}0,T;L^{p-1}_{ps}(\mathbb{R}^{N})\big{)}$ be a
weak solution of (1.1) in $\Omega_{T}$. Assume that
$(x_{0},t_{0})\in\Omega_{T}$,
$\hat{\mathcal{Q}}=B_{R}(x_{0})\times(t_{0}-2\hat{T},t_{0}+2\hat{T})\Subset\Omega_{T}$,
where $r\in(0,1)$ such that $0<r\leq\frac{R}{2}$, $\hat{T}>0$. Suppose $u\in
L^{\infty}(\hat{\mathcal{Q}})$ such that $u\geq 0$ in $\hat{\mathcal{Q}}$ and
$g(x,t,u)=h(x,t)$ in $\Omega_{T}\times\mathbb{R}$ where $h\in
L^{\infty}_{\mathrm{loc}}(\Omega_{T})$. Let $A,D>0$ and $B>1$ and set

$w=\min\left\\{\bigg{[}\frac{\log(A+D)}{\log(u+D)}\bigg{]}_{+},\log\,B\right\\}.$

Then, there exists a constant $C=C(N,p,s,\Lambda,C_{1},C_{2},C_{3},C_{4},h)>0$
such that

(4.14)
$\begin{split}&\int_{t_{0}-\hat{T}}^{t_{0}-\hat{T}}\fint_{B_{r}}|w-(w)_{B_{r}}(t)|^{p}\,dx\,dt\\\
&\leq
C\left(\hat{T}D^{1-p}\Big{(}\frac{r}{R}\Big{)}^{p}\mathrm{Tail}_{\infty}(u;x_{0},R,t_{0}-2\hat{T},t_{0}+2\hat{T})^{p-1}+\hat{T}+r^{p}D^{2-p}+\hat{T}r^{p}D^{1-p}\right),\end{split}$

where $B_{r}=B_{r}(x_{0})$.

###### Proof.

Using the Poincaré inequality from [26, Theorem $2$], for every
$t\in(t_{0}-\hat{T},t_{0}+\hat{T})$, we have

(4.15) $\fint_{B_{r}}|w(x,t)-(w)_{B_{r}}(t)|^{p}\,dx\leq
Cr^{p-N}\int_{B_{r}}|\nabla w(x,t)|^{p}\,dx,$

for some constant $C=C(N,p)>0$. Integrating (4.15)with respect to
$t\in(t_{0}-\hat{T},t_{0}+\hat{T})$, we get

(4.16)
$\int_{t_{0}-\hat{T}}^{t_{0}-\hat{T}}\fint_{B_{r}}|w(x,t)-(w)_{B_{r}}(t)|^{p}\,dx\,dt\leq
Cr^{p-N}\int_{t_{0}-\hat{T}}^{t_{0}-\hat{T}}\int_{B_{r}}|\nabla
w(x,t)|^{p}\,dx\,dt,$

for some constant $C=C(N,p)>0$. Notice that $w$ is a truncation of a constant
and $\log(u+D)$, thus, we have

(4.17) $\begin{split}&\int_{t_{0}-\hat{T}}^{t_{0}-\hat{T}}\int_{B_{r}}|\nabla
w|^{p}\,dx\,dt\leq\int_{t_{0}-\hat{T}}^{t_{0}-\hat{T}}\int_{B_{r}}|\nabla\log(u+D)|^{p}\,dx\,dt\\\
&\leq
C\left(\hat{T}r^{N-p}D^{1-p}\Big{(}\frac{r}{R}\Big{)}^{p}\mathrm{Tail}_{\infty}(u;x_{0},R,t_{0}-2\hat{T},t_{0}+\hat{T})^{p-1}+\hat{T}r^{N-p}+r^{N}D^{2-p}+\hat{T}r^{N}D^{1-p}\right),\end{split}$

for some constant $C=C(N,p,s,\Lambda,C_{1},C_{2},C_{3},C_{4},h)>0$, where to
obtain the last inequality above, we have used Lemma 4.3. Employing (4.17) in
(4.16), the estimate (4.14) holds. Hence the result follows. ∎

Next, we prove the oscillation Lemma 4.6 stated below, for weak solutions of
the problem (1.1). To this end, at first we set some parameters. Let
$2<p<\infty$, $(x_{0},t_{0})\in\Omega_{T}$ and $r\in(0,R]$ for some
$R\in(0,1)$ so that
$\mathcal{Q}_{R}=B_{R}(x_{0})\times(t_{0}-R^{p},t_{0}+R^{p})\Subset\Omega_{T}$.
Let $\eta_{*}=\eta_{*}(p)>0$ be such that
$\eta^{\frac{p}{p-1}}\leq\frac{1}{4}$ for every $\eta\in(0,\eta_{*}]$. We
define the corresponding sequences of decreasing radius by

(4.18) $r_{j}=\frac{\eta^{j}r}{2},\quad j\in\mathbb{N}\cup\\{0\\},$

where $\eta<\min\\{\eta_{*},\frac{1}{4}\\}$ will be chosen later.

Let $u\in
L^{p}_{\mathrm{loc}}\big{(}0,T;W^{1,p}_{\mathrm{loc}}(\Omega)\big{)}\cap
C_{\mathrm{loc}}\big{(}0,T;L^{2}_{\mathrm{loc}}(\Omega)\big{)}\cap
L^{\infty}_{\mathrm{loc}}\big{(}0,T;L^{p-1}_{ps}(\mathbb{R}^{N})\big{)}$ be a
weak solution of (1.1) in $\Omega_{T}$ and $g(x,t,u)=h(x,t)$ in
$\Omega_{T}\times\mathbb{R}$ where $h\in
L^{\infty}_{\mathrm{loc}}(\Omega_{T})$. Then choosing $l=p$, $m=\frac{1}{2}$
in Theorem 3.1 we can find a large enough constant
$C=C(N,p,s,\Lambda,C_{2},C_{2},C_{3},C_{4},h)\geq 1$ such that

(4.19) $\operatorname*{ess\,sup}_{\mathcal{Q}_{r_{0}}}\,u\leq\frac{L}{2},$

holds, where

(4.20)
$L=C\Bigg{[}\max\bigg{\\{}\bigg{(}\fint_{\mathcal{Q}_{r}}|u|^{p}\,dx\,dt\bigg{)}^{\frac{1}{2}},1\bigg{\\}}+\mathrm{Tail}_{\infty}(u;x_{0},\tfrac{r}{2},t_{0}-r^{p},t_{0}+r^{p})\Bigg{]}.$

Let $\gamma\in(0,\frac{p}{p-1})$ to be chosen below and define

(4.21)
$\omega(r_{0})=\omega\Big{(}\frac{r}{2}\Big{)}=L,\quad\omega(r_{j})=\Big{(}\frac{r_{j}}{r_{0}}\Big{)}^{\gamma}\omega(r_{0}),\quad
j\in\mathbb{N}.$

Let $b_{0}=1$ and $b_{j}=(\varepsilon\eta^{(j-1)\gamma}L)^{2-p}$ for
$j\in\mathbb{N}$, where $\varepsilon=\eta^{\frac{p}{p-1}-\gamma}$. We denote
$B_{j}=B_{r_{j}}(x_{0})$, $l_{j}=b_{j}r_{j}^{p}$ and

(4.22)
$\mathcal{Q}_{j}=\mathcal{Q}_{r_{j},b_{j}}(x_{0},t_{0})=B_{j}\times(t_{0}-l_{j},t_{0}+l_{j}),\quad
j\in\mathbb{N}\cup\\{0\\}.$

We know that $r_{j+1}<r_{j}$ and by [$\eqref{lj}$, Lemma 4.5] we have
$l_{j+1}<l_{j}$. Therefore, we have $\mathcal{Q}_{j+1}\subset\mathcal{Q}_{j}$
for all $j\in\mathbb{N}\cup\\{0\\}$. First, we prove the following auxiliary
result.

######  Lemma 4.5.

Let $2<p<\infty$, then for every $j\in\mathbb{N}\cup\\{0\\}$, we have

(4.23)
$b_{j+1}=(\varepsilon\omega(r_{j}))^{2-p}\quad\text{and}\quad(\varepsilon\omega(r_{j}))^{1-p}\leq
r_{j+1}^{-p}.$

Moreover, we have

(4.24) $4l_{j+1}\leq l_{j},\quad j\in\mathbb{N}\cup\\{0\\}.$

###### Proof.

By definition $b_{j}$ and $\omega(r_{j})$, it directly follows that

$b_{j+1}=(\varepsilon\omega(r_{j}))^{2-p},\quad j\in\mathbb{N}\cup\\{0\\}.$

Moreover, using that $\omega(r_{0})=L\geq 1$ and $r\in(0,1)$, we have

$(\varepsilon\omega(r_{j}))^{1-p}=\eta^{\gamma(p-1)-p}\Big{(}\Big{(}\frac{r_{j}}{r_{0}}\Big{)}^{\gamma}\omega(r_{0})\Big{)}^{1-p}\leq\omega(r_{0})^{1-p}\eta^{-(j+1)p}\leq
r_{j+1}^{-p}.$

Hence (4.23) follows. To obtain (4.24), we observe that

$\beta=\Big{(}\frac{p}{p-1}-\gamma\Big{)}(2-p)+p\geq\frac{p}{p-1}.$

Since $4\eta^{\frac{p}{p-1}}\leq 1$, we obtain $4\eta^{\beta}\leq
4\eta^{\frac{p}{p-1}}\leq 1$. Therefore, we get

(4.25)
$4\eta^{(\frac{p}{p-1}-\gamma)(2-p)}r_{1}^{p}=4\eta^{\beta}r_{0}^{p}\leq
r_{0}^{p}.$

Since $\gamma\in(0,\frac{p}{p-1})$ and $p>2$, we have
$\gamma(2-p)+p\geq\frac{p}{p-1}$. Thus, using the definition of $r_{j}$ and
again the fact that $4\eta^{\frac{p}{p-1}}\leq 1$, we obtain

(4.26) $4\eta^{\gamma(2-p)}r_{j+1}^{p}=4\eta^{\gamma(2-p)+p}r_{j}^{p}\leq
4\eta^{\frac{p}{p-1}}r_{j}^{p}\leq r_{j}^{p}.$

Let $j=0$. Then, using (4.25), $p>2$ and $L\geq 1$, we have

(4.27) $4l_{1}=4b_{1}r_{1}^{p}=4(\varepsilon
L)^{2-p}r_{1}^{p}=4\eta^{(\frac{p}{p-1}-\gamma)(2-p)}r_{1}^{p}L^{2-p}\leq
r_{0}^{p}=l_{0}.$

When $j\in\mathbb{N}$, notice that by the definition of $l_{j}$ and using
(4.26), we have

(4.28)
$4l_{j+1}=4b_{j+1}r_{j+1}^{p}=(4\eta^{\gamma(2-p)}r_{j+1}^{p})(b_{j+1}\eta^{-\gamma(2-p)})\leq
r_{j}^{p}(\epsilon\eta^{(j-1)\gamma}L)^{2-p}=r_{j}^{p}b_{j}=l_{j}.$

Hence from (4.27) and (4.28) we obtain (4.24). ∎

Now we are ready to prove our oscillation result.

