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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2018 (2018), No. 44, pp. 1--20.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2018 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2018/44\hfil Fractional $p$-Laplacian parabolic problems]
{Existence and global behavior of solutions to
fractional $p$-Laplacian parabolic problems}

\author[J. Giacomoni, S. Tiwari \hfil EJDE-2018/44\hfilneg]
{Jacques Giacomoni, Sweta Tiwari}

\dedicatory{Communicated by Vicentiu Radulescu}

\address{Jacques Giacomoni \newline
Universit\'e de Pau et des Pays de l'Adour, CNRS,
LMAP (UMR 5142), Bat. Ipra, avenue de l'Universit\'e, Pau, France}
\email{jacques.giacomoni@univ-pau.fr}

\address{Sweta Tiwari \newline
Department of Mathematics, IIT Guwahati, India}
\email{swetatiwari@iitg.ernet.in}

\thanks{Submitted July 29, 2017. Published February 8, 2018.}
\subjclass[2010]{35K59, 35K55, 35B40}
\keywords{p-fractional operator; existence and regularity of weak solutions;
\hfill\break\indent asymptotic behavior of global solutions; stabilization}

\begin{abstract}
 First, we discuss the existence, the uniqueness and the regularity
 of the weak solution to the following parabolic equation involving
 the fractional $p$-Laplacian,
 \begin{gather*}
 u_t+(-\Delta)_{p}^su +g(x,u)= f(x,u)\quad \text{in } Q_T:=\Omega\times (0,T), \\
 u = 0 \quad \text{in } \mathbb{R}^N\setminus\Omega\times(0,T),\\
 u(x,0) =u_0(x)\quad \text{in }\mathbb{R}^N.
 \end{gather*}
 Next, we deal with the asymptotic behavior of global weak solutions.
 Precisely, we prove under additional assumptions on $f$ and $g$ that global
 solutions converge to the unique stationary solution as $t\to \infty$.
\end{abstract}

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

\section{Introduction and Preliminaries}

In this article we study the  parabolic problem involving fractional
$p$-Laplacian,
\begin{equation}\label{PP}
 \begin{gathered}
 u_t+(-\Delta)_{p}^su +g(x,u)=f(x,u)\quad\text{in } Q_T:=\Omega\times (0,T), \\
 u= 0 \quad\text{in } \mathbb{R}^{ N}\setminus\Omega\times(0,T),\\
 u(x,0) =u_0(x)\quad\text{in }\mathbb{R}^{ N}
 \end{gathered}
\end{equation}
 where $\Omega$ is a smooth bounded domain in $\mathbb{R}^{ N}$
(at least $C^2$), $s\in(0,1)$, $1<p<N/s$, $u_0\in L^\infty(\Omega)$
 and $f(x,z)$, $g(x,z)$ are
 Carath\'eodory functions, locally Lipschitz with respect to $z$ uniformly
in $x$ and satisfying the following assumptions:
 \begin{itemize}
 \item[(A1)] $f(x,z),g(x,z)>0$ for $t>0$ and
 $f(x,0)=0$, $g(x,0)=0$ for a.e. $x\in\Omega$.

\item[(A2)] For a.e. $x\in\Omega$ and $z\geq 0$, $g(x,z)$ satisfy the
 growth condition:
 $$
g(x,z)\leq C_1+C_2z^{r-1},\quad 1<r< p^*_s:=\frac{ Np}{ N-sp}
$$
for some positive constants $C_1$ and $C_2$.

\item[(A3)] $f(x,z)/z^{p-1}$ is non-increasing and
$g(x,z)/z^{p-1}$ is non-decreasing in $z$ for a.e. $x\in\Omega$.

\item[(A4)]$ \limsup_{z\to 0^+}f(x,z)/z^{p-1}>\lambda_{1,s,p}$,
 $ \limsup_{z\to \infty}f(x,z)/z^{p-1}<\lambda_{1,s,p}$,
 where $\lambda_{1,s,p}$ is the first eigenvalue of  $(-\Delta)^s_p$,
 $ \limsup_{z\to 0^+}g(x,z)/z^{p-1}=0$
 and
 $ \limsup_{z\to \infty} g(x,z)/ z^{p-1}=\infty$, uniformly in $x\in \Omega$.

\end{itemize}
 For instance we can take $f(x,z)=a(x)z^{q-1}$ and $g(x,z)=b(x)z^{r-1}$
where $1<q<p$, $p<r<p_s^*$ and $a,b\in L^\infty(\Omega)$
 as these functions satisfy $(A1)-(A4)$.

 We recall that the fractional $p$-Laplacian operator $(-\Delta)^s_p$
(up to a normalizing constant) is defined as
$$
(-\Delta)^s_pu(x):= \lim_{\epsilon\to 0^+}\int_{\mathbb{R}^{ N}\setminus B_\epsilon(x)}
\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{ N+sp}}dy,\quad x\in\mathbb{R}^{ N}.
$$
The systematic study of the problems involving non-local
operators have found great interest in the recent years due to there occurrence
in concrete real-world applications, such as, the thin obstacle problem,
optimization, finance, phase transitions.
Elliptic theory of linear or quasilinear non-local operators
has been actively studied during last decades in the works of Caffarelli and
collaborators \cite{AC,ACS,CSS}, Kassmann \cite{KASSMANN},
Silvestre \cite{Silvestre} and many others. For further references,
we refer to the surveys \cite {Va2}, \cite{radu2} and in the nonlinear
diffusion of degenerate type case ($p$-fractional operators type) \cite{MoSq}.
 We also refer to \cite{paper1, paper2, paper3, paper4, paper5} on
related existence results for nonlocal problems driven by the fractional
Laplace operator.

Concerning the parabolic equation involving fractional Laplacian, the study of
anomalous diffusion equation has gained interest
for its occurrence in a number of
phenomena in several areas of physics, finance, biology, ecology, geophysics,
and many others which can be characterized as having non-Brownian scaling.

Contrary to the stationary version, there are quite few results about the
corresponding evolution equations involving quasilinear and nonlocal operators.
We can quote first that local existence and uniqueness of mild solutions are
investigated in \cite{MaRoTo} by semi-group theory. The homogeneous Dirichlet
problem for the fractional $p$-Laplacian evolution equation is studied also in
the recent work of V\'azquez where the author proved everywhere positivity
of weak solutions. This striking property contrasts with the finite propagation
property occurring in the local setting ($p$-Laplace operator).
In \cite{AABP}, authors have studied (\ref{PP}) with the nonlinearity $f$
depending only on $x$ and $t$ and prove the existence and some properties of
entropy solutions.
In particular, the questions related to the extinction in finite time and
the finite speed of propagation are analyzed.

In this article, we investigate different issues of the existence and the
regularity of energy weak solutions that in our knowledge are not discussed
in former works.
We also deal with the long-time behavior of weak solutions for a class of
sub-homogeneous nonlinearities $f$ and $g$ following the approach in \cite{BBG}
for the local homogeneous
$p$-Laplacian operator and in \cite{JSG} for the local non-homogeneous
$p(x)$-Laplacian operator.
We point out that using the results in \cite{JAS}, authors in \cite{JTS} have
studied similar questions for the semilinear version of \eqref{PP}
with singular nonlinearity.

\section{Preliminaries}

We consider the  function space
\begin{align*}
W^{s,p}(\mathbb{R}^ N):=\Big\{&u| u:\mathbb{R}^ N\to\mathbb{R}^ N
 \text{ is  measurable },u\in L^p(\mathbb{R}^ N) \\
&\text{and } \frac{(u(x)-u(y))}{|x-y|^{\frac{ N+sp}{p}}}
 \in L^p(\mathbb{R}^ N\times \mathbb{R}^ N)\Big\}.
\end{align*}
$W^{s,p}(\mathbb{R}^ N)$ is a Banach space endowed with the  norm
$$
\|u\|_{W^{s,p}(\mathbb{R}^ N)}:=\|u\|_{L^p(\mathbb{R}^ N)}
+\Big(\int_{\mathbb{R}^ N\times\mathbb{R}^ N}\frac{|u(x)-u(y)|^p}{|x-y|^{ N+sp}}dx
\,dy\Big)^{1/p}.
$$
Also define the closed linear subspace $X_0(\Omega)$ of $W^{s,p}(\mathbb{R}^ N)$ as
$$
X_0(\Omega):=\{u\in W^{s,p}(\mathbb{R}^ N)|u(x)=0 \text{ a.e. }
x\in\mathcal{C}\Omega\}
$$
endowed with the norm
$$
\|u\|_{X_0(\Omega)}:=\left(\frac{1}{2}
\int_Q\frac{|u(x)-u(y)|^p}{|x-y|^{ N+sp}}dx\,dy\right)^{1/p}
$$
where $Q=\mathbb{R}^ N\times\mathbb{R}^ N\backslash
\mathcal{C}\Omega\times \mathcal{C}\Omega$,
$\mathcal{C}\Omega=\mathbb{R}^ N\backslash\Omega$.
Then $X_0(\Omega)$ is a uniformly convex Banach space.
Also $C^\infty_0(\Omega)$ is dense in $X_0(\Omega)$ and $X_0(\Omega)$ is compactly
embedded in $L^r(\Omega)$ for $1\leq r<p^*_s$.

\begin{remark}\label{identity 1} \rm
Let $t^+=\max(t,0)$. If $v\in X_0(\Omega)$, then
\[
|v(x)-v(y)|^{p-2}(v^+(x)-v^+(y))(v(x)-v(y))\geq|v^+(x)-v^+(y)|^{p}.
\]
 \end{remark}

\begin{definition}\label{cone1} \rm
Set $d(x):=\text{ dist }(x,\partial\Omega)$.
 Define the  normed space
$$
C_{d(\Omega)}:=\{u \in C_0(\overline{\Omega}) :\exists c\geq0 \text{ such that }
|u(x)|\leq c d(x), \forall x\in\Omega\}
$$
\end{definition}

\begin{definition}\label{cone2} \rm
Define the  open convex subset of $C_{d(\Omega)}$,
$$
C_{d^s(\Omega)}^+:=\{u\in C_{d(\Omega)}:\inf_{x\in\Omega}\frac{u(x)}{d^s(x)}>0\}.
$$
\end{definition}

 Let $\phi_{1,s,p}$ be the eigenfunction corresponding to the first eigenvalue
$\lambda_{1,s,p}$ of the operator $(-\Delta)^s_p$.
Then $\phi_{1,s,p}\in C^+_{d^s}(\Omega)$.

