\documentclass[reqno]{amsart}
\usepackage{hyperref}
\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 45, pp. 1--5.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2016/45\hfil Non-extinction of solutions]
{Non-extinction of solutions to a fast diffusion system
with nonlocal sources}
\author[Haixia Li, Yuzhu Han \hfil EJDE-2016/45\hfilneg]
{Haixia Li, Yuzhu Han}
\address{Haixia Li \newline
School of Mathematics,
Changchun Normal University,
Changchun 130032, China}
\email{lihaixia0611@126.com}
\address{Yuzhu Han (corresponding author) \newline
School of Mathematics, Jilin University,
Changchun 130012, China}
\email{yzhan@jlu.edu.cn}
\thanks{Submitted October 28, 2015. Published February 10, 2016.}
\subjclass[2010]{35K40, 35K51}
\keywords{Fast diffusion system; nonlocal source; non-extinction}
\begin{abstract}
In this short article, we give a positive answer to the problem
proposed by Zheng et al \cite{Zheng2015}, and show that the fast diffusion system
\begin{gather*}
u_t=\operatorname{div}(|\nabla u|^{p-2}\nabla u)
+\int_\Omega v^\alpha\mathrm{d}x, \\
v_t =\operatorname{div}(|\nabla v|^{q-2}\nabla v)
+\int_\Omega u^\beta\mathrm{d}x
\end{gather*}
under homogeneous Dirichlet boundary condition admits at least one
non-extinction solution when $\alpha\beta<(p-1)(q-1)$ and the initial
data are strictly positive.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks
\section{Introduction}
This short note concerns the non-extinction properties of solutions
to the fast diffusion parabolic system
\begin{equation}\label{1.1}
\begin{gathered}
u_t=\operatorname{div}(|\nabla u|^{p-2}\nabla u)
+\int_\Omega v^\alpha\mathrm{d}x,\quad x\in\Omega,\; t>0,\\
v_t=\operatorname{div}(|\nabla v|^{q-2}\nabla v)
+\int_\Omega u^\beta\mathrm{d}x,\quad x\in\Omega,\; t>0,\\
u(x,t)=v(x,t)=0,\quad x\in\partial\Omega,\; t>0,\\
u(x,0)=u_0(x),\quad v(x,0)=v_0(x),\quad x\in\Omega,
\end{gathered}
\end{equation}
where $1
0$, $\Omega$ is a bounded domain in
$\mathbb{R}^N$ ($N\geq1$) with smooth boundary $\partial\Omega$ and
the initial data $u_0\in L^\infty({\Omega})\cap W^{1,p}_0(\Omega)$,
$v_0\in L^\infty({\Omega})\cap W^{1,q}_0(\Omega)$.
We refer to \cite{Zheng2015} and the references therein for the motivation
of studying problem \eqref{1.1}.
In particular, the authors in \cite{Zheng2015} investigated the extinction
properties of solutions to the above problem.
More precisely, by combining the methods of energy estimates with the
comparison principle they showed that if
$\alpha\beta>(p-1)(q-1)$, then every weak solution of problem \eqref{1.1}
vanishes in finite time when the initial data
are comparable in some sense; if $\alpha\beta=(p-1)(q-1)$ and the
diameter of the domain $\Omega$ is sufficiently small,
then problem \eqref{1.1} admits at least one extinction solution for
small initial data. However, for the case $\alpha\beta<(p-1)(q-1)$,
they did not give any result and conjectured that problem \eqref{1.1}
should admit at least one non-extinction solution for any nonnegative initial
data. Since to give some sufficient conditions for the non-extinction of
solutions to systems like \eqref{1.1} is much
more challenging, one can not expect a full answer to this problem.
In this short note, we give a partial answer to the problem
proposed by Zheng et al.
It is well known that the equations in \eqref{1.1} are
singular when $1
0,\; x\in\Omega\},
$$
and
$$
S_2=\{u\in L^\infty({\Omega})\cap W^{1,q}_0(\Omega): u(x)\geq k\phi_2(x)
\text{ for some } k>0,\ x\in\Omega\}.
$$
Our main result reads as follows.
\begin{theorem}\label{main}
Assume that $1
0$ will be fixed later.