######  Lemma 4.6.

(Oscillation Lemma) Let $0<s<1,\,2<p<\infty$ and $u\in
L^{p}_{\mathrm{loc}}\big{(}0,T;W^{1,p}_{\mathrm{loc}}(\Omega)\big{)}\cap
C_{\mathrm{loc}}\big{(}0,T;L^{2}_{\mathrm{loc}}(\Omega)\big{)}\cap
L^{\infty}_{\mathrm{loc}}\big{(}0,T;L^{p-1}_{ps}(\mathbb{R}^{N})\big{)}$ be a
weak solution of the problem (1.1) in $\Omega_{T}$. Assume that
$(x_{0},t_{0})\in\Omega_{T}$ and $r\in(0,R]$ for some $R\in(0,1)$ such that
the cylinder
$\mathcal{Q}_{R}=\mathcal{Q}_{R}(x_{0},t_{0})=B_{R}(x_{0})\times(t_{0}-R^{p},t_{0}+R^{p})\Subset\Omega_{T}$.
Let $g(x,t,u)=h(x,t)$ in $\Omega_{T}\times\mathbb{R}$ for some $h\in
L^{\infty}_{\mathrm{loc}}(\Omega_{T})$. Then, we have

(4.29) $\mathrm{osc}_{\mathcal{Q}_{j}}\,u\leq\omega(r_{j})\quad\text{for
every}\quad j\in\mathbb{N}\cup\\{0\\},$

where $\omega(r_{j})$ and $\mathcal{Q}_{j}$ are defined by (4.21) and (4.22)
respectively.

###### Proof.

We prove the result by induction. First, we observe that the estimate (4.29)
holds for $j=0$. Indeed, since $g(x,t,u)=h(x,t)$ and $u$ is a weak solution of
the problem (1.1), both $u$ and $-u$ are weak subsolutions of the problem
(1.1). Then, by applying (4.19) to $u$ and $-u$, the estimate (4.29) holds for
$j=0$.

Induction hypothesis: Suppose (4.29) holds for every $i=0,1,\ldots,j$ where
$j\geq 0$. Then, by induction, to obtain (4.29), it is enough to prove (4.29)
for $i=j+1$. We prove this fact in two steps below. In Step $1$, we obtain the
estimate (4.35) related to $u_{j}$, where $u_{j}$ is defined in (4.32). In
Step $2$, we use an iteration argument to conclude the proof of (4.29) for
$i=j+1$, where the estimate (4.35) in Step $1$ plays an important role.

We set $2B_{j+1}=B_{2r_{j+1}}(x_{0})$ and
$2\mathcal{Q}_{j+1}=\mathcal{Q}_{2r_{j+1},2^{1-p}b_{j+1}}(x_{0},t_{0})=2B_{j+1}\times(t_{0}-2l_{j+1},t_{0}+2l_{j+1})$,
where $l_{j+1}=b_{j+1}r_{j+1}^{p}$. Then, one of the following estimates

(4.30)
$\frac{\left|2\mathcal{Q}_{j+1}\cap\big{\\{}u\geq\operatorname*{ess\,inf}_{\mathcal{Q}_{j}}+\frac{\omega(r_{j})}{2}\big{\\}}\right|}{|2\mathcal{Q}_{j+1}|}\geq\frac{1}{2}$

or

(4.31)
$\frac{\left|2\mathcal{Q}_{j+1}\cap\big{\\{}u\leq\operatorname*{ess\,inf}_{\mathcal{Q}_{j}}+\frac{\omega(r_{j})}{2}\big{\\}}\right|}{|2\mathcal{Q}_{j+1}|}\geq\frac{1}{2}$

must hold. We define

(4.32)
$u_{j}=\begin{cases}u-\operatorname*{ess\,inf}_{\mathcal{Q}_{j}}u,\quad\text{if
}\eqref{p1}\text{ holds},\\\
\omega(r_{j})-(u-\operatorname*{ess\,inf}_{\mathcal{Q}_{j}}u),\quad\text{if
}\eqref{p2}\text{ holds}.\end{cases}$

Then by (4.30) and (4.31), we have

(4.33)
$\frac{\left|2\mathcal{Q}_{j+1}\cap\big{\\{}u_{j}\geq\frac{\omega(r_{j})}{2}\big{\\}}\right|}{|2\mathcal{Q}_{j+1}|}\geq\frac{1}{2}.$

Moreover, we notice that for every $i=0,1,2,\ldots,j$,

(4.34) $\operatorname*{ess\,sup}_{\mathcal{Q}_{i}}|u_{j}|\leq 4\omega(r_{i}).$

Step $1$: Here, we prove that

(4.35) $\frac{|2\mathcal{Q}_{j+1}\cap\\{u_{j}\leq
2\varepsilon\omega(r_{j})\\}|}{|2\mathcal{Q}_{j+1}|}\leq\frac{\hat{C}}{\log(\frac{1}{\eta})},$

holds, for some positive constant $\hat{C}$ depending on
$N,p,s,\Lambda,C_{1},C_{2},C_{3},C_{4},h$ and the difference between
$\frac{p}{p-1}$ and $\gamma$. To prove (4.35), we apply the logarithmic
estimate [(4.14), Corollary 4.4], where a tail quantity appears.
Tail estimate: First, we prove the following tail estimate given by

(4.36)
$\mathrm{Tail}_{\infty}(u_{j};x_{0},r_{j},t_{0}-l_{j},t_{0}+l_{j})^{p-1}\leq
C\eta^{-\gamma(p-1)}\omega(r_{j})^{p-1},$

for some positive constant $C$ depending on $s,p,N,\Lambda$ and the difference
between $\frac{p}{p-1}$ and $\gamma$. It is easy to see that (4.36) follows
for $j=0$. When $j\geq 1$, we infer that

(4.37)
$\begin{split}\mathrm{Tail}_{\infty}(u_{j};x_{0},r_{j},t_{0}-l_{j},t_{0}+l_{j})^{p-1}&=r_{j}^{p}\operatorname*{ess\,sup}_{t\in(t_{0}-l_{j},t_{0}+l_{j})}\sum_{i=1}^{j}\int_{B_{i-1}\setminus
B_{i}}\frac{|u_{j}(y,t)|^{p-1}}{|y-x_{0}|^{N+ps}}\,dy\\\
&\quad+r_{j}^{p}\operatorname*{ess\,sup}_{t\in(t_{0}-l_{j},t_{0}+l_{j})}\int_{\mathbb{R}^{N}\setminus
B_{0}}\frac{|u_{j}(y,t)|^{p-1}}{|y-x_{0}|^{N+ps}}\,dy\\\ &=I+J.\end{split}$

Estimate of $I$: We observe that

(4.38)
$\begin{split}I&=r_{j}^{p}\operatorname*{ess\,sup}_{t\in(t_{0}-l_{j},t_{0}+l_{j})}\sum_{i=1}^{j}\int_{B_{i-1}\setminus
B_{i}}\frac{|u_{j}(y,t)|^{p-1}}{|y-x_{0}|^{N+ps}}\,dy\\\ &\leq
r_{j}^{p}\sum_{i=1}^{j}\big{(}\operatorname*{ess\,sup}_{\mathcal{Q}_{i-1}}|u_{j}|\big{)}^{p-1}\int_{\mathbb{R}^{N}\setminus
B_{i}}\frac{dy}{|y-x_{0}|^{N+ps}}\\\ &\leq
C\sum_{i=1}^{j}\Big{(}\frac{r_{j}}{r_{i}}\Big{)}^{p}\omega(r_{i-1})^{p-1},\end{split}$

for some positive constant $C=C(N,p,s)$, where to deduce the last inequality
above, we have used the property (4.34) and the fact that $r\in(0,1)$.
Estimate of $J$: Noting (1.3), $r\in(0,1)$, the definition of $u_{j}$ in
(4.33) and again using the property (4.34), for some constant $C=C(N,p,s)>0$,
we obtain

(4.39)
$\begin{split}J&=r_{j}^{p}\operatorname*{ess\,sup}_{t\in(t_{0}-l_{j},t_{0}+l_{j})}\int_{\mathbb{R}^{N}\setminus
B_{0}}\frac{|u_{j}(y,t)|^{p-1}}{|y-x_{0}|^{N+ps}}\,dy\\\ &\leq
r_{j}^{p}\bigg{(}\operatorname*{ess\,sup}_{t\in(t_{0}-l_{j},t_{0}+l_{j})}\int_{\mathbb{R}^{N}\setminus
B_{0}}\frac{|u(y,t)|^{p-1}}{|y-x_{0}|^{N+ps}}\,dy+r_{0}^{-p}\omega(r_{0})^{p-1}+r_{0}^{-p}\operatorname*{ess\,sup}_{\mathcal{Q}_{0}}|u|^{p-1}\bigg{)}\\\
&\leq C\Big{(}\frac{r_{j}}{r_{1}}\Big{)}^{p}\omega(r_{0})^{p-1}.\end{split}$

Combining the estimates (4.38) and (4.39) in (4.37), we obtain

(4.40) $\mathrm{Tail}_{\infty}(u_{j};x_{0},t_{0}-l_{j},t_{0}+l_{j})^{p-1}\leq
CS,$

for some constant $C=C(N,p,s)>0$, where

$S=\sum_{i=1}^{j}\Big{(}\frac{r_{j}}{r_{i}}\Big{)}^{p}\omega(r_{i-1})^{p-1}.$

Using the definition of $\omega(r_{j})$ from (4.21) and the facts
$\eta\in(0,\frac{1}{4}),\gamma\in(0,\frac{p}{p-1})$, we get