 We also recall the following inequalities due to Simon \cite{S2}:
for all $u, v\in \mathbb{R}^ N$,
\begin{gather}\label{A0}
\left||u|^{p-2}u-|v|^{p-2}v\right|
\leq \begin{cases}
c |u-v|(|u|+|v|)^{p-2} & \text{if } p\geq 2,\\
c |u-v|^{p-1} &\text{if } p\leq 2,
\end{cases}\\
\label{A00}
\langle |u|^{p-2}u-|v|^{p-2}v, u-v\rangle
\geq \begin{cases}
\tilde c |u-v|^{p} & \text{if } p\geq 2,\\
\tilde c  {\frac{|u-v|^{2}}{(|u|+|v|)^{2-p}} }&\text{if } p\leq 2,
\end{cases}
\end{gather}
where $c, \tilde c$ are positive constants and $\langle\cdot,\cdot\rangle$
 is the canonical scalar product of $\mathbb{R}^ N$.

\section{Main results}\label{main results}

First we consider the  problem
\begin{equation} \label{ST}
\begin{gathered}
u_t+(-\Delta) _{p}^su = h(x,t) \quad\text{in } Q_T:=\Omega\times (0,T), \\
u(x,t) = 0 \quad\text{in } \mathbb{R}^{ N}\setminus\Omega\times(0,T),\\
u(x,0) =u_0(x)\quad\text{in }\Omega ,
\end{gathered}
\end{equation}
where $T>0$, $h\in L^\infty(Q_T)$.
Considering the initial data $u_0\in L^\infty(\Omega)$, we study the weak
solution of \eqref{ST} defined as follows:

\begin{definition}\label{defi1} \rm
A weak solution of \eqref{ST} is a function $u\in L^\infty(0,T;X_0(\Omega))$ such
that $u_t\in L^2(Q_T)$ and for any $\phi\in C^\infty_0(Q_T)$,
\begin{align*}
&\int_{Q_T} u_t\phi\,dx\,dt+\frac{1}{2}\int_0^T\int_{Q}\frac{|u(x)- u(y)|^{p-2}
(u(x)-u(y))}{|x-y|^{ N+sp}}(\phi(x)-\phi(y))\,dx\,dy\,dt\\
&=\int_{Q_T} h(x,t)\phi\,dx\,dt
\end{align*}
and $u(x,0)=u_0(x)$ for a.e. $x\in\Omega$.
\end{definition}

Next, we consider the initial data $u_0$ in $C^+_{d^s(\Omega)}$ and study
the evolution equation
\begin{equation} \label{PT}
 \begin{gathered}
 u_t+(-\Delta)_{p}^su +g(x,u)= f(x,u)\quad\text{in } Q_T:=\Omega\times (0,T), \\
 u(x,t) = 0\quad\text{in } \mathbb{R}^{ N}\setminus\Omega\times(0,T),\\
 u(x,0) = u_0(x)\quad\text{in }\Omega.
 \end{gathered}
\end{equation}

\begin{definition} \label{defi2} \rm
A solution of \eqref{PT} is a function $u\in L^\infty(0,T;X_0(\Omega))$ such
that $u_t\in L^2(Q_T)$ and for any $\phi\in C^\infty_0(Q_T)$,
\begin{align*}
&\int_{Q_T} u_t\phi\,dx\,dt+\frac{1}{2}\int_0^T\int_{Q}
 \frac{|u(x)- u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{ N+sp}}(\phi(x)-\phi(y))\,dx\,dy\,dt\\
&+\int_{Q_T} g(x,u)\phi\,dx\,dt \\
&=\int_{Q_T}f(x,u)\phi\,dx\,dt
\end{align*}
and $u(x,0)=u_0(x)$ for a.e. $x\in\Omega$.
\end{definition}

\begin{theorem}\label{sol S_T}
Let $T>0$, $h(x,t)\in L^\infty(Q_T)$ and $u_0\in L^\infty(\Omega)$.
Then there exists a unique weak solution $u$ to the problem \eqref{ST}.
Moreover $u\in C([0,T],X_0(\Omega))$ and satisfies for any $t\in[0,T]$:
\begin{equation}\label{equality of sol S_T}
\begin{aligned}
&\int_0^t\int_\Omega\Big(\frac{\partial u}{\partial t}\Big)^2\,dx\,ds
+\frac{1}{p}\|u(t)\|_{X_0(\Omega)}^p \\
&= \int_0^t\int_\Omega h(x,s)\Big(\frac{\partial u}{\partial t}\Big)\,dx\,ds
+\frac{1}{p}\|u_0\|_{X_0(\Omega)}^p
\end{aligned}
\end{equation}
\end{theorem}

Concerning problem \eqref{PT}, we deduce the following similar result.

\begin{theorem}\label{sol P_T}
Let $f,g$ be Carath\'eodory functions, locally Lipschitz with respect to
second variable uniformly in $x\in\Omega$
and satisfying the assumptions $(A1), (A2)$ and $(A4)$.
Let $u_0\in C^+_{d^s(\Omega)}$.
Then for any $T>0$, there exists a unique weak solution $u$ to
 problem \eqref{PT}.
Moreover $u\in C([0,T],X_0(\Omega))$ and satisfies for any $t\in[0,T]$:
\begin{equation}\label{equality of sol P_T}
\begin{aligned}
&\int_0^t\int_\Omega\Big(\frac{\partial u}{\partial t}\Big)^2\,dx\,ds
+\frac{1}{p}\|u(x,t)\|_{X_0(\Omega)}^p \\
&= \int_\Omega F(x,u(x,t))\,dx-\int_\Omega G(x,u(x,t))\,dx
 +\frac{1}{p}\|u_0(x)\|_{X_0(\Omega)}^p
\end{aligned}
\end{equation}
where $F(x,z)=\int_0^zf(x,s)ds$ and $G(x,z)=\int_0^zg(x,s)ds$.
\end{theorem}


 Next we observe that the operator $A:=(-\Delta)^s_p$, with Dirichlet
boundary conditions, is $m$-accretive in $L^\infty(\Omega)$.
Precisely, we have the following lemma.

\begin{lemma}\label{accretive lemma}
Consider ${\mathcal D}(A)=\{u\in X_0(\Omega)\cap L^\infty(\Omega):
 Au\in L^\infty(\Omega)\}$ as the domain of the operator $A$.
Then $A$ is $m$-accretive in $L^\infty(\Omega)$.
\end{lemma}

 Now by appealing the theory of maximal accretive operators in Banach spaces,
we obtain the following results for the
solutions of \eqref{ST} and \eqref{PT}, respectively.
\begin{theorem}\label{regularity S_T}
 Let $T>0$, $h\in L^\infty(Q_T)$ and let $u_0$ be in
$\overline{{\mathcal D}(A)}^{L^\infty}$. Then
\begin{itemize}
\item[(i)] the unique weak solution $u$ to \eqref{ST} obtained in Theorem
\ref{sol S_T} belongs to $\mathcal C([0,T]; \mathcal C_0(\overline{\Omega}))$.
\item[(ii)] If $v$ is another mild solution to \eqref{ST} with the initial
datum $v_0\in\overline{{\mathcal D}(A)}^{L^\infty}$
and the right-hand side $k(x,t)\in L^\infty(Q_T)$, then the following
estimate holds:
\begin{equation}
\|u(t)-v(t)\|_{L^\infty(\Omega)}\leq \|u_0-v_0\|_{L^\infty(\Omega)}
+\int_0^t\|h(s)-k(s)\|_{L^\infty(\Omega)}\, {\rm d}s,
\label{estimation2}
\end{equation}
for $0\leq t\leq T$.

\item[(iii)] If $u_0\in {\mathcal D}(A)$ and $h\in W^{1,1}(0,T;L^\infty(\Omega))$
then $u\in W^{1,\infty}(0,T;L^\infty(\Omega))$
and $(-\Delta)^s_{p} u\in L^\infty(Q_T)$, and the following estimate holds:
\begin{equation}
\|\frac{\partial u}{\partial t}(\cdot, t)\|_{L^\infty(\Omega)}
\leq \|(-\Delta)^s_{p} u_0+h(\cdot,0)\|_{L^\infty(\Omega)}
+\int_0^T\|\frac{\partial h}{\partial t}(\cdot,\tau)\|_{L^\infty(\Omega)}{\rm d}\tau.
\label{estimationbiss2}
\end{equation}
\end{itemize}
\end{theorem}

\begin{theorem}\label{regularity P_T}
Assume that conditions and hypotheses on $f,g$ in Theorem \ref{sol P_T}
are satisfied and $u_0\in\overline{{\mathcal D}(A)}^{L^\infty}$.
Then, the unique weak solution to \eqref{PT}
belongs to $C([0,T];C_0(\overline{\Omega}))$ and
\begin{itemize}
\item[(i)] there exists $\omega>0$ such that if $v$ is another weak solution
to \eqref{PT} with the initial datum
$v_0\in\overline{{\mathcal D}(A)}^{L^\infty}$ then the following estimate holds:
$$
\|u(t)-v(t)\|_{L^\infty(\Omega)}\leq
e^{\omega t} \|u_0-v_0\|_{L^\infty(\Omega)},\quad 0\leq t\leq T.
$$
\item[(ii)] If $u_0\in {\mathcal D}(A)$ then
$u\in W^{1,\infty}(0,T;L^\infty(\Omega))$ and $(-\Delta)_{p}^s u\in L^\infty(Q_T)$,
and the following estimate holds:
$$
\|\frac{\partial u}{\partial t}(t)\|_{L^\infty(\Omega)}
\leq e^{\omega t} \|(-\Delta)^s_{p} u_0+f(x,u_0)\|_{L^\infty(\Omega)}.
$$
\end{itemize}
\end{theorem}