By direct computation we see that
$(\underline{u},\underline{v})$ satisfies (in the weak sense)
\begin{gather}\label{e4}
\underline{u}_t-\operatorname{div}(|\nabla \underline{u}|^{p-2}
\nabla \underline{u})-\int_\Omega\underline{v}^\alpha \,\mathrm{d}x
=k^{\theta_1(p-1)}-k^{\theta_2\alpha}\mu_2,\\
\label{e5}
\underline{v}_t-\operatorname{div}(|\nabla \underline{v}|^{q-2}
\nabla \underline{v})-\int_\Omega\underline{u}^\beta \,\mathrm{d}x
=k^{\theta_2(q-1)}-k^{\theta_1\beta}\mu_1.
\end{gather}
Combining \eqref{e4}, \eqref{e5} with \eqref{e3} we know that there
exists a constant $k_1>0$ such that
for all $k\in(0,k_1]$, the following relations hold
\begin{equation}\label{e6}
\begin{gathered}
\underline{u}_t-\operatorname{div}(|\nabla \underline{u}|^{p-2}
\nabla \underline{u})-\int_\Omega\underline{v}^\alpha\,\mathrm{d}x\leq0,
\quad x\in\Omega,\; t>0,\\
\underline{v}_t-\operatorname{div}(|\nabla \underline{v}|^{q-2}\nabla
\underline{v})-\int_\Omega\underline{u}^\beta \,\mathrm{d}x\leq0,
\quad x\in\Omega,\; t>0.
\end{gathered}
\end{equation}
On the other hand, since $(u_0,v_0)\in S_1\times S_2$, there exists a
constant $k_2>0$ such that for all $k\in(0,k_2]$ we have
\begin{equation}\label{e7}
u_0(x)\geq k^{\theta_1}\phi_1(x),\quad
v_0(x)\geq k^{\theta_2}\phi_2(x),\quad x\in\Omega.
\end{equation}
Therefore, from \eqref{e6} and \eqref{e7} we know that
$(\underline{u},\underline{v})$ is a non-extinction weak subsolution
of \eqref{1.1} for all $00,\\
v_t=\operatorname{div}(|\nabla v|^{q-2}\nabla v)
+\int_\Omega(u_++1)^\beta\,\mathrm{d}x,\quad x\in\Omega,\; t>0,\\
u(x,t)=v(x,t)=0,\quad x\in\partial\Omega,\quad t>0,\\
u(x,0)=u_0(x),\quad v(x,0)=v_0(x),\quad x\in\Omega.
\end{gathered}
\end{equation}
Here $s_+=\max\{s,0\}$. By applying the standard regularization and
a priori estimates methods (see \cite{Han} for instance)
we know that problem \eqref{a1} admits a weak solution
$(\overline{u},\overline{v})$. By the weak maximum principle it is known
that $(\overline{u},\overline{v})$ is nonnegative. Moreover,
$(\overline{u},\overline{v})$
exists globally and is locally bounded if $\alpha\beta\leq1$.
If we can show that
$(\underline{u},\underline{v})\leq(\overline{u},\overline{v})$,
then there exists a solution $(u,v)$ of \eqref{1.1} satisfying
$(\underline{u},\underline{v})\leq(u,v)\leq(\overline{u},\overline{v})$.
\smallskip
\noindent\textbf{Step 3.}
We will show that $(\underline{u},\underline{v})
\leq(\overline{u},\overline{v})$. For this, fix $T\in(0,\infty)$.