(4.41)
$\begin{split}S&=\omega(r_{j})^{p-1}\eta^{-\gamma(p-1)}\sum_{i=0}^{j-1}\eta^{i(p-\gamma(p-1))}\leq\omega(r_{j})^{p-1}\frac{\eta^{-\gamma(p-1)}}{1-\eta^{p-\gamma(p-1)}}\\\
&\leq\frac{4^{p-\gamma(p-1)}}{\big{(}p-\gamma(p-1)\big{)}\log
4}\eta^{-\gamma(p-1)}\omega(r_{j})^{p-1}.\end{split}$

Therefore, using (4.41) in (4.40), our claim (4.36) follows. For
$\varepsilon>0$ small enough to be chosen below, we set $\mu>0$ by

(4.42)
$\mu=\log\left(\frac{\frac{\omega(r_{j})}{2}+\varepsilon\omega(r_{j})}{3\varepsilon\omega(r_{j})}\right)\approx\log\Big{(}\frac{1}{\varepsilon}\Big{)}$

and for $b=\varepsilon\omega(r_{j})$, we define

(4.43)
$\Theta=\min\left\\{\left[\log\left(\frac{\frac{\omega(r_{j})}{2}+b}{u_{j}+b}\right)\right]_{+},\mu\right\\}.$

Then, using (4.33), we have

(4.44)
$\begin{split}\frac{|2\mathcal{Q}_{j+1}\cap\\{\Theta=\mu\\}|}{|2\mathcal{Q}_{j+1}|}\mu&=\frac{|2\mathcal{Q}_{j+1}\cap\\{\Theta=\mu\\}|}{|2\mathcal{Q}_{j+1}|}\frac{1}{|2\mathcal{Q}_{j+1}\cap\\{u_{j}\geq\frac{\omega(r_{j})}{2}\\}|}\iint_{2\mathcal{Q}_{j+1}\cap\\{u_{j}\geq\frac{\omega(r_{j})}{2}\\}}\mu\,dx\,dt\\\
&=\frac{|2\mathcal{Q}_{j+1}\cap\\{\Theta=\mu\\}|}{|2\mathcal{Q}_{j+1}|}\frac{1}{|2\mathcal{Q}_{j+1}\cap\\{u_{j}\geq\frac{\omega(r_{j})}{2}\\}|}\iint_{2\mathcal{Q}_{j+1}\cap\\{\Theta=0\\}}\mu\,dx\,dt\\\
&\leq\frac{|2\mathcal{Q}_{j+1}\cap\\{\Theta=\mu\\}|}{|2\mathcal{Q}_{j+1}|}\frac{2}{|2\mathcal{Q}_{j+1}|}\iint_{2\mathcal{Q}_{j+1}}(\mu-\Theta)\,dx\,dt\\\
&=2\frac{|2\mathcal{Q}_{j+1}\cap\\{\Theta=\mu\\}|}{|2\mathcal{Q}_{j+1}|}(\mu-(\Theta)_{2\mathcal{Q}_{j+1}})\\\
&=\frac{2}{|2\mathcal{Q}_{j+1}|}\iint_{2\mathcal{Q}_{j+1}\cap\\{\Theta=\mu\\}}(\mu-(\Theta)_{2\mathcal{Q}_{j+1}})\,dx\,dt\\\
&=\frac{2}{|2\mathcal{Q}_{j+1}|}\iint_{2\mathcal{Q}_{j+1}\cap\\{\Theta=\mu\\}}\bigg{(}\mu-\frac{1}{4l_{j+1}}\int_{t_{0}-2l_{j+1}}^{t_{0}+2l_{j+1}}(\Theta)_{2B_{j+1}}(\tau)\,d\tau\bigg{)}\,dx\,dt\\\
&=\frac{1}{2l_{j+1}|2\mathcal{Q}_{j+1}|}\iint_{2\mathcal{Q}_{j+1}\cap\\{\Theta=\mu\\}}\int_{t_{0}-2l_{j+1}}^{t_{0}+2l_{j+1}}\big{(}\mu-(\Theta)_{2B_{j+1}}(\tau)\big{)}\,d\tau\,dx\,dt\\\
&\leq\frac{1}{2l_{j+1}|2\mathcal{Q}_{j+1}|}\int_{t_{0}-2l_{j+1}}^{t_{0}+2l_{j+1}}\iint_{2\mathcal{Q}_{j+1}}|\Theta-(\Theta)_{2B_{j+1}}|\,d\tau\,dx\,dt=I.\end{split}$

We claim that

(4.45) $I\leq C,$

for some positive constant $C$ depending on
$N,p,s,\Lambda,C_{1},C_{1},C_{3},C_{4},h$ and the difference of
$\frac{p}{p-1}$ and $\gamma$. Note that due to the induction hypothesis
$\text{essosc}_{\mathcal{Q}_{j}}\,u\leq\omega(r_{j})$, we obtain $u_{j}\geq 0$
in $\mathcal{Q}_{j}$. Choosing $A=\frac{\omega(r_{j})}{2}$, $B=e^{\mu}$ and
$D=b=\varepsilon\omega(r_{j})$ in [(4.14), Corollary 4.4], we obtain

(4.46)
$\begin{split}&\int_{t_{0}-2l_{j+1}}^{t_{0}+2l_{j+1}}\fint_{2B_{j+1}}|\Theta(x,t)-(\Theta)_{2B_{j+1}}(t)|^{p}\,dx\,dt\\\
&\leq
C\Big{(}l_{j+1}b^{1-p}\Big{(}\frac{r_{j+1}}{r_{j}}\Big{)}^{p}\mathrm{Tail}_{\infty}(u_{j};x_{0},r_{j},t_{0}-4l_{j+1},t_{0}+4l_{j+1})^{p-1}\\\
&\quad\quad\quad+b^{2-p}r_{j+1}^{p}+l_{j+1}b^{1-p}r_{j+1}^{p}+l_{j+1}\Big{)},\\\
&=I+J+K+L,\end{split}$

for some positive constant $C=(N,p,s,\Lambda,C_{1},C_{2},C_{3},C_{4},h)$.
Estimate of $I$: From [(4.24), Lemma 4.5] we know that $4l_{j+1}\leq l_{j}$
for every $j\in\mathbb{N}\cup\\{0\\}$ and thus, using the tail estimate
(4.36), we obtain

(4.47)
$\begin{split}\mathrm{Tail}_{\infty}(u_{j};x_{0},r_{j},t_{0}-4l_{j+1},t_{0}+4l_{j+1})^{p-1}&\leq\mathrm{Tail}_{\infty}(u_{j};x_{0},r_{j},t_{0}-l_{j+1},t_{0}+l_{j+1})^{p-1}\\\
&\leq C\eta^{-\gamma(p-1)}\omega(r_{j})^{p-1},\end{split}$

for some positive constant $C$ depending on $N,p,s,\Lambda$ and the difference
between $\frac{p}{p-1}$ and $\gamma$. Since $b=\varepsilon\omega(r_{j})$,
$r_{j+1}=\eta^{j+1}r_{0}$ and $\varepsilon=\eta^{\frac{p}{p-1}-\gamma}$, using
(4.47), we obtain

(4.48)
$I=Cl_{j+1}b^{1-p}\Big{(}\frac{r_{j+1}}{r_{j}}\Big{)}^{p}\mathrm{Tail}_{\infty}(u_{j};x_{0},r_{j},t_{0}-4l_{j+1},t_{0}+4l_{j+1})^{p-1}\leq
Cl_{j+1},$

for some positive constant $C$ depending on
$N,p,s,\Lambda,C_{1},C_{2},C_{3},C_{4},h$ and the difference between
$\frac{p}{p-1}$ and $\gamma$.
Estimate of $J$: Using the first estimate in [(4.23), Lemma 4.5] we know that
$b^{2-p}=b_{j+1}$ and therefore, we have

(4.49) $J=Cb^{2-p}r_{j+1}^{p}=Cl_{j+1},$

for some positive constant $C=(N,p,s,\Lambda,C_{1},C_{2},C_{3},C_{4},h)$.
Estimate of $K$: Using the second estimate in [(4.23), Lemma 4.5] we have
$b^{1-p}\leq r_{j+1}^{-p}$. Hence, we get

(4.50) $K=Cl_{j+1}b^{1-p}r_{j+1}^{p}\leq Cl_{j+1},$

for some positive constant $C=(N,p,s,\Lambda,C_{1},C_{2},C_{3},C_{4},h)$.

Combining the estimates (4.48), (4.49) and (4.50) in (4.46), we obtain

(4.51)
$\int_{t_{0}-2l_{j+1}}^{t_{0}+2l_{j+1}}\fint_{2B_{j+1}}|\Theta(x,t)-(\Theta)_{2B_{j+1}}(t)|^{p}\,dx\,dt\leq
Cl_{j+1},$

for some positive constant $C$ depending on
$N,p,s,\Lambda,C_{1},C_{2},C_{3},C_{4},h$ and the difference between
$\frac{p}{p-1}$ and $\gamma$. Therefore, using the estimate (4.51), the claim
(4.45) follows.