Next, we investigate the asymptotic behavior of global solution of \eqref{PT},
in particular the convergence to a stationary solution. For this first
we study the following stationary problem corresponding to \eqref{PT}.
\begin{equation} \label{eP}
 \begin{gathered}
 (-\Delta)_p^su+g(x,u) = f(x,u)\quad \text{in } \Omega, \\
 u >0 \quad\text{in }\Omega,\\
 u = 0 \quad \text{in } \mathbb{R}^{ N}\setminus\Omega.
 \end{gathered}
\end{equation}

 \begin{definition}\label{weak sol of P} \rm
 A function $u\in X_0(\Omega)$ is said to be a weak solution of \eqref{eP}
if $u>0$ in $\Omega$ and
\begin{align*}
&\int_{Q}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y)) (\phi(x)-\phi(y))}{|x-y|^{ N+sp}}
 dx\,dy
 +\int_\Omega g(x,u)\phi(x) dx \\
&=\int_\Omega f(x,u)\phi(x) dx,
\end{align*}
 for all $\phi\in X_0(\Omega)$.
\end{definition}

\begin{theorem}\label{Existence of sol P}
Let $f,g$ be Carath\'eodary functions, locally Lipschitz with respect to
second variable uniformly in $x\in\Omega$
and satisfying the assumptions {\rm (A1)--(A4)}.
Then there exists a unique weak solution $u_\infty$ of \eqref{eP}.
Moreover, $u_\infty\in C^+_{d^s(\Omega)}$.
\end{theorem}

\begin{theorem}\label{stabilization}
Assume that $f$ satisfies {\rm (A1)--(A4)}. Then
the weak solution $u$ to \eqref{PT} is defined in $(0,\infty)\times \Omega$ and
$$
u(t)\to u_\infty\quad\text{in } L^\infty(\Omega)\quad\text{as }t\to\infty
$$
where $u_\infty$ is the unique solution to the stationary problem \eqref{eP}.
\end{theorem}

\section{Proofs of the main results}

In this section we give the proofs of the results stated in
Section \ref{main results}. We begin with the following sequence of results.

\begin{lemma}\label{Existence P}
Let $f,g$ be Carath\'eodory functions, locally Lipschitz with respect to second
variable uniformly in $x\in\Omega$ and satisfying the assumptions {\rm (A1)--(A3)}.
Then there exists a non-negative and non trivial weak solution $u\in X_0(\Omega)$
to the equation in \eqref{eP}.
\end{lemma}

\begin{proof}
Consider the energy functional $J$ corresponding to \eqref{eP}, given by
\[
J(u)=\frac{1}{p}\int_{Q}\frac{ |u(x)-u(y)|^p}{|x-y|^{ N+sp}}\,dx\,dy
+\int_\Omega G(x,u)\,dx-\int_\Omega F(x,u)\,dx.
\]
Note that $J$ is coercive in $X_0(\Omega)$.
Indeed, by the assumption (A1), (A2) and the Sobolev embedding theorem we have
 \begin{align*}
 J(u)&=\frac{1}{p}\int_{Q}\frac{ |u(x)-u(y)|^p}{|x-y|^{ N+sp}}\,dx\,dy
+\int_\Omega G(x,u)\,dx-\int_\Omega F(x,u)\,dx \\
&\geq\frac{1}{p}\|u\|_{X_0(\Omega)}^{p}-C_1^*\|u\|_{X_0(\Omega)}-C_2^*\|u\|_{X_0(\Omega)}^{q}
\end{align*}
which tends to $\infty$ for $\|u\|_{X_0(\Omega)}$ large enough.
Thus $J$ is coercive. Furthermore $J\in C^1(X_0(\Omega))$ and
weakly lower semi-continuous in $X_0(\Omega)$
and therefore admits a global minimizer which is a weak solution to \eqref{eP}.
From (A4), we obtain easily
$ \inf_{X_0(\Omega)}J<0$ and then $u\not\equiv 0$. Also
$J(|u|)\leq J(u)$. Indeed, as by triangle inequality, for
$x,y\in\mathbb{R}^{ N}$, we have
$\|u(x)|-|u(y)\|\leq |u(x)-u(y)|$.
Also from (A1), $F(x,|u(x)|)=F(x,u(x))$ and $G(x,|u(x)|)=G(x,u(x))$ for all
$x\in\Omega$. Thus we have $J(|u|)\leq J(u)$ for $u\in X_0(\Omega)$, and hence
the minimizer of $J$ in $X_0(\Omega)$ can be assumed non-negative.
This establishes the existence of a non-negative, non-trivial weak solution
$u\in X_0(\Omega)$ of \eqref{eP}.\end{proof}

Now we prove that $u\in L^\infty(\Omega)$ and is unique.
For proving these properties, first we recall
the following Picone inequality (see \cite{BF}).

 \begin{lemma}\label{Picone identity}
For every $a_1 , a_2\geq 0$ and $b_1 , b_2 > 0$
$$
|a_1-a_2|^p\geq |b_1-b_2|^{p-2}(b_1-b_2)
\Big(\frac{a_1^p}{b_1^{p-1}}-\frac{a_2^p}{b_2^{p-1}}\Big).
$$
The equality holds if and only if $(a_1 , a_ 2 ) = k(b_1 , b_2 )$
for some constant $k$.
 \end{lemma}

\begin{proposition}\label{L-infinity bound for P}
Let $u\in X_0(\Omega)$ be a weak solution of \eqref{eP}. Then $u\in L^\infty(\Omega)$.
\end{proposition}

\begin{proof}
We adapt arguments from \cite{FrPa}. First we note that due to the homogeneity
of the problem \eqref{eP}, it suffices to prove that
\begin{equation}\label{claim1}
\|u^+\|_{L^\infty(\Omega)}\leq 1 \text{ whenever }\|u^+\|_{L^p(\Omega)}\leq \delta
 \text{ for some }\delta>0.
\end{equation}
A similar assertion can be established for $u^{-}$ where
$u^+(x)=\max\{u(x),0\}$ and $u^-(x)=\max\{-u(x),0\}$.
Therefore $u\in L^\infty(\Omega)$.
For $k\geq 1$, set $w_k(x)=(u(x)-(1-2^{-k}))^+$.
Then first we make the following observations about $w_k(x)$.
\begin{itemize}
 \item[(i)] $w_{k+1}(x)\leq w_k(x)$ for all $x\in\Omega$,

 \item[(ii)] $u(x)<(2^{k+1}+1)w_k(x)$ for $x\in\{w_{k+1}(x)>0\}$.
 For proving this, take $x\in\{w_{k+1}(x)>0\}$.
Then $w_k(x)=u(x)-(1-2^{-k})$. This implies
 \[
(2^{k+1}+1)w_k(x)=u(x)+2^{k+1}u(x)-(2^{k+1}+1)(1-2^{-k}).
\]
 Now as for $x\in\{w_{k+1}(x)>0\}$, $u(x)>(1-2^{-(k+1)})=1-2^{-k}+2^{-(k+1)}$.
This implies
 \begin{align*}
 2^{k+1}u(x)
&>2^{k+1}(1-2^{-k})+1\\
 &>2^{k+1}(1-2^{-k})+(1-2^{-k})\\
 &=(2^{k+1}+1)(1-2^{-k}).
 \end{align*}
Therefore $(2^{k+1}+1)w_k(x)=u(x)+2^{k+1}u(x)-(2^{k+1}+1)(1-2^{-k})>u(x)$
for $x\in\{w_{k+1}(x)>0\}$.

\item[(iii)] $\{w_{k+1}>0\}\subset\{w_k>2^{-(k+1)}\}$.
\end{itemize}
Now set $U_k:=\|w_k\|_{L^p}^p$. Taking $v=u-(1-2^{-(k+1)})$ in
Lemma \ref{identity 1} we obtain
\[
|u(x)-u(y)|^{p-2}(w_{k+1}(x)-w_{k+1}(y))(u(x)-u(y))
\geq|w_{k+1}(x)-w_{k+1}(y)|^{p}.
\]
Therefore, using (i)-(ii) above, we obtain
\begin{align*}
\|w_{k+1}\|_{X_0(\Omega)}^p
&=\int_{Q}\frac{|w_{k+1}(x)-w_{k+1}(y)|^p}{|x-y|^{ N+sp}}dx\,dy\\
 &\leq\int_{Q}\frac{|u(x)-u(y)|^{p-2}(w_{k+1}(x)-w_{k+1}(y))
(u(x)-u(y))}{|x-y|^{ N+sp}}dx\,dy\\
 &\leq\int_{\Omega}|f(x,u)|w_{k+1} dx\\
 &\leq \int_{\{w_{k+1}(x)>0\}}(C_1+C_2|u|^{p-1})w_{k+1} dx\\
 &= C_1\int_{\{w_{k+1}(x)>0\}}w_{k+1} dx
 +C_2\int_{\{w_{k+1}(x)>0\}}|u|^{p-1}w_{k+1} dx\\
 &\leq C_1|\{x\in\Omega:w_{k+1}(x)>0\}|^{1-1/p}U_k^{1/p}\\
&\quad +C_2\int_{\{w_{k+1}(x)>0\}} (2^{k+1}+1)^{p-1}w_{k}^p dx\\
 &\leq C_1|\{x\in\Omega:w_{k+1}(x)>0\}|^{1-1/p}U_k^{1/p}+C_2(2^{k+1}+1)^{p-1}U_k.
 \end{align*}
Now as for (iii) we have
\[
U_k=\int_\Omega w_k^pdx\geq\int_{\{w_{k+1}>0\}}w_k^p
\geq2^{-(k+1)p}|\{x\in\Omega:w_{k+1}(x)>0\}|.
\]
 Therefore,
 \begin{equation}\label{Peq1}
 \|w_{k+1}\|_{X_0(\Omega)}^p\leq
 (C_1 2^{(k+1)(p-1)}+C_2(2^{k+1}+1)^{p-1})U_k\leq C_3(2^{k+1}+1)^{p-1}U_k.
 \end{equation}
Also from H\"older's inequality we have
\begin{equation} \label{Peq2}
\begin{aligned}
U_{k+1}&=\int_{\{w_{k+1}(x)>0\}} w_{k+1}^p dx \\
&\leq \Big(\int_{\{w_{k+1}(x)>0\}} w_{k+1}^{\frac{ N}{ N-sp}}\Big)^{\frac{ N-sp}{ N}}
|\{x\in\Omega:w_{k+1}(x)>0\}|^{sp/N} \\
&\leq C_4\| w_{k+1}\|_{X_0(\Omega)}^p(2^{(k+1)p}U_k)^{sp/N}.
\end{aligned}
\end{equation}
Hence,
\begin{equation} \label{Peq3}
\begin{aligned}
U_{k+1}
&\leq C_5(2^{k+1}+1)^{p-1}U_k(2^{(k+1)p}U_k)^{sp/N} \\
&\leq C_5(2^{k+1}+1)^{p(1+\frac{sp}{ N})}U_k^{1+\frac{sp}{ N}} \\
&\leq C_5C^kU_k^{1+\alpha}
\end{aligned}
\end{equation}
where $C>1$ and $\alpha=\frac{sp}{ N}$. This will imply that
\begin{equation}\label{Peq4}
 \lim_{k\to\infty} U_k = 0
\end{equation}
provided that $\|u^+\|^p_{L^p(\Omega)}=U_0\leq C^{-\frac{1}{\alpha^2}} =:\delta^p$.
As $w_k(x)\to (u(x)-1)^+$ for a.e. $x\in\mathbb{R}^{ N}$,
\eqref{claim1} follows from \eqref{Peq4}.
\end{proof}