From the definition of weak super and subsolutions, we obtain, for any
$0\leq\varphi_1\in E_{p0}$ and $0\leq\varphi_2\in E_{q0}$,
\begin{gather}
\label{c1}
\begin{aligned}
&\iint_{Q_T}\Big(\frac{\partial \underline{u}}{\partial t}
-\frac{\partial \overline{u}}{\partial t}\Big)\varphi_1\,\mathrm{d}x\,
\mathrm{d}\tau
+\iint_{Q_T}(|\nabla \underline{u}|^{p-2}\nabla \underline{u}
-|\nabla \overline{u}|^{p-2}\nabla \overline{u})\nabla\varphi_1
\,\mathrm{d}x\,\mathrm{d}\tau \\
&\leq\iint_{Q_T}\int_\Omega\Big[\underline{v}^\alpha(y,\tau)
-(\overline{v}_+(y,\tau)+1)^\alpha\Big]\,\mathrm{d}y\varphi_1\,\mathrm{d}x
\,\mathrm{d}\tau,
\end{aligned}\\
\label{c2}
\begin{aligned}
&\iint_{Q_T}\Big(\frac{\partial \underline{v}}{\partial t}
-\frac{\partial \overline{v}}{\partial t}\Big)\varphi_2\,\mathrm{d}x
\,\mathrm{d}\tau
+\iint_{Q_T}(|\nabla \underline{v}|^{q-2}\nabla \underline{v}
-|\nabla \overline{v}|^{q-2}\nabla \overline{v})\nabla\varphi_2\
,\mathrm{d}x\,\mathrm{d}\tau \\
&\leq\iint_{Q_T}\int_\Omega\Big[\underline{u}^\beta(y,\tau)
-(\overline{u}_+(y,\tau)+1)^\beta\Big]\,\mathrm{d}y\varphi_2
\,\mathrm{d}x\,\mathrm{d}\tau.
\end{aligned}
\end{gather}
By Lagrange mean value theorem we know that if $0<\alpha<1$,
then there exists a $\xi$ between $\underline{v}$ and $\overline{v}_++1$
such that
\begin{equation}\label{c3}
[\underline{v}^\alpha(y,\tau)-(\overline{v}_+(y,\tau)+1)^\alpha]_+
=\alpha\xi^{\alpha-1}[\underline{v}-(\overline{v}_++1)]_+
\leq\alpha(\underline{v}-\overline{v})_+;
\end{equation}
if $\alpha=1$, then
\begin{equation}\label{c4}
\underline{v}-(\overline{v}_++1)\leq(\underline{v}-\overline{v})_+;
\end{equation}
if $\alpha>1$, then
there exists an $\eta$ between $\underline{v}$ and $\overline{v}_++1$ such that
\begin{equation}\label{c5}
\begin{aligned}{}
[\underline{v}^\alpha(y,\tau)-(\overline{v}_+(y,\tau)+1)^\alpha]_+
&=\alpha\eta^{\alpha-1}[\underline{v}-(\overline{v}_++1)]_+\\
&\leq\alpha k^{\theta_2(\alpha-1)}M_2^{\alpha-1}(\underline{v}-\overline{v})_+.
\end{aligned}
\end{equation}
Noticing that both $\underline{u}$ and $\overline{u}$ belong to $E_p$
and $\underline{u}\leq0\leq \overline{u}$ on $\partial\Omega\times(0,T)$,
it is not hard to check that
$\varphi_1=\chi_{[0,t]}(\underline{u}-\overline{u})_+\in E_{p0}$
for any $t\in(0,T)$.
Taking $\varphi_1=\chi_{[0,t]}(\underline{u}-\overline{u})_+$
for any $t\in(0,T)$
and noticing \eqref{c3}-\eqref{c5}, we see by simple computation that
there exists a constant $C_1>0$ depending only on $\alpha$, $k$, $\theta_2$
and $M_2$ such that
\begin{align*}
&\int_{\Omega}(\underline{u}-\overline{u})^2_+\,\mathrm{d}x
+2\iint_{Q_t}(|\nabla \underline{u}|^{p-2}\nabla \underline{u}-|\nabla \overline{u}|^{p-2}\nabla \overline{u})\nabla(\underline{u}-\overline{u})_+\,\mathrm{d}x\,\mathrm{d}\tau \\
&\leq C_1\int_0^t\Big(\int_\Omega(\underline{v}-\overline{v})_+\,\mathrm{d}x
\int_\Omega(\underline{u}-\overline{u})_+\,\mathrm{d}x\Big)\,\mathrm{d}\tau\\
&\leq \frac{C_1|\Omega|}{2}\Big(\iint_{Q_t}(\underline{v}-\overline{v})^2_+
\,\mathrm{d}x\,\mathrm{d}\tau+\iint_{Q_t}(\underline{u}-\overline{u})^2_+
\,\mathrm{d}x\,\mathrm{d}\tau\Big).