Using the definition of $\Theta$ from (4.43) and employing the estimate (4.45)
in (4.44), we obtain

(4.52) $\frac{|2\mathcal{Q}_{j+1}\cap\\{u_{j}\leq
2\varepsilon\omega(r_{j})\\}|}{|2\mathcal{Q}_{j+1}|}=\frac{|2\mathcal{Q}_{j+1}\cap\\{\Theta=\mu\\}|}{|2\mathcal{Q}_{j+1}|}\leq\frac{\hat{C}}{\log(\frac{1}{\eta})},$

for some positive constant $\hat{C}$ depending on
$N,p,s,\Lambda,C_{1},C_{2},C_{3},C_{4},h$ and the difference between
$\frac{p}{p-1}$ and $\gamma$. Hence the estimate (4.35) follows.
Step $2$: Now we use the iteration argument to obtain (4.29). To this end, for
$k\in\mathbb{N}\cup\\{0\\}$ we set

(4.53)
$\begin{split}\delta_{k}&=(1+2^{-k})r_{j+1},\quad\hat{\delta}_{k}=\frac{\delta_{k}+\delta_{k+1}}{2},\\\
\gamma_{k}&=(1+2^{-k})l_{j+1},\quad\hat{\gamma}_{k}=\frac{\gamma_{k}+\gamma_{k+1}}{2},\\\
\mathcal{Q}^{k}&=B^{k}\times\Gamma^{k}=B_{\delta_{k}}(x_{0})\times(t_{0}-\gamma_{k},t_{0}+\gamma_{k}),\\\
\mathcal{\hat{Q}}^{k}&=\hat{B}^{k}\times\hat{\Gamma}^{k}=B_{\hat{\delta}_{k}}\times(t_{0}-\hat{\gamma}_{k},t_{0}+\hat{\gamma}_{k}).\end{split}$

Recall the definition of $\mathcal{Q}_{j}$ from (4.22) and $u_{j}$ from
(4.32). We observe that $\mathcal{Q}^{k}\subset\mathcal{Q}_{j}$ for all
$k\in\mathbb{N}\cup\\{0\\}$. Thus, since $u_{j}\geq 0$ in $\mathcal{Q}_{j}$,
we have $u_{j}\geq 0$ in $\mathcal{Q}^{k}$ for every
$k\in\mathbb{N}\cup\\{0\\}$. Note that

(4.54) $\Gamma^{k+1}\subset\hat{\Gamma}^{k}\subset\Gamma^{k},\quad
B^{k+1}\subset\hat{B}^{k}\subset B^{k},\quad k\in\mathbb{N}\cup\\{0\\}.$

We define
$\theta_{k}=(1+2^{-k})\varepsilon\omega(r_{j}),\,\hat{\theta}_{k}=\frac{\theta_{k}+\theta_{k+1}}{2}$
and the level sets

$A^{k}=\mathcal{Q}^{k}\cap\\{u_{j}\leq\theta_{k}\\},\quad
k\in\mathbb{N}\cup\\{0\\}.$

Then, noting $\hat{\theta}_{k}<\theta_{k}$, over the set $A^{k+1}$, we have

(4.55)
$0<\hat{\theta}_{k}-\theta_{k+1}\leq\hat{\theta}_{k}-u_{j}\leq(\hat{\theta}_{k}-u_{j})_{+}\leq(\theta_{k}-u_{j})_{+}=w_{k}.$

Let $\\{\Phi_{k}\\}_{k=0}^{\infty}\subset
C_{c}^{\infty}(\hat{\mathcal{Q}}^{k})$ be a sequence of cutoff functions such
that

(4.56) $\begin{split}0\leq\Phi_{k}\leq 1,\quad|\nabla\Phi_{k}|\leq
C\frac{2^{k}}{r_{j+1}}\text{ in }\hat{\mathcal{Q}}^{k}\text{ with
}\Phi_{k}\equiv 1\text{ in }\mathcal{Q}^{k+1},\end{split}$

for some positive constant $C=C(N,p)$. Then noting (4.55), by Hölder’s
inequality along with Lemma 2.9 (b), for some constant $C=C(N,p)>0$, we arrive
at

(4.57)
$\begin{split}&2^{-k-2}\varepsilon\omega(r_{j})|A^{k+1}|=\int_{A^{k+1}}(\hat{\theta}_{k}-\theta_{k+1})\,dx\,dt\leq\int_{A^{k+1}}{w}_{k}\,dxdt\leq\int_{\mathcal{Q}^{k+1}}{w}_{k}\,dx\,dt\\\
&\leq\int_{\hat{\mathcal{Q}}^{k}}w_{k}\Phi_{k}\,dx\,dt\leq\bigg{(}\int_{\hat{\mathcal{Q}}^{k}}(w_{k}\Phi_{k})^{p(1+\frac{p}{N})}\,dx\,dt\bigg{)}^{\frac{N}{p(N+p)}}|A^{k}|^{1-\frac{N}{p(N+p)}}\\\
&\leq
C\bigg{(}\int_{\hat{\mathcal{Q}}^{k}}|\nabla(w_{k}\Phi_{k})|^{p}\,dx\,dt\bigg{)}^{\frac{N}{p(N+p)}}\bigg{(}\operatorname*{ess\,sup}_{t\in\hat{\Gamma}^{k}}\int_{\hat{B}^{k}}|w_{k}\Phi^{k}|^{p}\,dx\bigg{)}^{\frac{1}{N+p}}|A^{k}|^{1-\frac{N}{p(N+p)}}\\\
&\leq C\bigg{(}\int_{\hat{\mathcal{Q}}^{k}}(|\nabla
w_{k}|^{p}+w_{k}^{p}|\nabla\Phi^{k}|^{p})\,dx\,dt\bigg{)}^{\frac{N}{p(N+p)}}\bigg{(}\operatorname*{ess\,sup}_{t\in\hat{\Gamma}^{k}}\int_{\hat{B}^{k}}|w_{k}|^{p}\,dx\bigg{)}^{\frac{1}{N+p}}|A^{k}|^{1-\frac{N}{p(N+p)}}\\\
&=C(I+J)^{\frac{N}{p(N+p)}}K^{\frac{1}{N+p}}|A^{k}|^{1-\frac{N}{p(N+p)}},\end{split}$

where

$I=\int_{\hat{\mathcal{Q}}^{k}}|\nabla w_{k}|^{p}\,dx\,dt,\quad
J=\int_{\hat{\mathcal{Q}}^{k}}w_{k}^{p}|\nabla\Phi^{k}|^{p}\,dx\,dt\quad\text{and}\quad
K=\operatorname*{ess\,sup}_{t\in\hat{\Gamma}^{k}}\int_{\hat{B}^{k}}|w_{k}|^{p}\,dx.$

Estimate of $J$: Notice that $w_{k}\leq\theta_{k}\leq
2\varepsilon\omega(r_{j})$. Thus, using the property of $\Phi^{k}$ from
(4.56), for some constant $C=C(N,p)>0$, we have

(4.58)
$\begin{split}J&=\int_{\hat{\mathcal{Q}}^{k}}w_{k}^{p}|\nabla\Phi^{k}|^{p}\,dx\,dt\leq
Cr_{j+1}^{-p}(\varepsilon\omega(r_{j}))^{p}2^{kp}|A^{k}|.\end{split}$

Estimate of $I$: Let $\xi_{k}=\psi_{k}\eta_{k}$, where
$\\{\psi_{k}\\}_{k=0}^{\infty}\subset C_{c}^{\infty}(B^{k})$ and
$\\{\eta_{k}\\}_{k=0}^{\infty}\in C_{c}^{\infty}(\Gamma^{k})$ such that

(4.59) $\begin{split}0&\leq\psi_{k}\leq 1,\quad|\nabla\psi_{k}|\leq
C\frac{2^{k}}{r_{j+1}}\text{ in }B^{k},\quad\psi_{k}\equiv 1\text{ in
}\hat{B}^{k},\quad\mathrm{dist}(\mathrm{supp}\,\psi_{k},\mathbb{R}^{N}\setminus
B^{k})\geq C\frac{\delta_{k}}{2^{k}},\\\ 0&\leq\eta_{k}\leq
1,\quad|\partial_{t}\eta_{k}|\leq C\frac{2^{pk}}{l_{j+1}}\text{ in
}\Gamma^{k},\quad\eta_{k}\equiv 1\text{ in }\hat{\Gamma}^{k},\end{split}$

for some constant $C=C(N,p)>0$. Since $g(x,t,u)=h(x,t)$, noting Lemma 2.10,
Remark 3.4 and setting $r=\delta_{k}$, $\tau_{1}=t_{0}-\hat{\gamma}_{k}$,
$\tau_{2}=t_{0}+\gamma_{k}$, $\tau=\gamma_{k}-\hat{\gamma}_{k}$ in Lemma 3.3,
we obtain

(4.60)
$\begin{split}&\operatorname*{ess\,sup}_{\hat{\Gamma}^{k}}\int_{\hat{B}^{k}}w_{k}^{2}\,dx+\int_{\hat{\Gamma}^{k}}\int_{\hat{B}^{k}}|\nabla
w_{k}|^{p}\,dx\,dt\leq I_{0}+I_{1}+I_{2}+I_{3}+I_{4},\end{split}$

where

$\begin{split}I_{0}&=C\int_{\Gamma^{k}}\int_{B^{k}}w_{k}^{p}|\nabla\psi_{k}|^{p}\eta_{k}^{p}\,dx\,dt,\\\
I_{1}&=C\int_{\Gamma^{k}}\int_{B^{k}}\int_{B^{k}}\big{(}\max\\{w_{k}(x,t),w_{k}(y,t)\\}\big{)}^{p}|\psi_{k}(x)-\psi_{k}(y)|^{p}\eta_{k}^{p}\,d\mu\,dt,\\\
I_{2}&=C\operatorname*{ess\,sup}_{x\in\mathrm{supp}\,\psi_{k},\,t\in\Gamma^{k}}\int_{\mathbb{R}^{N}\setminus
B^{k}}\frac{w_{k}(y,t)^{p-1}}{|x-y|^{N+ps}}\,dy\int_{\Gamma^{k}}\int_{B^{k}}w_{k}\psi_{k}^{p}\eta_{k}^{p}\,dx\,dt,\\\
I_{3}&=C\int_{\Gamma^{k}}\int_{B^{k}}w_{k}^{2}\psi_{k}^{p}|\partial_{t}\eta_{k}^{p}|\,dx\,dt\quad\text{
and }\\\
I_{4}&=\int_{\Gamma^{k}}\int_{B^{k}}h(x,t)w_{k}\psi_{k}^{p}\eta_{k}^{p}\,dx\,dt,\end{split}$

for some constant $C=C(N,p,s,\Lambda,C_{1},C_{2},C_{3},C_{4},h)>0$.
Estimate of $I_{0}$: Noting that $w_{k}\leq\theta_{k}\leq
2\varepsilon\omega(r_{j})$ and using the properties of $\psi_{k}$ and
$\eta_{k}$, we obtain

(4.61)
$I_{0}=C\int_{\Gamma^{k}}\int_{B^{k}}w_{k}^{p}|\nabla\psi_{k}|^{p}\eta_{k}^{p}\,dx\,dt\leq
C2^{kp}\big{(}\varepsilon\omega(r_{j})\big{)}^{p}r_{j+1}^{-p}|A_{k}|,$

for some positive constant $C=C(N,p,s,\Lambda,C_{1},C_{2},C_{3},C_{4},h)$.
Estimate of $I_{1}$: Using the properties of $\psi_{k}$ and $\eta_{k}$ and the
fact that $r\in(0,1)$, we obtain