Now we recall \cite[Theorem 1.1]{IMS} that provides the
$C^\alpha(\overline\Omega)$ regularity of weak solution of \eqref{eP}.

\begin{theorem}\label{C-alpha regularity}
 There exist $\alpha=\alpha( N,p,s)\in(0,s]$ and
$C=C( N,p,s,\Omega,\|u\|_{L^\infty}(\Omega) )$,
 such that, for all weak solutions
 $u\in X_0(\Omega)$ of \eqref{eP}, $u\in C^\alpha(\overline\Omega)$ and
 $ \|u\|_{C^\alpha(\overline\Omega)}\leq C$.
\end{theorem}

Next, we have the following Hopf Lemma from
\cite[Theorems 1.4 and 1.5, p. 778]{PQ}.

\begin{lemma}\label{Hopf lemma}
 Let $\Omega$ satisfy the interior ball condition and
$u\in X_0(\Omega)\cap C(\overline\Omega)$ be a non-trivial,
 non-negative weak super-solution of
\begin{equation*}
(-\Delta)^s_pu=c(x)|u|^{p-1}\quad\text{in }\Omega
\end{equation*}
with $c\in L^1_{\rm loc}(\Omega)$ and non-positive.
 Then $u>0$ in $\Omega$ and
 \begin{equation}\label{eq hopf}
  \liminf_{B_R\ni x\to x_0}\frac{u(x)}{d_R(x)^s}>0
 \end{equation}
where $B_R$ is a ball such that $x_0\in B_R\subset\Omega$ and $d_R(x)$
is distance from $x$ to $\partial B_R$.
\end{lemma}

Writing $g(x,u)=c(x)u^{p-1}$ and using (A1) and (A4), we obtain that
any nonnegative and non trivial weak solution $u$
to the equation in \eqref{eP} is positive and satisfies $u\geq kd(x)$
for some $k>0$. Next, using \cite[Theorem 4.4]{IMS},
we obtain that any nonnegative and non trivial weak solution $u$ to the equation
in \eqref{eP} belongs to $C^+_{d^s(\Omega)}$.
Then it follows that any couple of non trivial and nonnegative weak solutions
$u$, $v$ to the equation in \eqref{eP} satisfy $u/v$, $v/u\in L^\infty(\Omega)$.
We use this property
to prove the uniqueness of the solution of \eqref{eP}.

\begin{theorem}\label{uniqueness of sol of P}
Let $u,v\in X_0(\Omega)$ be two non trivial and nonnegative weak solutions to
the equation in \eqref{eP}. The $u= v$ for a.e. in $\Omega$.
\end{theorem}

\begin{proof}
Set $u_n=u+\frac{1}{n}$ and $v_n=v+\frac{1}{n}$ and define
$$
\tilde v_n:=\frac{u^p}{v_n^{p-1}},\quad \tilde u_n:=\frac{v^p}{u_n^{p-1}}.
$$
First we claim that $\tilde v_n,\;\tilde u_n\in X_0(\Omega)$.
Note that since $u,v>0$ in $\Omega$, $\tilde v_n,\;\tilde u_n>0$ in $\Omega$,
$\tilde v_n,\tilde u_n=0$ in $\mathbb{R}^{ N}\setminus\Omega$ for all $n\in\mathbb N$.
Also since $u,v\in L^\infty(\Omega)$, we have that
$\tilde v_n,\tilde u_n\in L^p(\Omega)$ for all $n\in\mathbb N$.
Also as
\begin{equation} \label{Peq5}
\begin{aligned}
&|\tilde v_n(x)-\tilde v_n(y)|\\
&=\big|\frac{u^p(x)}{v_n^{p-1}(x)}-\frac{u^p(y)}{v_n^{p-1}(y)}\big| \\
&\leq\big|\frac{u^p(x)}{v_n^{p-1}(x)}-\frac{u^p(y)}{v_n^{p-1}(x)}\big|
 + \big|\frac{u^p(y)}{v_n^{p-1}(x)}-\frac{u^p(y)}{v_n^{p-1}(y)}\big| \\
&=\big|\frac{u^p(x)-u^p(y)}{v_n^{p-1}(x)}\big|
 + |u^p(y)|\big|\frac{v_n^{p-1}(y)-v_n^{p-1}(x)}{v_n^{p-1}(x)v_n^{p-1}(y)}\big| \\
&\leq n^{p-1}|u_n^p(x)-u_n^p(y)|+
 \|u\|^p_{L^\infty(\Omega)}\big|\frac{v_n^{p-1}(y)-v_n^{p-1}(x)}{v_n^{p-1}(x)
 v_n^{p-1}(y)}\big| \\
&\leq 2n^{p-1}p\|u_n\|^{p-1}_{L^\infty(\Omega)}|u(x)-u(y)|\\
&\quad +(p-1)\|u\|^p_{L^\infty(\Omega)}
 \big|\frac{v_n^{p-2}(y)+v_n^{p-2}(x)}{v_n^{p-1}(x)v_n^{p-1}(y)}\big||v_n(y)-v_n(x)| \\
&= 2n^{p-1}p\|u\|^{p-1}_{L^\infty(\Omega)}|u(x)-u(y)| \\
&\quad +(p-1)\|u\|^p_{L^\infty(\Omega)}
\big|\frac{1}{v_n^{p-1}(x)v_n(y)}+\frac{1}{v_n^{p-1}(y)v_n(x)}\big||v(y)-v(x)| \\
&\leq 2n^{p-1}p\|u\|_{L^\infty(\Omega)}^{p-1}|u(x)-u(y)|+
2n^p(p-1)\|u\|_{L^\infty(\Omega)}^p|v(y)-v(x)| \\
&\leq C(n,p,\|u\|_{L^\infty(\Omega)})\left(|u(x)-u(y)|+|v(y)-v(x)|\right)
\end{aligned}
\end{equation}
for all $(x,y)\in\mathbb R^{2 N}$. Thus
$\tilde v_n\in X_0(\Omega)$ for all $n\in\mathbb N$.
Similarly $\tilde u_n\in X_0(\Omega)$ for all $n\in\mathbb N$.
As $u$ and $v$ solve \eqref{eP}, we have
\begin{gather}\label{u solves P}
\langle(-\Delta)_p^su,u-\tilde u_n\rangle
=\left(f(x,u)-g(x,u)\right)(u-\tilde u_n), \\
\label{v solves P}
\langle(-\Delta)_p^sv,v-\tilde v_n\rangle
=\left(f(x,v)-g(x,v)\right)(u-\tilde u_n).
\end{gather}
Set
\begin{align*}
& L(u,v)(x,y) \\
&=|u(x)-u(y)|^p-|v(x)-v(y)|^{p-2}(v(x)-v(y))
\big(\frac{u^p(x)}{v^{p-1}(x)}-\frac{u^p(x)}{v^{p-1}(y)}\big).
\end{align*}
Now using \eqref{u solves P}, \eqref{v solves P} and Lemma
\ref{Picone identity} we have the estimate
\begin{align}
0&\leq \int_{Q}L(u,v_n)(x,y)+L(v,u_n)(x,y)\,dx\,dy \nonumber \\
&=\int_{Q}\frac{|u(x)-u(y)|^p}{|x-y|^{ N+sp}}-\frac{|v_n(x)-v_n(y)|^{p-2}
 (v_n(x)-v_n(y))}{|x-y|^{ N+sp}} \nonumber \\
&\quad\times\Big(\frac{u^p(x)}{v_n^{p-1}(x)}-\frac{u^p(y)}{v_n^{p-1}(y)}\Big)
 dx\,dy \nonumber \\
&\quad +\int_{Q}\frac{|v(x)-v(y)|^p}{|x-y|^{ N+sp}}
-\frac{|u_n(x)-u_n(y)|^{p-2}(u_n(x)-u_n(y))}{|x-y|^{ N+sp}} \nonumber \\
&\quad\times \Big(\frac{v^p(x)}{u_n^{p-1}(x)}-\frac{v^p(y)}{u_n^{p-1}(y)}\Big)
  \,dx\,dy \nonumber \\
&= \int_{Q}\frac{|u(x)-u(y)|^p}{|x-y|^{ N+sp}}
 -\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{ N+sp}} \nonumber \\
&\quad\times \Big(\frac{v^p(x)}{u_n^{p-1}(x)}-\frac{v^p(y)}{u_n^{p-1}(y)}\Big)
 dx\,dy \nonumber \\
&\quad +\int_{Q}\frac{|v(x)-v(y)|^p}{|x-y|^{ N+sp}}
-\frac{|v(x)-v(y)|^{p-2}(v(x)-v(y))}{|x-y|^{ N+sp}} \nonumber \\
&\quad\times \Big(\frac{u^p(x)}{v_n^{p-1}(x)}
-\frac{u^p(y)}{v_n^{p-1}(y)}\Big) dx\,dy \nonumber \\
&=\int_{Q}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{ N+sp}}
\Big(\frac{u^p(x)}{u^{p-1}(x)}-\frac{u^p(y)}{u^{p-1}(y)}\Big) dx\,dy  \nonumber \\
&\quad-\int_{Q}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{ N+sp}}
\Big(\frac{v^p(x)}{u_n^{p-1}(x)}-\frac{v^p(y)}{u_n^{p-1}(y)}\Big) dx\,dy \nonumber \\
&\quad +\int_{Q}\frac{|v(x)-v(y)|^{p-2}(v(x)-v(y))}{|x-y|^{ N+sp}}
\Big(\frac{v^p(x)}{v^{p-1}(x)}-\frac{v^p(y)}{v^{p-1}(y)}\Big) dx\,dy \nonumber \\
&\quad -\int_{Q}\frac{|v(x)-v(y)|^{p-2}(v(x)-v(y))}{|x-y|^{ N+sp}}
\Big(\frac{u^p(x)}{v_n^{p-1}(x)}-\frac{u^p(y)}{v_n^{p-1}(y)}\Big) dx\,dy \nonumber \\
&=\int_\Omega\left(f(x,u)-g(x,u)\right)(u-\tilde u_n)\,dx
+\int_{\Omega}\left(f(x,v)-g(x,v)\right)(v-\tilde v_n)\,dx. \label{Peq7}
\end{align}
Also using the Monotone convergence theorem
we estimate the right-hand side of \eqref{Peq7} for large $n$ as follows.
\begin{align*} %\label{Peq8}
&\int_\Omega\left(f(x,u)-g(x,u)\right)(u-\tilde u_n)\,dx
 +\int_{\Omega}\left(f(x,v)-g(x,v)\right)(v-\tilde v_n)\,dx \\
&=\int_\Omega\left(f(x,u)-g(x,u)\right)u\,dx
 -\int_\Omega\left(f(x,u)-g(x,u)\right)\frac{v^p}{(u+\frac{1}{n})^{p-1}}dx \\
&\quad +\int_\Omega\left(f(x,v)-g(x,v)\right)v\,dx
 -\int_\Omega\left(f(x,v)-g(x,v)\right)\frac{u^p}{(v+\frac{1}{n})^{p-1}}dx\\
&\quad +o_n(1) \\
&=\int_\Omega\left(f(x,u)-g(x,u)\right)u\,dx
 -\int_\Omega\left(f(x,u)-g(x,u)\right)\frac{v^p}{u^{p-1}}dx \\
&\quad +\int_\Omega\left(f(x,v)-g(x,v)\right)v\,dx
 -\int_\Omega\left(f(x,v)-g(x,v)\right)\frac{u^p}{v^{p-1}}dx+o_n(1) \\
&=\int_\Omega\Big(\frac{f(x,u)-g(x,u)}{u^{p-1}}
 -\frac{f(x,v)-g(x,v)}{v^{p-1}}\Big)(u^p-v^p)\,dx+o_n(1) \\
 &\leq o_n(1).
\end{align*}
Thus from this inequality and \eqref{Peq7}, and passing to the limit as
$n\to\infty$ together with $u/v,v/u\in L^\infty(\Omega)$, we infer that