\end{align*}
Symmetrically, we also have
\begin{align*}
&\int_{\Omega}(\underline{v}-\overline{v})^2_+\,\mathrm{d}x
+2\iint_{Q_t}(|\nabla \underline{v}|^{q-2}\nabla \underline{v}
-|\nabla \overline{v}|^{q-2}\nabla \overline{v})\nabla(\underline{v}
-\overline{v})_+\,\mathrm{d}x\,\mathrm{d}\tau \\
&\leq C_2\int_0^t\Big(\int_\Omega(\underline{v}-\overline{v})_+\,\mathrm{d}x
\int_\Omega(\underline{u}-\overline{u})_+\,\mathrm{d}x\Big)\,\mathrm{d}\tau\\
&\leq \frac{C_2|\Omega|}{2}\Big(\iint_{Q_t}(\underline{v}-\overline{v})^2_+
\,\mathrm{d}x\,\mathrm{d}\tau+\iint_{Q_t}(\underline{u}-\overline{u})^2_+
\,\mathrm{d}x\,\mathrm{d}\tau\Big),
\end{align*}
for some $C_2>0$ depending only on $\beta$, $k$, $\theta_1$ and $M_1$.
Noticing the monotonicity of $p$-Laplace operator we obtain that
$$
\int_{\Omega}[(\underline{u}-\overline{u})^2_++(\underline{v}
-\overline{v})^2_+]\,\mathrm{d}x
\leq C\iint_{Q_t}[(\underline{u}-\overline{u})^2_++(\underline{v}
-\overline{v})^2_+]\,\mathrm{d}x\,\mathrm{d}\tau.
$$
Thus, the desired result follow from the above inequality and Gronwall's
inequality.
\smallskip
\noindent\textbf{Step 4.}
Define $(u_1,v_1)=(\underline{u},\underline{v})$ and $\{(u_k,v_k)\}_{k\geq2}$
iteratively to be a solution of the following problem
\begin{equation}\label{3.23}
\begin{gathered}
u_{kt}=\operatorname{div}(|\nabla u_k|^{p-2}\nabla u_k)
+\int_\Omega v_{k-1}^\alpha\,\mathrm{d}x,\quad x\in\Omega,\; t>0,\\
v_{kt}=\operatorname{div}(|\nabla v_k|^{p-2}\nabla v_k)
+\int_\Omega u_{k-1}^\beta\,\mathrm{d}x,\quad x\in\Omega,\; t>0,\\
u(x,t)=v(x,t)=0,\quad x\in\partial\Omega,\; t>0,\\
u(x,0)=u_0(x),\quad v(x,0)=v_0(x),\quad x\in\Omega.
\end{gathered}
\end{equation}
By induction we can prove that $(u_k,v_k)\leq(u_{k+1},v_{k+1})$ and
$(u_k,v_k)\leq(\overline{u},\overline{v})$ for all $k\geq1$.
Thus the limits $u(x,t)=\lim_{k\rightarrow\infty}u_k(x,t)$ and
$v(x,t)=\lim_{k\rightarrow\infty}v_k(x,t)$ exist for every $x\in\Omega$
and $t>0$ and it is not hard to show that $(u,v)$ is a weak solution of
\eqref{1.1} by the regularities of $\{(u_k,v_k)\}_{k\geq2}$.
Therefore, $(u,v)$ is a non-extinction solution of \eqref{1.1} since
$(u,v)\geq(\underline{u},\underline{v})$. The proof is complete.
\end{proof}
\subsection*{Acknowledgments}
The first author is supported by Natural Science Foundation of Changchun
Normal University.
The second author is supported by NSFC (11271154, 11401252),
by Science and Technology Development Project of Jilin Province
(20150201058NY) and by the scientific research project of The
Education Department of Jilin Province (2015-463).
The authors highly appreciate the referees' valuable comments and
suggestions which improve the original manuscript.
They would also like to express their sincere gratitude to Professor
Wenjie Gao for his enthusiastic guidance and constant encouragement.
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\end{document}