(4.62)
$\begin{split}I_{1}&=C\int_{\Gamma^{k}}\int_{B^{k}}\int_{B^{k}}\big{(}\max\\{w_{k}(x,t),w_{k}(y,t)\\}\big{)}^{p}|\psi_{k}(x)-\psi_{k}(y)|^{p}\eta_{k}^{p}\,d\mu\,dt\\\
&\leq C2^{kp}r_{j+1}^{-p}\theta_{k}^{p}\operatorname*{ess\,sup}_{x\in
B^{k}}\int_{B^{k}}\frac{dy}{|x-y|^{N+ps-p}}\int_{\Gamma^{k}}\int_{B^{k}}\chi_{\\{u_{j}\leq\theta_{k}\\}}\,dx\,dt\\\
&\leq
C2^{kp}r_{j+1}^{-p}\theta_{k}^{p}\frac{\delta_{k}^{p-ps}}{p-ps}|A_{k}|\leq
C2^{kp}\big{(}\varepsilon\omega(r_{j})\big{)}^{p}r_{j+1}^{-p}|A^{k}|,\end{split}$

for some positive constant $C=C(N,p,s,\Lambda,C_{1},C_{2},C_{3},C_{4},h)$.
Estimate of $I_{2}$: Without loss of generality, we assume that $x_{0}=0$. For
$x\in\mathrm{supp}\,\psi_{k}$ and $y\in\mathbb{R}^{N}\setminus B^{k}$, we have

$\frac{1}{|x-y|}=\frac{1}{|y|}\frac{|y|}{|x-y|}\leq\frac{1}{|y|}\frac{|x|+|x-y|}{|x-y|}\leq\frac{2^{k+4}}{|y|}.$

Thus, for some constant $C=C(N,p,s)>0$, we observe that

(4.63)
$\begin{split}&\operatorname*{ess\,sup}_{x\in\mathrm{supp}\psi_{k}\,t\in\Gamma^{k}}\int_{\mathbb{R}^{N}\setminus
B^{k}}\frac{w_{k}(y,t)^{p-1}}{|x-y|^{N+ps}}\,dy\\\ &\leq
C2^{k(N+ps)}\operatorname*{ess\,sup}_{t\in\Gamma^{k}}\int_{\mathbb{R}^{N}\setminus
B^{k}}\frac{w_{k}(y,t)^{p-1}}{|y|^{N+ps}}\,dy\\\ &\leq
C2^{k(N+p)}\Bigg{(}\operatorname*{ess\,sup}_{t\in\Gamma^{k}}\int_{B_{j}\setminus
B_{j+1}}\frac{w_{k}(y,t)^{p-1}}{|y|^{N+ps}}\,dy+\operatorname*{ess\,sup}_{t\in\Gamma^{k}}\int_{\mathbb{R}^{N}\setminus
B_{j}}\frac{w_{k}(y,t)^{p-1}}{|y|^{N+ps}}\,dy\Bigg{)}\\\
&=C2^{k(N+p)}(I_{2}^{1}+I_{2}^{2}).\end{split}$

Estimate of $I_{2}^{1}$: Since $w_{k}\leq 2\varepsilon\omega(r_{j})$ in
$B_{j}\times(t_{0}-2l_{j+1},t_{0}+2l_{j+1})$, we have

(4.64)
$I_{2}^{1}=\operatorname*{ess\,sup}_{t\in\Gamma^{k}}\int_{B_{j}\setminus
B_{j+1}}\frac{w_{k}(y,t)^{p-1}}{|y|^{N+ps}}\,dy\leq
Cr_{j+1}^{-p}(\varepsilon\omega(r_{j}))^{p-1},$

for some constant $C=C(N,p,s)>0$.
Estimate of $I_{2}^{2}$: Due to the fact that
$w_{k}\leq|u_{j}|+2\varepsilon\omega(r_{j})$ in
$\mathbb{R}^{N}\times(t_{0}-2l_{j+1},t_{0}+2l_{j+1})$, we have

(4.65)
$\begin{split}I_{2}^{2}&=\operatorname*{ess\,sup}_{t\in\Gamma^{k}}\int_{\mathbb{R}^{N}\setminus
B_{j}}\frac{w_{k}(y,t)^{p-1}}{|y|^{N+ps}}\,dy\\\ &\leq
C\operatorname*{ess\,sup}_{t\in\Gamma^{k}}\int_{\mathbb{R}^{N}\setminus
B_{j}}\frac{|u_{j}(y,t)|^{p-1}}{|y|^{N+ps}}\,dy+C\operatorname*{ess\,sup}_{t\in\Gamma^{k}}\int_{\mathbb{R}^{N}\setminus
B_{j}}\frac{(\varepsilon\omega(r_{j}))^{p-1}}{|y|^{N+ps}}\,dy\\\ &\leq
Cr_{j}^{-p}\mathrm{Tail}_{\infty}(u_{j};0,r_{j},t_{0}-2l_{j+1},t_{0}+l_{j+1})^{p-1}+C(\varepsilon\omega(r_{j}))^{p-1}r_{j}^{-ps}\\\
&\leq Cr_{j+1}^{-p}(\varepsilon\omega(r_{j}))^{p-1},\end{split}$

where we have also used that $r\in(0,1)$ and the tail estimate below from
(4.36), for some positive constant $C$ depending on $N,p,s,\Lambda$ and the
difference between $\frac{p}{p-1}$ and $\gamma$. Combining (4.64) and (4.65)
in (4.63), we obtain

(4.66)
$\begin{split}I_{2}&=C\operatorname*{ess\,sup}_{x\in\mathrm{supp}\,\psi_{k},\,t\in\Gamma^{k}}\int_{\mathbb{R}^{N}\setminus
B_{k}}\frac{w_{k}(y,t)^{p-1}}{|x-y|^{N+ps}}\,dy\int_{\Gamma^{k}}\int_{B^{k}}w_{k}\psi_{k}^{p}\eta_{k}^{p}\,dx\,dt\\\
&\leq
C2^{k(N+p)}(\varepsilon\omega(r_{j}))^{p}r_{j+1}^{-p}|A^{k}|,\end{split}$

for some positive constant $C$ depending on
$N,p,s,\Lambda,C_{1},C_{2},C_{3},C_{4},h$ and the difference between
$\frac{p}{p-1}$ and $\gamma$.
Estimate of $I_{3}$: Noting the property (4.23) and the properties of
$\psi_{k}$ and $\eta_{k}$, we have

(4.67)
$I_{3}=C\int_{\Gamma^{k}}\int_{B^{k}}w_{k}^{2}\psi_{k}^{p}|\partial_{t}\eta_{k}^{p}|\,dx\,dt\leq
C\frac{2^{pk}}{l_{j+1}}\theta_{k}^{2}|A^{k}|\leq
C2^{pk}(\varepsilon\omega(r_{j}))^{p}r_{j+1}^{-p}|A^{k}|,$

for some positive constant $C=C(N,p,s,\Lambda,C_{1},C_{2},C_{3},C_{4},h)$.
Estimate of $I_{4}$: First we observe that

$\omega(r_{j})=\Big{(}\frac{r_{j}}{r_{0}}\Big{)}^{\gamma}\omega(r_{0})=\eta^{j\gamma}\omega(r_{0})\geq\frac{1}{\varepsilon}\Big{(}\frac{r_{j+1}}{r_{0}}\Big{)}^{\frac{p}{p-1}}\omega(r_{0}),$

which gives
$r_{j+1}^{-p}(\varepsilon\omega(r_{j}))^{p-1}\geq\omega(r_{0})^{p-1}=L^{p-1}\geq
1$. Employing this fact, we obtain

(4.68)
$I_{4}=C\int_{\Gamma^{k}}\int_{B^{k}}h(x,t)w_{k}\psi_{k}^{p}\eta_{k}^{p}\,dx\,dt\leq
C\varepsilon\omega(r_{j})|A^{k}|\leq
Cr_{j+1}^{-p}(\varepsilon\omega(r_{j}))^{p}|A^{k}|,$

for some positive constant $C=C(N,p,s,\Lambda,C_{1},C_{2},C_{3},C_{4},h)$.
Thus, combining (4.61), (4.62), (4.66), (4.67) and (4.68) in (4.60), we get

(4.69) $I=\int_{\hat{\mathcal{Q}}^{k}}|\nabla w_{k}|^{p}\,dx\,dt\leq
C2^{kp}r_{j+1}^{-p}(\varepsilon\omega(r_{j}))^{p}|A^{k}|,$

and

(4.70)
$\hat{K}=\operatorname*{ess\,sup}_{t\in\hat{\Gamma}^{k}}\int_{\hat{B}^{k}}|w_{k}|^{2}\,dx\leq
Cr_{j+1}^{-p}(\varepsilon\omega(r_{j}))^{p}|A^{k}|,$

for some positive constant $C$ depending on
$N,p,s,\Lambda,C_{1},C_{2},C_{3},C_{4},h$ and the difference between
$\frac{p}{p-1}$ and $\gamma$.
Estimate of $K$: Since $p>2$ and $\theta_{k}\leq 2\varepsilon\omega(r_{j})$,
using (4.70) and [(4.23), Lemma 4.5], we observe that

(4.71)
$\begin{split}K&=\operatorname*{ess\,sup}_{t\in\hat{\Gamma}^{k}}\int_{\hat{B}^{k}}|w_{k}|^{p}\,dx=\operatorname*{ess\,sup}_{t\in\hat{\Gamma}^{k}}\int_{\hat{B}^{k}}|w_{k}|^{2}|w_{k}|^{p-2}\,dx\\\
&\leq\theta_{k}^{p-2}\operatorname*{ess\,sup}_{t\in\hat{\Gamma}^{k}}\int_{\hat{B}^{k}}|w_{k}|^{2}\,dx\leq\frac{C}{b_{j+1}}r_{j+1}^{-p}(\varepsilon\omega(r_{j}))^{p}|A^{k}|,\end{split}$

for some positive constant $C$ depending on
$N,p,s,\Lambda,C_{1},C_{2},C_{3},C_{4},h$ and the difference between
$\frac{p}{p-1}$ and $\gamma$.