\begin{equation*}
\int_Q(L(u,v)+L(v,u)) dx=0.
\end{equation*}
Using Lemma \ref{Picone identity} this implies $ku(x)=v(x)$ for a.e.
$x\in\Omega$ for some $k>0$.
Assume that $k\neq 1$. Then, without loss of generality, we can take $k<1$.
Therefore, using (A3),
\begin{align*}
(-\Delta)^s_p(ku)
&=k^{p-1}(-\Delta)^s_pu=k^{p-1}(f(x,u)-g(x,u)) \\
&<f(x,ku)-g(x,ku)=(-\Delta)^s_p(v)
\end{align*}
from which we obtain a contradiction.
Hence $k=1$ and $u=v$.
\end{proof}

Now we proceed to prove Theorem \ref{sol S_T}.
First, we consider the following stationary problem.
\begin{equation}\label{eE}
\begin{gathered}
u+\lambda(-\Delta )^s_{p}u = \tilde g\quad\text{in }\Omega \\
u = 0 \quad\text{in }\mathbb{R}^{ N}\setminus\Omega
\end{gathered}
\end{equation}
where $\lambda>0$ and $\tilde g\in L^\infty(\Omega)$. We have the following
existence result for the problem \eqref{eE}.

\begin{lemma}\label{sol of P}
For any $\lambda>0$, \eqref{eE} admits a unique weak solution $u$ in the
sense that $u\in X_0(\Omega)$ satisfies
$$
\int_\Omega u\varphi\,dx
+\lambda\int_{Q} \frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{ N+sp}}
(\varphi(x)-\varphi(y))\,dx\,dy
=\int_\Omega \tilde g\varphi\,dx,
$$
for all $\varphi \in X_0(\Omega)$.
Moreover, $u\in C_0(\overline\Omega)$.
\end{lemma}

\begin{proof}
The proof follows using the similar arguments as above.
Precisely, for the existence of a weak solution we can argue as in the
proof of Lemma~\ref{Existence P}.
From the weak comparison principle, we obtain
$\|u\|_{L^\infty(\Omega)}\leq \|\tilde g\|_{L^\infty(\Omega)}$ and
from Theorem~\ref{C-alpha regularity}, $u\in C_0(\overline\Omega)$.
The uniqueness of the weak solution is a consequence
of the monotonicity of the operator $(-\Delta)^s_p$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{sol S_T}]
Let $\mathcal{N}\in\mathbb N$ and $T>0$. We set
$\Delta_t=\frac{T}{\mathcal{N}}$.
For $0\leq n\leq \mathcal{N}$, we define $t_n=n\Delta_t$.
We perform the proof along four steps.
\smallskip

\noindent\textbf{Step 1.} Approximation of $h$.
For $n\in\{1,\dots,\mathcal{N}\}$, we define for $t\in[t_{n-1}, t_n)$ and
 $x \in \Omega$,
$$
h_{\Delta_t}(t,x)=h^n(x):=\frac{1}{\Delta_t}\int_{t_{n-1}}^{t_n} h(s,x)ds.$$
Then by Jensen's Inequality for any $1<q<\infty$,
\begin{align*}
\|h_{\Delta_{t}}\|_{L^q(Q_T)}^q
&=\Delta_t \sum_{n=1}^\mathcal{N}\|h^n\|^q_{L^{q}}
 =\Delta_t \sum_{n=1}^\mathcal{N}\|\frac{1}{\Delta_t}
 \int_{t_{n-1}}^{t_n}h(s,x)ds\|^q_{L^q} \\
&\leq C(\Omega,T)\|h\|_{L^\infty(Q_T)}^q.
\end{align*}
Thus $h_{\Delta_t}\in L^q(Q_T)$. Also note that $h_{\Delta_t}\to h$ in $L^q(Q_T)$.
\smallskip

\noindent\textbf{Step 2.} Approximation of \eqref{ST}.
We define the  iterative scheme:
$u^0=u_0$ and for $1\leq n\leq \mathcal{N}$, $u^n$ is solution of
\begin{equation}\label{eq1}
\begin{gathered}
\frac{u^{n}-u^{n-1}}{\Delta_{t}}+(-\Delta)^s _{p}u^n = h^n
 \quad\text{in }\Omega, \\
u^n = 0 \quad\text{on }\mathbb{R}^ N\backslash\Omega.
\end{gathered}
\end{equation}
Note that the sequence $(u^n)_{n\in\{1,\dots,\mathcal{N}\}}$ is well defined.
Indeed, we apply Lemma \ref{sol of P} with $g=\Delta_t h^1+u^0\in
L^\infty(\Omega)$ to prove the existence of $u^1\in X_0(\Omega)\cap L^\infty(\Omega)$.
Inductively we obtain the existence of $(u^n)$, for any $n=2,\dots,\mathcal{N}$.
Defining the functions $u_{\Delta_t}$ and $\tilde u_{\Delta_t}$,
for $n=1,\dots,\mathcal{N}$ and $t\in[t_{n-1},t_n)$ as
\begin{equation}\label{uu}
u_{\Delta_t}(t)=u^n \quad \text{and}\quad \tilde u_{\Delta_t}(t)
=\frac{(t-t_{n-1})}{\Delta_t}(u^n-u^{n-1})+u^{n-1}
\end{equation}
we obtain
\begin{equation}\label{eq2}
 \frac{\partial \tilde u_{\Delta_t}}{\partial t} +(-\Delta)_{p}^su_{\Delta_t}=h_{\Delta_t} \quad\text{in }
Q_T.
\end{equation}
\smallskip

\noindent\textbf{Step 3.} A priori estimates for $u_{\Delta_t}$ and $\tilde u_{\Delta_t}$.
Multiplying the equation in \eqref{eq1} by $(u^n-u^{n-1})$ and summing from
$n=1$ to $N'\leq \mathcal{N}$, we obtain
\begin{equation} \label{eq3}
\begin{aligned}
&\sum_{n=1}^{N'}\Delta_t\int_\Omega\Big(\frac{u^n-u^{n-1}}{\Delta_t}\Big)^2dx \\
&+\sum_{n=1}^{N'}\int_{Q}\frac{|u^n(x)-u^n(y)|^{p-2}(u^n(x)-u^n(y))}{|x-y|^{ N+sp}}
\left((u^n-u^{n-1})(x)\right)\,dx\,dy \\
&= \sum_{n=1}^{N'}\int_\Omega h^n(u^n-u^{n-1})\,dx.
\end{aligned}
\end{equation}
Hence by Young's inequality and using the  convexity property
\begin{equation} \label{convexity}
\begin{aligned}
\frac{1}{p}(\|u^n\|^p_{X_0}-\|u^{n-1}\|^p_{X_0})
&\leq \frac{1}{2}\int_{Q}\frac{|u^n(x)-u^n(y)|^{p-2}(u^n(x)-u^n(y))}{|x-y|^{ N+sp}}\\
&\quad\times \left((u^n-u^{n-1})(x)-(u^n-u^{n-1})(y)\right)\,dx\,dy
\end{aligned}
\end{equation}
we obtain
\begin{align*} %\label{eq6}
&\frac{1}{2}\sum_{n=1}^{N'}\Delta_t\int_\Omega\Big(\frac{u^n-u^{n-1}}{\Delta_t}\Big)^2dx\\
&+\sum_{n=1}^{N'}\int_{Q} \frac{1}{p}\Big(\frac{|u^n(x)-u^n(y)|^{p}}{|x-y|^{ N+sp}}
 -\frac{|u^{n-1}(x)-u^{n-1}(y)|^{p}}{|x-y|^{ N+sp}}\Big)\,dx\,dy\\
&\leq \frac{C(\Omega,T)}{2}\|h\|^2_{L^\infty(Q_T)}.
\end{align*}
This implies that
\begin{gather}\label{b1}
 \Big(\frac{\partial\tilde u_{\Delta t}}{\partial t}\Big)_{\Delta t} \text{ is bounded in }
L^2(Q_T) \text{ uniformly in } \Delta_t, \\
\label{b2}
\begin{aligned}
&\text{$(u_{\Delta_t}) \text{ and }(\tilde u_{\Delta_t})$ are bounded in
 $L^\infty(0,T,X_0(\Omega))\cap L^\infty(Q_T)$}\\
&\text{and  uniformly in }\Delta_t.
\end{aligned}
\end{gather}
Furthermore, we have
\begin{equation}\label{eq7}
\|u_{\Delta_t}-\tilde u_{\Delta_t}\|_{L^\infty(0,T:L^2(\Omega))}
\leq  \max_{n=1,\dots,N}\|u^n-u^{n-1}\|_{L^2(\Omega)}
\leq C\Delta_t^{1/2}.
\end{equation}
Therefore for $\Delta_t\to 0$, there exist
$u,v\in L^\infty(0,T,X_0(\Omega))\cap L^\infty(Q_T)$ such that (up to a subsequence)
\begin{gather}\label{convweak*}
\tilde u_{\Delta_t}\stackrel{*}{\rightharpoonup} u \text{ in }L^\infty(0,T,X_0(\Omega)),\quad
u_{\Delta_t}\stackrel{*}{\rightharpoonup} v \text{ in }L^\infty(0,T,X_0(\Omega)), \\
\label{convL2}
\frac{\partial\tilde u_{\Delta_t}}{\partial t}\rightharpoonup \frac{\partial u}{\partial t}
\text { in }L^2(Q_T).
\end{gather}
 It follows from \eqref{eq7} that $u\equiv v$.
\smallskip


\noindent\textbf{Step 4.} $u$ satisfies \eqref{ST}.
 Plugging \eqref{b1}, \eqref{b2} and since $X_0(\Omega)\hookrightarrow L^2(\Omega)$
compactly, the Aubin-Simon's result
implies that $\{u_{\Delta_t}\}$ is compact in $C([0,T];L^2(\Omega))$.
Now using interpolation we obtain, up to a subsequence
\begin{equation}\label{new 1}
\tilde u_{\Delta_t}\to u\in C([0,T],L^q(\Omega)),\quad \text{for all } q>1.
\end{equation}
and hence, from \eqref{eq7}, we have
\begin{equation}\label{new 2}
u_{\Delta_t}\to u\in L^\infty([0,T],L^q(\Omega)),\quad \text{for all } q>1.
\end{equation}
Multiplying \eqref{eq2} by $(u_{\Delta_t}-u)$ we obtain
\begin{equation*}%\label{eq8}
\int_0^T\!\!\!\int_\Omega\frac{\partial\tilde u_{\Delta_t}}{\partial t}(u_{\Delta_t}-u)\,dx\,dt
+\int_0^T\langle(-\Delta)_{p}^su_{\Delta_t},u_{\Delta_t}-u\rangle\, dt
 =\int^T_0\int_\Omega h_{\Delta_t}(u_{\Delta_t}-u)\,dx\,dt.
\end{equation*}
Rearranging the terms in the above equation and using
\eqref{eq7}-\eqref{convweak*} we have
\begin{align*} %\label{eq9}
&\int_0^T\!\!\!\int_\Omega\Big(\frac{\partial\tilde u_{\Delta_t}}{\partial t}-\frac{\partial u}{\partial t}\Big)
(\tilde u_{\Delta_t}-u)\,dx\,dt \\
&+\int_0^T\langle(-\Delta)^s_{p}u_{\Delta_t}-(-\Delta)^s_{p}u,u_{\Delta_t}-u\rangle dt
=o_{\Delta_t}(1).
\end{align*}
Thus we obtain
\begin{equation*}%\label{eq10}
 \frac{1}{2}\int_\Omega|\tilde u_{\Delta_t}(T)-u(T)|^2
 dx+\int_0^T\langle(-\Delta)^s_{p}u_{\Delta_t}-(-\Delta)^s_{p}u,u_{\Delta_t}-u\rangle
=o_{\Delta_t}(1).
\end{equation*}
Using \eqref{new 1}, we obtain
\[
\int_0^T\langle(-\Delta)^s_{p}u_{\Delta_t}-(-\Delta)^s_{p}u,u_{\Delta_t}-u\rangle dt
=o_{\Delta_t}(1).
\]
This implies
\begin{align*}
&\int_0^T\int_{Q}
\Big(\big(|u_{\Delta_t}(x)-u_{\Delta_t}(y)|^{p-2}(u_{\Delta_t}(x)-u_{\Delta_t}(y))
-|u(x)-u(y)|^{p-2}(u(x)-u(y))\big)\\
&\times \big(u_{\Delta_t}(x)-u_{\Delta_t}(y)-u(x)
+u(y)\big)\Big) \frac{1}{|x-y|^{ N+sp}} \, dx\,dy\,dt
=o_{\Delta_t}(1).
\end{align*}
Thus by \eqref{A00}, for $p\geq2$ we conclude that
$$
\int_0^T\int_{Q}\frac{|u_{\Delta_t}(x)-u_{\Delta_t}(y)-u(x)+u(y)|^{p}}{|x-y|^{ N+sp}}
=o_{\Delta_t}(1).
$$
Also for $1<p\leq2$, \eqref{A00} together with the H\"older inequality in
$\mathbb{R}^2$ imply
$$
\int_0^T\int_{Q}\frac{|u_{\Delta_t}(x)-u_{\Delta_t}(y)|^{2}}
{\left(|u_{\Delta_t}(x)-u_{\Delta_t}(y)|^p+|u(x)-u(y)|^p\right)^{\frac{2-p}{p}}|x-y|^{ N+sp}}
=o_{\Delta_t}(1).
$$
Therefore using H\"older's inequality we obtain
\begin{align*}
0&\leq
\int_0^T\int_{Q}\frac{| (u_{\Delta_t}(x)-u_{\Delta_t}(y)-u(x)+u(y))|^{p}}{|x-y|^{ N+sp}}
 \,dx\,dy\,dt\\
&=\int_0^T\int_{Q}
\Big(\big| \big(u_{\Delta_t}(x)-u_{\Delta_t}(y)-u(x)+u(y)\big)\big|^{p} \\
&\quad \times\big(|u_{\Delta_t}(x)-u_{\Delta_t}(y)|^p+|u(x)-u(y)|^p\big)^{\frac{2-p}{2}}\Big) \\
&\quad \times\frac{1}
{\big(|u_{\Delta_t}(x)-u_{\Delta_t}(y)|^p+|u(x)-u(y)|^p\big)^{\frac{2-p}{2}}
|x-y|^{(N+sp)(\frac{p}{2}+\frac{2-p}{2})} } \,dx\,dy\,dt\\
&\leq \Big(\int_0^T\int_{Q}\frac{|u_{\Delta_t}(x)-u_{\Delta_t}(y)+u(y)|^{2}}
{\left(|u_{\Delta_t}(x)-u_{\Delta_t}(y)|^p+|u(x)-u(y)|^p\right)^{\frac{(2-p)}{p}}
|x-y|^{ N+sp}}\Big)^{p/2}\\
&\quad \times\Big(\int_0^T\int_{Q}\frac{\left(|u_{\Delta_t}(x)-u_{\Delta_t}(y)|^p
 +|u(x)-u(y)|^p\right)}{|x-y|^{ N+sp}}\Big)^{\frac{2-p}{2}}\\
&\leq \Big(\int_0^T\int_{Q}\frac{|u_{\Delta_t}(x)-u_{\Delta_t}(y)+u(y)|^{2}}
{\left(|u_{\Delta_t}(x)-u_{\Delta_t}(y)|^p+|u(x)-u(y)|^p\right)^{\frac{(2-p)}{p}}
|x-y|^{ N+sp}}\Big)^{p/2} \\
&\quad\times \Big(\|u_{\Delta_t}\|^p_{X_0(\Omega)}+\|u\|^p_{X_0(\Omega)}\Big)^{\frac{2-p}{2}}
=o_{\Delta_t}(1).
\end{align*}
Thus in both cases we have
$$
\int_0^T\int_{Q}\frac{| (u_{\Delta_t}(x)-u_{\Delta_t}(y)-u(x)+u(y))|^{p}}{|x-y|^{ N+sp}}
\,dx\,dy\,dt\to 0.
$$
This implies $u_{\Delta_t}$ converges to $ u$ in $L^{p}(0,T,X_0(\Omega))$.
Therefore, for $\phi\in C^\infty_0(Q)$,
\begin{align*}
&\int_0^T\int_{Q}\frac{|u_{\Delta_t}(x)-u_{\Delta_t}(y)|^{p-2}( u_{\Delta_t}(x)
-u_{\Delta_t}(y))(\phi(x)-\phi(y))\,dx\,dy\,dt}{|x-y|^{ N+sp}}\\
&\to\int_0^T\int_{Q}\frac{|u(x)-u(y)|^{p-2}( u(x)-u(y))(\phi(x)-\phi(y))
\,dx\,dy\,dt}{|x-y|^{ N+sp}}
\end{align*}
Hence, we conclude passing to the limit, in the distribution sense,
in equation \eqref{eq2} that $u$ is a weak solution of \eqref{ST}.
Also $u$ is the unique weak solution of \eqref{ST}.
Indeed, assume that there exists $v$ a weak solution of \eqref{ST}.
Then, we have for any arbitrary $t_0\in (0,T]$
$$
\int_0^{t_0}\int_\Omega \frac{\partial(u-v)}{\partial t}(u-v)(x,t)\,dx\,dt+
\int_0^{t_0}\langle(-\Delta)^s_pu-(-\Delta)^s_pv,u-v\rangle dt=0.
$$
 Since $(-\Delta)^s_p$ is monotone, this together with $u(0)=v(0)$ imply
\begin{align*}
\frac{1}{2}\int_\Omega(u(t_0)-v(t_0))^2dx
&=\int_0^{t_0}\frac{\partial}{\partial t}\int_\Omega\frac{1}{2}(u-v)^2dx\,dt\\
&=\int_{(0,t_0)\times\Omega}
\frac{\partial(u-v)}{\partial t}(u-v)\,dx\,dt\leq 0
\end{align*}
from which it follows that $u\equiv v$.
Next we claim that $u\in C([0,T];X_0(\Omega))$ and satisfies
 \eqref{equality of sol S_T}.
Using \eqref{new 1} and the compact embedding of $X_0(\Omega)$ into $L^p(\Omega)$,
it is easy to check that $u(\cdot,t)\in X_0(\Omega)$ and the map
$[0,T]\ni t\to u(\cdot,t) \in X_0(\Omega)$,
is weakly continuous. Therefore,
$\|u(\cdot,t_0)\|_{X_0(\Omega)}\leq \lim\inf_{t\to t_0}\| u(\cdot,t_0)\|_{X_0(\Omega)}$.
Now multiplying \eqref{eq1} by $u^n-u^{n-1}$,
taking integration over $\mathbb{R}^{ N}$ both sides, summing from $1\leq n=N''$
to $N'\leq \mathcal{N}$, and using \eqref{convexity} we obtain
\begin{equation} \label{right conti1}
\begin{aligned}
 &\Delta_t \sum_{n=N''}^{n=N'}\Big(\frac{u^n-u^{n-1}}{\Delta_t}\Big)^2
+\frac{1}{p}\left(\|u^{N'}\|_{X_0(\Omega)}-\|u^{N''-1}\|_{X_0(\Omega)}\right) \\
 &\leq  \sum_{n=N''}^{n=N'}\Delta_t\int_\Omega h_{\Delta_t}
\Big(\frac{u^{n}-u^{n-1}}{\Delta_t}\Big)\,dx.
\end{aligned}
\end{equation}
For any $t\in[t_0,T]$, choose $N''$ and $N'$ such that $N''\Delta_t\to t$ and
$N'\Delta_t\to t_0$.
Then \eqref{right conti1} gives
\begin{equation}\label{right conti2}
\begin{aligned}
& \int_{t_0}^t\int_\Omega\Big(\frac{\partial u}{\partial t}\Big)^2dx\,dt
 +\frac{1}{p}\|u(\cdot,t)\|_{X_0(\Omega)} \\
&\leq \int_{t_0}^t\int_\Omega h\Big(\frac{\partial u}{\partial t}\Big)\,dx\,dt
 +\frac{1}{p}\|u(\cdot,t_0)\|_{X_0(\Omega)}.
\end{aligned}
\end{equation}
Now from the above inequality and \eqref{new 2} we infer that
$$
\lim\sup_{t\to t_0^+}\|u(\cdot,t)\|_{X_0(\Omega)}\leq \|u(\cdot,t_0)\|_{X_0(\Omega)},
$$
and hence the map $[0,T]\ni t\to u(\cdot,t) \in X_0(\Omega)$ is right continuous.
Now for proving the left continuity, take $0<k\leq t-t_0$ and multiply \eqref{ST}
by $\tau_k(u)(s)=\frac{u(x,s+k)-u(x,s)}{k}$ and
integrate over $(t_0,t)\times \mathbb{R}^{ N}$. Using \eqref{convexity}, we obtain
\begin{equation}\label{leftconti1}
\begin{split}
&\int_{t_0}^t\int_\Omega \tau_k(u)\frac{\partial u}{\partial t}\,dxd\theta+\frac{1}{pk}\int_{t_0}^{t}\|u(\theta+k)\|_{X_0(\Omega)}^p-\|u(\theta)\|_{X_0(\Omega)}^p d\theta\\
&\geq \int_{t_0}^t\int_\Omega\tau_k(u)h\,dxd\theta.
\end{split}
\end{equation}
It follows that
\begin{equation}\label{leftconti1b}
\begin{split}
&\int_{t_0}^t\int_\Omega \tau_k(u)\frac{\partial u}{\partial t}\,dxd\theta
+\frac{1}{pk}(\int_{t}^{t+k}\|u(\theta)\|_{X_0(\Omega)}^p d\theta
-\int_{t_0}^{t_0+k}\|u(\theta)\|_{X_0(\Omega)}^p d\theta)\\
&\geq \int_{t_0}^t\int_\Omega\tau_k(u)h\,dxd\theta.
\end{split}
\end{equation}
By the right continuity of $t\mapsto u(\cdot,t)$, as $k\to 0^+$, we have
\begin{gather*}
\frac{1}{pk}\int_{t}^{t+k}\|u(\theta)\|_{X_0(\Omega)}^p d\theta
 \to\frac{1}{p}\|u(t)\|_{X_0(\Omega)}^p,\\
\frac{1}{pk}\int_{t_0}^{t_0+k}\|u(\theta)\|_{X_0(\Omega)}^p d\theta
 \to\frac{1}{p}\|u(t_0)\|_{X_0(\Omega)}^p.
\end{gather*}
Hence as $k\to 0^+$, \eqref{leftconti1b} becomes
\begin{equation}\label{leftcont2}
\int_{t_0}^t\int_\Omega \Big(\frac{\partial u}{\partial t}\Big)^2\,dxd\theta
+\frac{1}{p}\|u(\cdot,t)\|^p_{X_0(\Omega)}
\geq \int_{t_0}^t\int_\Omega h\frac{\partial u}{\partial t}\,dx\,ds
+\frac{1}{p}\|u(\cdot,t_0)\|^p_{X_0(\Omega)}.
\end{equation}
From the above inequality, we deduce that we have the equality in
\eqref{right conti2} and hence the claim.
This completes the proof of the Theorem \ref{sol S_T}.
\end{proof}