Inserting (4.58), (4.69) and (4.71) in (4.57), we obtain

(4.72) $2^{-k-2}\varepsilon\omega(r_{j})|A^{k+1}|\leq
C2^{\frac{kN}{p}}\frac{\varepsilon\omega(r_{j})}{r_{j+1}}\Big{(}\frac{1}{b_{j+1}}\Big{)}^{\frac{1}{N+p}}|A^{k}|^{1+\frac{1}{N+p}}$

Dividing both sides of (4.72) by $|\mathcal{Q}^{k+1}|$, we have

$\frac{|A^{k+1}|}{|\mathcal{Q}^{k+1}|}\leq
C\frac{2^{(\frac{N}{p}+1)k}}{r_{j+1}}\Big{(}\frac{1}{b_{j+1}}\Big{)}^{\frac{1}{N+p}}\frac{|A^{k}|}{|\mathcal{Q}^{k}|}^{1+\frac{1}{N+p}}\frac{|\mathcal{Q}^{k}|^{1+\frac{1}{N+p}}}{|\mathcal{Q}^{k+1}|},$

for some positive constant $C$ depending on
$N,p,s,\Lambda,C_{1},C_{2},C_{3},C_{4},h$ and the difference between
$\frac{p}{p-1}$ and $\gamma$. Denoting by
$Y_{k}=\frac{|A^{k}|}{|\mathcal{Q}^{k}|}$ for $k=0,1,2,\ldots$ we obtain from
the above estimate that

$Y_{k+1}\leq C_{5}2^{(\frac{N}{p}+1)k}Y_{k}^{1+\frac{1}{N+p}},$

for some positive constant $C_{5}$ depending on
$N,p,s,\Lambda,C_{1},C_{2},C_{3},C_{4},h$ and the difference between
$\frac{p}{p-1}$ and $\gamma$. By (4.35), we have

$Y_{0}\leq\frac{\hat{C}}{\log(\frac{1}{\eta})},$

for some positive constant $C_{5}$ depending on
$N,p,s,\Lambda,C_{1},C_{2},C_{3},C_{4},h$ and the difference between
$\frac{p}{p-1}$ and $\gamma$. Let

$K=\frac{C_{5}}{2},\quad
b=2^{(\frac{N}{p}+1)},\quad\delta_{2}=\delta_{1}=\delta=\frac{1}{N+p},$

in Lemma 2.11 and denote by
$\eta_{1}=(2K)^{-\frac{1}{\delta}}b^{-\frac{1}{\delta^{2}}}$. We choose
$\eta=\frac{1}{2}\min\big{\\{}\frac{1}{4},\eta^{*},e^{-\frac{\hat{C}}{\eta_{1}}}\big{\\}}$
then $Y_{0}\leq\eta_{1}$ holds. Thus by Lemma 2.11 we deduce that
$\lim_{k\to\infty}Y_{k}=0$ and therefore,

$u_{j}\geq\varepsilon\omega(r_{j})\text{ in }\mathcal{Q}_{j+1}.$

Now using the definition of $u_{j}$ from (4.32), we obtain

$\text{osc}_{\mathcal{Q}_{j+1}}\,u\leq(1-\varepsilon)\omega(r_{j})=(1-\varepsilon)\eta^{-\gamma}\omega(r_{j+1})\leq\omega(r_{j+1}),$

where we have chosen $\gamma\in(0,\frac{p}{p-1})$ small enough such that

(4.73) $\eta^{\gamma}\geq 1-\varepsilon=1-\eta^{\frac{p}{p-1}-\gamma}.$

This proves the induction estimate (4.29) for $i=j+1$. Thus the result
follows. ∎

##  5\. Semicontinuity and pointwise behavior of subsolutions and
supersolutions

In this section, we obtain lower semicontinuity as well as upper
semicontinuity results for weak supersolution and subsolution of (1.1)
respectively. We also discuss the pointwise behavior of such semicontinuous
functions. First, we define the lower and upper semicontinuous representatives
of a measurable function.

Let $u$ be a measurable function which is locally essentially bounded below in
$\Omega_{T}$. Suppose that $(x,t)\in\Omega_{T}$ and $r\in(0,1)$, $\theta>0$
such that $\mathcal{Q}_{r,\theta}(x,t)=B_{r}(x)\times(t-\theta r^{p},t+\theta
r^{p})\Subset\Omega_{T}$. The lower semicontinuous regularization $u_{*}$ of
$u$ is defined as

$u_{*}(x,t)=\operatorname*{ess\,lim\,inf}_{(y,\hat{t})\to(x,t)}\,u(y,\hat{t})=\lim_{r\to
0}\operatorname*{ess\,inf}_{\mathcal{Q}_{r,\theta}(x,t)}\,u\text{ for
}(x,t)\in\Omega_{T}.$

Analogously, for a locally essentially bounded above measurable function $u$
in $\Omega_{T}$, we define an upper semicontinuous regularization $u^{*}$ of
$u$ by

$u^{*}(x,t)=\operatorname*{ess\,lim\,sup}_{(y,\hat{t})\to(x,t)}\,u(y,\hat{t})=\lim_{r\to
0}\operatorname*{ess\,sup}_{\mathcal{Q}_{r,\theta}(x,t)}\,u\text{ for
}(x,t)\in\Omega_{T}.$

It is easy to see that $u_{*}$ is lower semicontinuous and $u^{*}$ is upper
semicontinuous in $\Omega_{T}$.

Let $u\in L^{1}_{\mathrm{loc}}(\Omega_{T})$ and define the set of Lebesgue
points of $u$ by

$\mathcal{F}=\bigg{\\{}(x,t)\in\Omega_{T}:|u(x,t)|<\infty,\,\lim_{r\to
0}\fint_{\mathcal{Q}_{r,\theta}(x,t)}|u(x,t)-u(y,\hat{t})|\,dy\,d\hat{t}=0\bigg{\\}}.$

From the Lebesgue differentiation theorem we have
$|\mathcal{F}|=|\Omega_{T}|$.

The following lower semicontinuity result for weak supersolutions of (1.1)
follows by combining Lemma 5.6 and Theorem 5.5.

######  Theorem 5.1.

(Lower semicontinuity) Let $0<s<1,\,1<p<\infty,\,g\equiv 0$ in
$\Omega_{T}\times\mathbb{R}$. Suppose that $u\in
L^{p}_{\mathrm{loc}}\big{(}0,T;W^{1,p}_{\mathrm{loc}}(\Omega)\big{)}\cap
C_{\mathrm{loc}}\big{(}0,T;L^{2}_{\mathrm{loc}}(\Omega)\big{)}\cap
L^{\infty}_{\mathrm{loc}}\big{(}0,T;L^{p-1}_{ps}(\mathbb{R}^{N})\big{)}$ is a
weak supersolution of (1.1) in $\Omega_{T}$ such that $u$ is essentially
bounded below in $\mathbb{R}^{N}\times(0,T)$. Then $u(x,t)=u_{*}(x,t)$ at
every Lebesgue point $(x,t)\in\Omega_{T}$. In particular, $u_{*}$ is a lower
semicontinuous representative of $u$ in $\Omega_{T}$.

As a consequence of Remark 2.8 and Theorem 5.1 we obtain the following result
for weak subsolutions.

######  Corollary 5.2.

(Upper semicontinuity) Let $0<s<1,\,1<p<\infty,\,g\equiv 0$ in
$\Omega_{T}\times\mathbb{R}$. Suppose that $u\in
L^{p}_{\mathrm{loc}}\big{(}0,T;W^{1,p}_{\mathrm{loc}}(\Omega)\big{)}\cap
C_{\mathrm{loc}}\big{(}0,T;L^{2}_{\mathrm{loc}}(\Omega)\big{)}\cap
L^{\infty}_{\mathrm{loc}}\big{(}0,T;L^{p-1}_{ps}(\mathbb{R}^{N})\big{)}$ is a
weak subsolution of (1.1) in $\Omega_{T}$ such that $u$ is essentially bounded
above in $\mathbb{R}^{N}\times(0,T)$. Then $u(x,t)=u^{*}(x,t)$ at every
Lebesgue point $(x,t)\in\Omega_{T}$. In particular, $u^{*}$ is an upper
semicontinuous representative of $u$ in $\Omega_{T}$.

Our final result asserts that the lower semicontinuous representative $u_{*}$,
given by Theorem 5.1, is determined by previous times. The proof follows by a
combination of Lemma 5.7 below and the proof of [35, Theorem 3.1].

######  Theorem 5.3.

(Pointwise behavior) Let $0<s<1,\,1<p<\infty,\,g\equiv 0$ in
$\Omega_{T}\times\mathbb{R}$. Suppose that $u\in
L^{p}_{\mathrm{loc}}\big{(}0,T;W^{1,p}_{\mathrm{loc}}(\Omega)\big{)}\cap
C_{\mathrm{loc}}\big{(}0,T;L^{2}_{\mathrm{loc}}(\Omega)\big{)}\cap
L^{\infty}_{\mathrm{loc}}\big{(}0,T;L^{p-1}_{ps}(\mathbb{R}^{N})\big{)}$ is a
weak supersolution of (1.1) in $\Omega_{T}$ such that $u$ is essentially
bounded below in $\mathbb{R}^{N}\times(0,T)$. Assume that $u_{*}$ is the lower
semicontinuous representative of $u$ given by Theorem 5.1. Then for every
$(x,t)\in\Omega_{T}$, we have

$u_{*}(x,t)=\inf_{\theta>0}\lim_{r\to
0}\operatorname*{ess\,inf}_{\mathcal{Q}^{\prime}_{r,\theta}(x,t)}\,u,$

where $\mathcal{Q}^{\prime}_{r,\theta}(x,t)=B_{r}(x)\times(t-2\theta
r^{p},t-\theta r^{p}),\,r\in(0,1)$. In particular, we have

$u_{*}(x,t)={\operatorname*{ess\,lim\,inf}_{(y,\hat{t})\to(x,t),\,\hat{t}<t}}\,u(y,\hat{t})$

at every point $(x,t)\in\Omega_{T}$.

###  5.1. Preliminaries

The following measure theoretic property from [35] will be useful for us.

######  Definition 5.4.

Let $u$ be a measurable function which is locally essentially bounded below in
$\Omega_{T}$. Assume that $(x_{0},t_{0})\in\Omega_{T}$ and $r\in(0,1)$,
$\theta>0$ such that
$\mathcal{Q}_{r,\theta}(x_{0},t_{0})=B_{r}(x_{0})\times(t_{0}-\theta
r^{p},t_{0}+\theta r^{p})\Subset\Omega_{T}$. Suppose

(5.1) $a,c\in(0,1),\quad
M>0,\quad\mu^{-}\leq\operatorname*{ess\,inf}_{\mathcal{Q}_{r,\theta}(x_{0},t_{0})}\,u.$

We say that $u$ satisfies the property $(\mathcal{D})$ if there exists a
constant $\nu\in(0,1)$, which depends only on $a,M,\mu^{-}$ and other data,
but independent of $r$, such that if

(5.2)
$|\\{u\leq\mu^{-}+M\\}\cap\mathcal{Q}_{r,\theta}(x_{0},t_{0})|\leq\nu|\mathcal{Q}_{r,\theta}(x_{0},t_{0})|,$

then

(5.3) $u\geq\mu^{-}+aM\text{ a.e. in }\mathcal{Q}_{cr,\theta}(x_{0},t_{0}).$

Next, we state a result from Liao in [35, Theorem 2.1] that shows that any
such function with the property $(\mathcal{D})$ has a lower semicontinuous
representative.