\begin{proof}[Proof of Theorem \ref{Existence of sol P}]
First we show that there exists a sub-solution $\underbar u$ and a super solution
$\bar u$ of \eqref{eP} such that $\underbar u,\bar u\in C^+_{d^s(\Omega)}$.
Since $f$ and $g$ satisfy $(A4)$ and using the fact that
$\phi_{1,s,p}\in C_{d^s}(\Omega)$, we can choose $\epsilon>0$ small enough such that
\begin{equation}\label{sub-sol}
(-\Delta)_p^s(\epsilon\phi_{1,s,p})
=\lambda_{1,s,p}\epsilon^{p-1}\phi_{1,s,p}^{p-1}\leq f(x,\epsilon\phi_{1,s,p})-g(x,\epsilon\phi_{1,s,p}).
\end{equation}
and $\epsilon\phi_{1,s,p}\leq u_0$.
Also let
$w$ be the solution of the following problem
\begin{gather*}%\label{(E)}
(-\Delta )^s_{p}w = \beta w^{p-1}+C\quad\text{in } \Omega \\
w  > 0 \quad\text{in } \Omega\\
w  = 0 \quad\text{in } \mathbb{R}^{ N}\setminus\Omega,
\end{gather*}
where $ \limsup_{\theta\to\infty}\frac{f(x,\theta)}{\theta^{p-1}}
\leq\beta<\lambda_{1,s,p}$ and $C>0$. Then arguing as in the proof of
Proposition \ref{L-infinity bound for P}, we obtain $w\in L^\infty(\Omega)$ and hence
by \cite[Theorem 1.1]{IMS},
$w\in C^\alpha(\overline\Omega)$ with $\alpha\in(0,s]$. Furthermore from
\cite[Theorem 4.4]{IMS} we have for some constant $C_0>0$,
$|w(x)|\leq C_0 d^s(x)$ a.e. $x\in\Omega$. Then, $w\in C_{d^s}(\Omega)$ and from the
Hopf lemma (see Lemma~\ref{Hopf lemma}), we obtain $w\in C^+_{d^s}(\Omega)$.
Again using the fact that $f$ and $g$ satisfy $(A4)$, we have that for some
constant $C'>0$,
\begin{equation*}
f(x,\theta)-g(x,\theta)\leq\beta\theta+C'.
\end{equation*}
Then for $M>0$ large enough, we obtain
 \begin{equation}\label{sup-sol}
(-\Delta)_p^s(Mw)=\beta (Mw)^{p-1}+CM^{p-1}\geq f(x,Mw)-g(x,Mw)
\end{equation}
and $u_0\leq Mw$. Then $\underbar u:=\epsilon\phi_{1,s,p}$ and $\bar u:=Mw$
are the required sub-solution and the super-solution of \eqref{eP}, respectively,
such that $\underbar u,\bar u\in C^+_{{d^s(\Omega)}}$.
We define the sequence $(u^n)$ by the  iterative scheme:
$u^0=u_0$ and
\begin{gather*}
u^{n}+(-\Delta)^s_{p}u^n +Ku^n= u^{n-1}+f(x, u^{n-1})-g(x,u^{n-1})+Ku^{n-1}
\quad\text{in }\Omega, \\
u^n = 0 \quad\text{on }\mathbb{R}^{ N}\setminus\Omega.
\end{gather*}
where $K>0$ is chosen such that the map $t\mapsto Kt+f(x,t)-g(x,t)$
is nondecreasing in $[0,\|\bar u\|_{X_0}]$, for a.e. $x\in\Omega$.
Then the existence of a weak solution $u_\infty\in [\underline{u},\overline{u}]$
to \eqref{eP} is obtained by the standard arguments of the monotone iteration method.
Also we have $\underbar u\leq u_\infty\leq\bar u$ in $\Omega$ and
$u_\infty\in C^+_{d^s}(\Omega)$.
The uniqueness of the solution to \eqref{eP} follows from
Theorem~\ref{uniqueness of sol of P}.
\end{proof}

\begin{proof}[Proof of Theorem \ref{sol P_T}]
Now we proceed as in the proof of Theorem \ref{sol S_T}.
Set $\Delta_t:=\frac{T}{\mathcal{N}}$, $\mathcal{N}\in\mathbb N$
and let $\underbar u$ and $\bar u$
be as defined in the proof of Theorem \ref{Existence of sol P}.
We define the sequence $\{u^n\}\in X_0(\Omega)$ as the solutions
to the  iterative scheme: $u^0 = u_0$ and
\[
u^{n}+\Delta_t((-\Delta)^s_{p}u^n+Ku^n)
= u^{k-1}+\Delta_t(f(x, u^{n-1})-g(x,u^{n-1})+Ku^{n-1})\quad\text{in }\Omega,
\]
The existence of $u^n\in C^+_{d^s}(\Omega)$, for any $n\geq 1$ follows
from Lemma~\ref{sol of P}
 and the Hopf Lemma. Note that from Theorem \ref{Existence of sol P},
we have $\underbar u\leq u_0\leq \bar u$, a.e. in $\Omega$.
We claim that $\underbar u\leq u^k\leq\bar u$. Indeed for $k=1$, we have
\[
\underline u-u^1+\Delta_t((-\Delta)_{p}^s\underline u-(-\Delta)^s_{p}u^1)
\leq \underbar u-u^0+\Delta_t(f(x,\underbar u)-f(x,u^0)-(g(x,\underbar u)-g(x,u^0))).
\]
Therefore,
\[
\underline u-u^1+\Delta_t((-\Delta)_{p}^s\underline u-(-\Delta)^s_{p}u^1
+K(\underline{u}-u^1))\leq 0.
\]
Thus by comparison principle given in \cite[Theorem 2.10]{IMS},
we have $\underline u\leq u^1$. Similarly we prove $u^1\leq \bar u$.
The rest of the claim follows by induction. Now we define
$u_{\Delta_t}$ and $\tilde u_{\Delta_t}$ as in the proof of Theorem \ref{sol S_T},
and
$$
h_{\Delta_t}(t,x):=f(x, u_{\Delta_t}(t-\Delta_t,x))-g(x, u_{\Delta_t}(t-\Delta_t,x)).
$$
Then clearly as $\underbar u\leq u_{\Delta_t}\leq\bar u$,
$h_{\Delta_t}(t,x)\in L^\infty(Q)$.

Therefore following a similar arguments as in the proof of Theorem \ref{sol S_T},
for $\Delta_t\to 0$, there exist $u\in L^\infty(0,T,X_0(\Omega))$ such that
(up to a subsequence)
\begin{gather}\label{P_Tconvweak*}
\tilde u_{\Delta_t},u_{\Delta_t}\stackrel{*}{\rightharpoonup} u \quad
\text{in }L^\infty(0,T,X_0(\Omega))\text{ and }L^\infty(Q_T),\\
\label{P_TconvL2}
\frac{\partial\tilde u_{\Delta_t}}{\partial t}\rightharpoonup \frac{\partial u}{\partial t}\quad
\text {in }L^2(Q_T).
\end{gather}
Again using a similar arguments as in the proof of Theorem \ref{sol S_T},
we have
\begin{equation}\label{P_Tnew 1}
\tilde u_{\Delta_t}\to u\in C([0,T],L^q(\Omega))\;\text{ and }
\tilde u_{\Delta_t}\to u\in L^\infty([0,T],L^q(\Omega)),
\end{equation}
 for all $q>1$.
Also using the Lipschitz continuity of $f$ and $g$ we have
\begin{equation} \label{P_Tnew 2}
\begin{aligned}
&\|h_{\Delta_t}(\cdot,t)-(f-g)(\cdot,u(\cdot,t))\|_{L^2(\Omega)} \\
&=\|(f-g)(\cdot,u_{\Delta_t}(\cdot,t-\Delta_t))-(f-g)(\cdot,u(\cdot,t))\|_{L^2(\Omega)} \\
&\leq C \|u_{\Delta_t}(\cdot,t-\Delta_t)-u(\cdot,t)\|_{L^2(\Omega)}\
 \end{aligned}
\end{equation}
Thus \eqref{P_Tnew 1}-\eqref{P_Tnew 2}, we deduce that
$h_{\Delta_t}(x,t)\to f(x,u(x))$ in $L^\infty(0,T;L^2(\Omega))$.
The rest of the proof follows using step 4 of the Theorem \ref{sol S_T}.
\end{proof}

Now we study the regularity of the solutions of \eqref{ST} and \eqref{PT}
given in Theorem \ref{regularity S_T} and Theorem \ref{regularity P_T}.


\begin{proof}[Proof of Lemma \ref{accretive lemma}]
 Let $h_1,h_2\in L^\infty(\Omega)$ and $u,v\in X_0(\Omega)$, respectively,
be the solutions to
\begin{gather*}%\label{m-accretivity}
u+(-\Delta)^s_p(u)=h_1~\quad\text{in }\Omega, \\
v+(-\Delta)^s_p(v)=h_2~\quad\text{in }\Omega
\end{gather*}
For $w\in L^\infty(\Omega)$, define $w^+(x)=\max\{w(x),0\}$.
Setting
$$
\Omega_+=\{x\in\Omega:(u-v-\|h_1-h_2\|_{L^\infty(\Omega)})^+(x)>0\},
$$
and noting that for $x\in\Omega_+$ and
$y\in\mathbb{R}^{ N}\setminus\Omega_+$, $u(x)-u(y)\ge v(x)-v(y)$,
 we obtain
\begin{align*}
 &\langle(-\Delta)^s_pu-(-\Delta)_p^sv,(u-v-\|h_1-h_2\|_{L^\infty(\Omega)})^+\rangle \\
 &=\int_{\Omega_+}\Big(\int_{\mathbb{R}^{ N}}
\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{ N+sp}}dy\Big)
 (u-v-\|h_1-h_2\|_{L^\infty(\Omega)})(x)\,dx \\
 &\quad -\int_{\Omega_+}\Big(\int_{\mathbb{R}^{ N}}
\frac{|v(x)-v(y)|^{p-2}(v(x)-v(y))}{|x-y|^{ N+sp}}dy\Big)
 (u-v-\|h_1-h_2\|_{L^\infty(\Omega)})(x)\,dx \\
 &=\int_{\Omega_+}\Big(\int_{\Omega_+}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}
{|x-y|^{ N+sp}}dy\Big)\\
&\quad\times  (u-v-\|h_1-h_2\|_{L^\infty(\Omega)})(x)\,dx \\
&\quad +\int_{\Omega_+}\Big(\int_{\mathbb{R}^{ N}\setminus\Omega_+}
 \frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{ N+sp}}dy\Big)\\
&\quad\times (u-v-\|h_1-h_2\|_{L^\infty(\Omega)})(x)\,dx \\
 &\quad -\int_{\Omega_+}\Big(\int_{\Omega_+}\frac{|v(x)-v(y)|^{p-2}(v(x)-v(y))}
 {|x-y|^{ N+sp}}dy\Big) \\
&\quad (u-v-\|h_1-h_2\|_{L^\infty(\Omega)})(x)\,dx \\
 &\quad -\int_{\Omega_+}\Big(\int_{\mathbb{R}^{ N}\setminus\Omega_+}
 \frac{|v(x)-v(y)|^{p-2}(v(x)-v(y))}{|x-y|^{ N+sp}}dy\Big) \\
&\quad\times (u-v-\|h_1-h_2\|_{L^\infty(\Omega)})(x)\,dx \\
 &\geq\int_{\Omega_+}\Big(\int_{\Omega_+}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}
{|x-y|^{ N+sp}}dy\Big)
 (u-v-\|h_1-h_2\|_{L^\infty(\Omega)})(x)\,dx \\
 &\quad -\int_{\Omega_+}\Big(\int_{\Omega_+}\frac{|v(x)-v(y)|^{p-2}(v(x)-v(y))}
{|x-y|^{ N+sp}}dy\Big)
 (u-v-\|h_1-h_2\|_{L^\infty(\Omega)})(x)\,dx \\
 &=\int_{\Omega_+}\int_{\Omega_+}
\Big(\big(|u(x)-u(y)|^{p-2}(u(x)-u(y))-|v(x)-v(y)|^{p-2}(v(x)-v(y))\big)\\
&\quad\times \big((u-v)(x)-(u-v)(y)\big)\Big)
\frac{1}{2|x-y|^{ N+sp}} \,dy\,dx
 \geq 0.
\end{align*}
This implies the $m$-accretivity of $A$ in $L^\infty(\Omega)$.
\end{proof}

Now Theorem \ref{regularity S_T} and Theorem \ref{regularity P_T}
follow using the approach as in \cite[Theorem 4.2 and 4.4]{Barbu1}.
Next we prove the asymptotic behavior of the solution of \eqref{PT}
as given in the Theorem \ref{stabilization}.

\begin{proof}[Proof of Theorem \ref{stabilization}]
 Let $\underbar u$ and $\bar u$ be the sub and super solutions respectively to
 \eqref{eP} as constructed in the proof of the Theorem \ref{Existence of sol P}
such that $\underbar u\leq u_0\leq\bar u$.
 Let $u_1$ and $u_2$ be the unique and global solution to \eqref{PT}
 with the initial data $\underline u$ and $\bar u$ respectively.
 Note that using the approach as in proof
 of \cite[Theorem 0.15]{BBG} we have
$\underbar u,\bar u\in \overline{\mathcal{D}(A)}^{L^\infty(\Omega)}$ and
 $\underline{u}\leq u_1(t)\leq u(t)\leq u_2(t)\leq \overline{u}$ and
$t\mapsto u_1(t)$  ($t\mapsto u_2(t)$) is non-decreasing
(non-increasing respectively) and converges a.e. to $u_1^\infty$ ($u_2^\infty$
respectively), as $t\to\infty$.
 Now from the semi-group theory we have
\begin{gather*}
u_1^\infty= \lim_{t'\to\infty}S(t'+t)(\underbar u)
=S(t)\big( \lim_{t'\to\infty}S(t')\underbar u\big)=S(t)u_1^\infty\\
u_2^\infty= \lim_{t'\to\infty}S(t'+t)(\bar u)=S(t)
\big( \lim_{t'\to\infty}S(t')\bar u\big)=S(t)u_2^\infty
\end{gather*}
 where $S(t)$ is the semi-group on $L^\infty(\Omega)$ generated by the given
evolution equation. This implies that $u_1^\infty$ and $u^\infty_2$ are
the stationary solutions  to \eqref{PT}.
From the uniqueness of the solution in Theorem \ref{Existence of sol P}
we obtain that $u_1^\infty=u_\infty=u_2^\infty$.
 Thus $u(t)\to u_\infty$ in $L^\infty(\mathbb{R}^ N)$.
\end{proof}

\subsection*{Acknowledgements} 
The authors were partially funded by IFCAM (Indo-French Centre for
 Applied Mathematics) UMI CNRS 3494 under the project 
``Singular phenomena in reaction diffusion equations and in conservation laws''.

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\end{thebibliography}

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