######  Theorem 5.5.

Let $u$ be a measurable function in $\Omega_{T}$ which is locally integrable
and locally essentially bounded below in $\Omega_{T}$. Assume that $u$
satisfies the property $(\mathcal{D})$. Then $u(x,t)=u_{*}(x,t)$ for every
$x\in\mathcal{F}$. In particular, $u_{*}$ is a lower semicontinuous
representative of $u$ in $\Omega_{T}$.

Assume that $(x_{0},t_{0})\in\Omega_{T}$ and $r\in(0,1)$, $\theta>0$ such that
$\mathcal{Q}_{r,\theta}(x_{0},t_{0})=B_{r}(x_{0})\times(t_{0}-\theta
r^{p},t_{0}+\theta r^{p})\Subset\Omega_{T}$. Let $a,\mu^{-}$ and $M$ be
defined as in (5.1). Then the following De Giorgi type lemma shows that a weak
supersolution satisfies the property $(\mathcal{D})$ in Definition 5.4.

######  Lemma 5.6.

Let $0<s<1,\,1<p<\infty$ and $g\equiv 0$ in $\Omega_{T}\times\mathbb{R}$.
Suppose that $u\in
L^{p}_{\mathrm{loc}}\big{(}0,T;W^{1,p}_{\mathrm{loc}}(\Omega)\big{)}\cap
C_{\mathrm{loc}}\big{(}0,T;L^{2}_{\mathrm{loc}}(\Omega)\big{)}\cap
L^{\infty}_{\mathrm{loc}}\big{(}0,T;L^{p-1}_{ps}(\mathbb{R}^{N})\big{)}$ is a
weak supersolution of (1.1) in $\Omega_{T}$ such that $u$ is essentially
bounded below in $\mathbb{R}^{N}\times(0,T)$ and let

(5.4)
$\lambda^{-}{}\leq\operatorname*{ess\,inf}_{\mathbb{R}^{N}\times(0,T)}\,u.$

Then there exists a constant
$\nu=\nu(a,M,\mu^{-},\lambda^{-},\theta,N,p,s,\Lambda,C_{1},C_{2},C_{3},C_{4})\in(0,1)$,
such that if

$|\\{u\leq\mu^{-}+M\\}\cap\mathcal{Q}_{r,\theta}(x_{0},t_{0})|\leq\nu|\mathcal{Q}_{r,\theta}(x_{0},t_{0})|,$

then

$u\geq\mu^{-}+aM\text{ a.e. in
}\mathcal{Q}_{\frac{3r}{4},\theta}(x_{0},t_{0}).$

###### Proof.

For $j\in\mathbb{N}\cup\\{0\\}$ let

(5.5)
$\begin{split}&k_{j}=\mu^{-}+aM+\frac{(1-a)M}{2^{j}},\quad\hat{k}_{j}=\frac{k_{j}+k_{j+1}}{2},\\\
&r_{j}=\frac{3r}{4}+\frac{r}{2^{j+2}},\quad\hat{r}_{j}=\frac{r_{j}+r_{j+1}}{2},\\\
&B_{j}=B_{r_{j}}(x_{0}),\quad\hat{B}_{j}=B_{\hat{r}_{j}}(x_{0}),\\\
&\Gamma_{j}=(t_{0}-\theta r_{j}^{p},t_{0}+\theta
r_{j}^{p}),\quad\hat{\Gamma}_{j}=(t_{0}-\theta\hat{r}_{j}^{p},t_{0}+\theta\hat{r}_{j}^{p})\\\
&\mathcal{Q}_{j}=B_{j}\times\Gamma_{j},\quad\mathcal{\hat{Q}}_{j}=\hat{B}_{j}\times\hat{\Gamma}_{j},\quad
A_{j}=\mathcal{Q}_{j}\cap\\{u\leq k_{j}\\}.\end{split}$

Notice that $r_{j+1}<\hat{r}_{j}<r_{j}$, $k_{j+1}<\hat{k}_{j}<k_{j}$ for all
$j\in\mathbb{N}\cup\\{0\\}$ and therefore, we have
$B_{j+1}\subset\hat{B}_{j}\subset B_{j}$ and
$\Gamma_{j+1}\subset\hat{\Gamma}_{j}\subset\Gamma_{j}$. Let
$\\{\Phi_{j}\\}_{j=0}^{\infty}\subset C_{c}^{\infty}(\hat{\mathcal{Q}}_{j})$
be such that

(5.6) $0\leq\Phi_{j}\leq 1,\quad|\nabla\hat{\mathcal{Q}}_{j}|\leq
C\frac{2^{j}}{r}\text{ in
}\hat{\mathcal{Q}}_{j}\quad\text{and}\quad\Phi_{j}\equiv 1\text{ in
}\mathcal{Q}_{j+1},$

for some constant $C=C(N,p)>0$. Notice that, over the set
$A_{j+1}=\mathcal{Q}_{j+1}\cap\\{u\leq k_{j+1}\\}$, we have
$\hat{k}_{j}-k_{j+1}\leq\hat{k}_{j}-u$. By integrating over the set $A_{j+1}$
and using Lemma 2.9 (b), we obtain

(5.7)
$\begin{split}(1-a)\frac{M}{2^{j+3}}|A_{j+1}|&\leq\int_{A_{j+1}}(\hat{k}_{j}-k_{j+1})\,dx\,dt\leq\int_{\mathcal{Q}_{j+1}}(\hat{k}_{j}-u)\,dx\,dt\leq\int_{\mathcal{\hat{Q}}_{j}}(u-\hat{k}_{j})_{-}\Phi\,dx\,dt\\\
&\leq\bigg{(}\int_{\hat{\mathcal{Q}}_{j}}\big{(}(u-\hat{k}_{j})_{-}\Phi_{j}\big{)}^{p(1+\frac{2}{N})}\,dx\,dt\bigg{)}^{\frac{N}{p(N+2)}}|A_{j}|^{1-\frac{N}{p(N+2)}}\\\
&\leq
C(I+J)^{\frac{N}{p(N+2)}}\hat{K}^{\frac{1}{N+2}}|A_{j}|^{1-\frac{N}{p(N+2)}},\end{split}$

for some constant $C=C(N,p)>0$, where

$I=\int_{\hat{\mathcal{Q}}_{j}}|\nabla(u-\hat{k}_{j})_{-}|^{p}\,dx\,dt,\quad
J=\int_{\hat{\mathcal{Q}}_{j}}(u-\hat{k}_{j})_{-}^{p}|\nabla\Phi_{j}|^{p}\,dx\,dt\quad\text{and}\quad\hat{K}=\operatorname*{ess\,sup}_{t\in\hat{\Gamma}_{j}}\int_{\hat{B}_{j}}(u-\hat{k}_{j})_{-}^{2}\,dx.$

Note that, due to the assumption (5.4), we know
$\lambda^{-}\leq\operatorname*{ess\,inf}_{\mathbb{R}^{N}\times(0,T)}\,u$,
which gives

(5.8)
$(u-\hat{k}_{j})_{-}\leq(\mu^{-}+M-\lambda^{-})_{+}:=L\quad\text{in}\quad\mathbb{R}^{N}\times(0,T).$

Estimate of $J$: Using the properties of $\Phi_{j}$ and (5.8), we get

(5.9)
$\begin{split}J&=\int_{\hat{\mathcal{Q}}_{j}}(u-\hat{k}_{j})_{-}^{p}|\nabla\Phi_{j}|^{p}\,dx\,dt\leq
C\frac{2^{jp}}{r^{p}}L^{p}|A_{j}|,\end{split}$

for some constant $C=C(N,p)>0$.
Estimate of $I$ and $\hat{K}$: Let $\xi_{j}=\psi_{j}\eta_{j}$, where
$\\{\psi_{j}\\}_{j=0}^{\infty}\subset C_{c}^{\infty}(B_{j})$ and
$\\{\eta_{j}\\}_{j=0}^{\infty}\subset C_{c}^{\infty}(\Gamma_{j})$ be such that

(5.10) $\begin{split}&0\leq\psi_{j}\leq 1,\quad|\nabla\psi_{j}|\leq
C\frac{2^{j}}{r}\text{ in }B_{j},\quad\psi_{j}\equiv 1\text{ in
}\hat{B}_{j},\quad\mathrm{dist}(\mathrm{supp}\,\psi_{j},\mathbb{R}^{N}\setminus
B_{j})\geq 2^{-j-1}r,\\\ &0\leq\eta_{j}\leq 1,\quad|\partial_{t}\eta_{j}|\leq
C\frac{2^{pj}}{\theta r^{p}}\text{ in }\Gamma_{j},\quad\eta_{j}\equiv 1\text{
in }\hat{\Gamma}_{j},\end{split}$

for some constant $C=C(N,p)>0$. Note that $g\equiv 0$ in
$\Omega_{T}\times\mathbb{R}$ and for $\hat{k}_{j}<k_{j}$, we have
$(u-k_{j})_{-}\geq(u-\hat{k}_{j})_{-}$. Therefore, noting Lemma 2.10 and
Remark 3.5, we set $r=r_{j}$, $\tau_{1}=t_{0}-\theta\hat{r}_{j}^{p}$,
$\tau_{2}=t_{0}+\theta r_{j}^{p}$, $\tau=\theta
r_{j}^{p}-\theta\hat{r}_{j}^{p}$ in Lemma 3.3 to obtain

(5.11)
$\begin{split}&I+\hat{K}=\int_{\hat{\Gamma}_{j}}\int_{\hat{B}_{j}}|\nabla(u-\hat{k}_{j})_{-}|^{p}\,dxdt+\operatorname*{ess\,sup}_{\hat{\Gamma}_{j}}\int_{\hat{B}_{j}}(u-\hat{k}_{j})_{-}^{2}\,dx\leq
I_{1}+I_{2}+I_{3}+I_{4},\end{split}$

where

$\begin{split}I_{1}&=C\int_{\Gamma_{j}}\int_{B_{j}}\int_{B_{j}}{\max\\{(u-k_{j})_{-}(x,t),(u-k_{j})_{-}(y,t)\\}^{p}|\xi_{j}(x,t)-\xi_{j}(y,t)|^{p}}\,d\mu\,dt,\\\
I_{2}&=C\int_{\Gamma_{j}}\int_{B_{j}}(u-k_{j})_{-}^{p}|\nabla\xi_{j}|^{p}\,dx\,dt,\\\
I_{3}&=C\operatorname*{ess\,sup}_{x\in\mathrm{supp}\,\psi_{j},\,t\in\Gamma_{j}}\int_{{\mathbb{R}^{N}\setminus
B_{j}}}{\frac{(u-k_{j})_{-}(y,t)^{p-1}}{|x-y|^{N+ps}}}\,dy\int_{\Gamma_{j}}\int_{B_{j}}(u-k_{j})_{-}\xi_{j}^{p}\,dx\,dt\quad\text{and}\\\
I_{4}&=C\int_{\Gamma_{j}}\int_{B_{j}}(u-k_{j})_{-}^{2}|\partial_{t}\xi_{j}^{p}|\,dx\,dt,\end{split}$

for some constant $C=C(N,p,s,\Lambda,C_{1},C_{2},C_{3},C_{4})>0$.
Estimate of $I_{1}$: Using (5.8), the properties of $\xi_{j}$ and $r\in(0,1)$,
we have

(5.12)
$\begin{split}I_{1}&=C\int_{\Gamma_{j}}\int_{B_{j}}\int_{B_{j}}{\max\\{(u-k_{j})_{-}(x,t),(u-k_{j})_{-}(y,t)\\}^{p}|\xi_{j}(x,t)-\xi_{j}(y,t)|^{p}}\,d\mu\,dt\\\
&\leq C\frac{2^{jp}}{r^{ps}}L^{p}|A_{j}|\leq
C\frac{2^{jp}}{r^{p}}L^{p}|A_{j}|,\end{split}$

$C=C(N,p,s,\Lambda,C_{1},C_{2},C_{3},C_{4})>0$.
Estimate of $I_{2}$: Again, using (5.8) and the properties of $\xi_{j}$, we
deduce that

(5.13)
$\begin{split}I_{2}&=C\int_{\Gamma_{j}}\int_{B_{j}}(u-k_{j})_{-}^{p}|\nabla\xi_{j}|^{p}\,dx\,dt\leq
C\frac{2^{jp}}{r^{p}}L^{p}|A_{j}|,\end{split}$

$C=C(N,p,s,\Lambda,C_{1},C_{2},C_{3},C_{4})>0$.
Estimate of $I_{3}$: Without loss of generality, we assume that $x_{0}=0$.
Then for every $x\in\mathrm{supp}\,\psi_{j}$ and every
$y\in\mathbb{R}^{N}\setminus B_{j}$, we observe that

(5.14)
$\frac{1}{|x-y|}=\frac{1}{|y|}\frac{|x-(x-y)|}{|x-y|}\leq\frac{1}{|y|}(1+2^{j+3})\leq\frac{2^{j+4}}{|y|}.$

Using (5.8), $r\in(0,1)$ and $0\leq\xi_{j}\leq 1$, we have

(5.15)
$\begin{split}I_{3}&=C\operatorname*{ess\,sup}_{(x,t)\in\mathrm{supp}\,\psi_{j},\,t\in\Gamma_{j}}\int_{{\mathbb{R}^{N}\setminus
B_{j}}}{\frac{(u-k_{j})_{-}(y,t)^{p-1}}{|x-y|^{N+ps}}}\,dy\int_{\Gamma_{j}}\int_{B_{j}}(u-k_{j})_{-}\xi_{j}^{p}\,dx\,dt\\\
&\leq C\frac{2^{j(N+p)}}{r^{p}}L^{p}|A_{j}|,\end{split}$

for some constant $C=C(N,p,s,\Lambda,C_{1},C_{2},C_{3},C_{4})>0$.
Estimate of $I_{4}$: Using (5.8) along with the properties of $\xi_{j}$, we
have

(5.16)
$\begin{split}I_{4}&=C\int_{\Gamma_{j}}\int_{B_{j}}(u-k_{j})_{-}^{2}|\partial_{t}\xi_{j}^{p}|\,dx\,dt\\\
&\leq C\frac{2^{jp}}{\theta
r^{p}}\int_{\Gamma_{j}}\int_{B_{j}}(u-k_{j})_{-}^{2}\,dx\,dt\leq
C\frac{2^{jp}}{\theta r^{p}}L^{2}|A_{j}|,\end{split}$

$C=C(N,p,s,\Lambda,C_{1},C_{2},C_{3},C_{4})>0$. Inserting (5.12), (5.13),
(5.15) and (5.16) in (5.11), we arrive at

(5.17) $\begin{split}I+\hat{K}\leq
C\frac{2^{j(N+p)}}{r^{p}}L^{p}\Big{(}1+\frac{L^{2-p}}{\theta}\Big{)}|A_{j}|,\end{split}$

for some constant $C=C(N,p,s,\Lambda,C_{1},C_{2},C_{3},C_{4})>0$. Using (5.9)
and (5.17) in (5.7), we obtain

(5.18) $\begin{split}|A_{j+1}|\leq
C\frac{2^{j((N+p)^{2}+1)}L^{\frac{N+p}{N+2}}}{(1-a)M}\Big{(}1+\frac{L^{2-p}}{\theta}\Big{)}^{\frac{N+p}{p(N+2)}}\frac{|A_{j}|^{1+\frac{1}{N+2}}}{r^{\frac{N+p}{N+2}}},\end{split}$

for some constant $C=C(N,p,s,\Lambda,C_{1},C_{2},C_{3},C_{4})>0$. Dividing
both sides of (5.18) by $|\mathcal{Q}_{j+1}|$ and denoting by
$Y_{j}=\frac{|A_{j}|}{|\mathcal{Q}_{j}|}$, we get

$Y_{j+1}\leq 2Kb^{j}Y_{j}^{1+\delta},$

where

$K=\frac{C(\theta
L^{N+p})^{\frac{1}{N+2}}}{2(1-a)M}\Big{(}1+\frac{L^{2-p}}{\theta}\Big{)}^{\frac{N+p}{p(N+2)}},\quad\delta=\frac{1}{N+2}\quad\text{and}\quad
b=2^{(N+p)^{2}+1}>1.$

We define $\nu=(2K)^{-\frac{1}{\delta}}b^{-\frac{1}{\delta^{2}}}$, which
depends on
$a,M,\mu^{-},\lambda^{-},\theta,N,p,s,\Lambda,C_{1},C_{2},C_{3},C_{4}$ such
that if $Y_{0}\leq\nu$, then by Lemma 2.11, we have
$\lim_{j\to\infty}Y_{j}=0$. This completes the proof. ∎

Recalling that $a,\mu^{-}$ and $M$ are defined as in (5.1), we prove our
second De Giorgi Lemma.

######  Lemma 5.7.

Let $0<s<1,\,1<p<\infty$ and $g\equiv 0$ in $\Omega_{T}\times\mathbb{R}$.
Suppose that $u\in
L^{p}_{\mathrm{loc}}\big{(}0,T;W^{1,p}_{\mathrm{loc}}(\Omega)\big{)}\cap
C_{\mathrm{loc}}\big{(}0,T;L^{2}_{\mathrm{loc}}(\Omega)\big{)}\cap
L^{\infty}_{\mathrm{loc}}\big{(}0,T;L^{p-1}_{ps}(\mathbb{R}^{N})\big{)}$ is a
weak supersolution of (1.1) in $\Omega_{T}$ such that $u$ is essentially
bounded below in $\mathbb{R}^{N}\times(0,T)$ and let

(5.19)
$\lambda^{-}\leq\operatorname*{ess\,inf}_{\mathbb{R}^{N}\times(0,T)}\,u.$

Then there exists a constant
$\theta=\theta(a,M,\mu^{-},\lambda^{-},N,p,s,\Lambda,C_{1},C_{2},C_{3},C_{4})>0$
such that if $t_{0}$ is a Lebesgue point of $u$ and

$u(\cdot,t_{0})\geq\mu^{-}+M\text{ a.e. in }B_{r}(x_{0}),$

then

$u\geq\mu^{-}+aM\text{ a.e. in
}\mathcal{Q}^{+}_{\frac{3r}{4},\theta}(x_{0},t_{0})=B_{\frac{3r}{4}}(x_{0})\times\big{(}t_{0},t_{0}+\theta\big{(}\tfrac{3r}{4}\big{)}^{p}\big{)}.$

###### Proof.

For $j\in\mathbb{N}\cup\\{0\\}$, we define
$k_{j},\hat{k}_{j},r_{j},\hat{r}_{j},B_{j},\hat{B}_{j}$ as in (5.5) and for
$\theta>0$, let us set

(5.20) $\begin{split}\Gamma_{j}=(t_{0},t_{0}+\theta
r_{j}^{p}),\quad\mathcal{Q}_{j}=B_{j}\times\Gamma_{j},\quad\mathcal{\hat{Q}}_{j}=\hat{B}_{j}\times\Gamma_{j},\quad
A_{j}=\mathcal{Q}_{j}\cap\\{u\leq k_{j}\\}.\end{split}$

Therefore, for all $j\in\mathbb{N}\cup\\{0\\}$ we have
$B_{j+1}\subset\hat{B}_{j}\subset B_{j},\,\Gamma_{j+1}\subset\Gamma_{j}.$ Let
$\\{\Phi_{j}\\}_{j=0}^{\infty}\subset C_{c}^{\infty}(\hat{\mathcal{Q}}_{j})$
be as defined in (5.6). Notice that, over the set
$A_{j+1}=\mathcal{Q}_{j+1}\cap\\{u\leq k_{j+1}\\}$, we have
$\hat{k}_{j}-k_{j+1}\leq\hat{k}_{j}-u$. Hence integrating over the set
$A_{j+1}$ as in the proof of (5.7), we obtain

(5.21) $\begin{split}(1-a)\frac{M}{2^{j+3}}|A_{j+1}|&\leq
C(I+J)^{\frac{N}{p(N+2)}}\hat{K}^{\frac{1}{N+2}}|A_{j}|^{1-\frac{N}{p(N+2)}},\end{split}$

for some constant $C=C(N,p,s)>0$, where

$I=\int_{\hat{\mathcal{Q}}_{j}}|\nabla(u-\hat{k}_{j})_{-}|^{p}\,dx\,dt,\quad