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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 263, pp. 1--13.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/263\hfil Critical exponent and blow-up rate]
{Critical exponent and blow-up rate for the $\omega$-diffusion
  equations on graphs with Dirichlet boundary conditions}

\author[W. Zhou, M. Chen, W. Liu \hfil EJDE-2014/263\hfilneg]
{Weican Zhou, Miaomiao Chen, Wenjun Liu}  % in alphabetical order

\address{Weican Zhou \newline
College of Mathematics and Statistics,
Nanjing University of Information Science and Technology,
Nanjing 210044, China}
\email{000496@nuist.edu.cn}

\address{Miaomiao Chen \newline
College of Mathematics and Statistics,
Nanjing University of Information Science and Technology,
Nanjing 210044, China}
\email{mmchennuist@163.com} 

\address{Wenjun Liu (corresponding author)\newline
College of Mathematics and Statistics,
Nanjing University of Information Science and Technology,
Nanjing 210044, China}
\email{wjliu@nuist.edu.cn}

\thanks{Submitted July 10, 2014. Published December 22, 2014.}
\subjclass[2000]{35R02, 35B44,  35B51}
\keywords{$\omega$-diffusion equation; critical exponent; blow up; 
blow-up rate; graph}

\begin{abstract}
 In this article, we study the $\omega$-diffusion equation on a graph
 with Dirichlet boundary conditions
 \begin{gather*}
 u_t(x,t)=\Delta_{\omega}u(x,t)+e^{\beta t}u^{p}(x,t), \quad
 (x,t)\in S\times(0,\infty), \\
 u(x,t)=0, \quad (x,t)\in \partial S\times[0,\infty), \\
 u(x,0)=u_0(x)\geq0, \quad x\in V,
 \end{gather*}
 where  $\Delta_{\omega}$ is the discrete weighted Laplacian operator.
 First, we prove the  existence and uniqueness of the local solution
 via Banach fixed point theorem. Then, by the method of supersolutions
 and subsolutions we prove that the $\omega$-diffusion problem has a
 critical exponent $p_{\beta}$: when $p>p_{\beta}$, the solution becomes
 global; while when $1<p<p_{\beta}$, the solution blows up in finite time.
 Under appropriate hypotheses, we estimate the blow-up rate  in the
 $L^{\infty}$-norm. Some numerical experiments illustrate our results.
\end{abstract}

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

%\newcommand{\abs}[1]{\lvert#1\rvert}

\section{Introduction}\label{s1}

In this article we are concerned with the blow-up properties of the problem
\begin{equation}
\begin{gathered}
u_t(x,t)=\Delta_{\omega}u(x,t)+e^{\beta t}u^{p}(x,t), \quad
 (x,t)\in S\times(0,\infty), \\
u(x,t)=0, \quad (x,t)\in \partial S\times[0,\infty), \\
u(x,0)=u_0(x)\geq0, \quad x\in V,
\end{gathered}\label{1.1}
\end{equation}
where $p>1$, $\beta>0$,  $u_0(x) $ is nonnegative and nontrivial,
 and $V$ is the set of vertices of a graph $G(V,E,\omega)$.
 In general, we can split the set of vertexes $V$ into two disjoint subjects
$S$ and $\partial S$ such that $V=S\cup\partial S$, which are called the
interior and the boundary of $V$, respectively. We write $(x,y)\in E$
when two vertices $x, y$ are adjacent and connected by an edge.
Throughout this paper, all the graphs in our concern are assumed to be simple,
finite, connected, undirected and weighted. Besides, $\Delta_\omega u(x,t)$
is defined as
$$
\sum_{y\in V}\omega(x,y)\big(u(y,t)-u(x,t)\big),\quad \forall x\in V,
$$
where $\omega:V\times V\to \mathbb{R} $ denotes the weighted function,
which has the following properties:
\begin{itemize}
\item[(1)] $\omega(x,x)=0$, for any $x\in V$,

\item[(2)] $\omega(x,y)=\omega(y,x)$, for any $x, y\in V$,

\item[(3)] $\omega(x,y)=0$, if and only if $(x,y)\notin E$.
\end{itemize}

A function on a graph is understood as a function defined just on the set of 
vertices of the graph. The integration of a function $f: V\to \mathbb{R} $ 
on a graph $G $ is defined by
$$
\int_{V}f=\sum_{x\in V}f(x).
$$

As usual, the set $C\big(V\times(0,\infty)\big) $ consists of all function
 $u $ defined on $V\times (0,\infty) $ which satisfies $u(x,t)\in C(0,\infty) $ 
for each $x\in V$.

The $\omega$-diffusion equation of the form $u_t(x,t)=\Delta_{\omega}u(x,t) $
and its variations can be used to model diffusion process. 
Recently, more diffusion equations are taken into account to graphs 
\cite{cck2007,pkc2009,t2003}. In \cite{clc2011,lc2012,xml20114}, the extinction 
and positivity of the solution of the $\omega$-diffusion equation with 
absorption and its variations was considered. In \cite{pmd2013}, the decay 
rates for evolution equations of the averaging operator $\partial t-\Delta_{F} $ 
were studied. In \cite{f2011}, the quenching phenomena for a non-local diffusion 
equation with a singular absorption was discussed. Meanwhile, 
Xin et al \cite{xxm2013} considered the blow-up for the $\omega$-heat equation 
with Dirichlet boundary conditions and a reaction term $u^{p}(x,t)  $ on graphs
\begin{equation}
\begin{gathered}
u_t(x,t)=\Delta_{\omega}u(x,t)+u^{p}(x,t), \quad (x,t)\in S\times(0,\infty), \\
u(x,t)=0, \quad (x,t)\in \partial S\times(0,\infty), \\
u(x,0)=u_0(x), \quad  x\in V.
\end{gathered}\label{1.4}
\end{equation}
They proved that when $p>1$, the solution of problem \eqref{1.4} blows up
in finite time under some suitable conditions and when $p\leq1$, every solution
is global.

Blow-up phenomenon has been widely studied in various evolution equations 
\cite{cb2005,fl2012,kwz2012,l1973,lr2009,t2013,w2011,wcz2013,w2013,z2012,zmmz2013}.
 Meier \cite{m1990} considered the  problem
\begin{equation}
\begin{gathered}
u_t=\Delta u(x,t)+e^{\beta t}u^{p}(x,t), \quad (x,t)\in \Omega\times(0,\infty), \\
u(x,t)=0, \quad (x,t)\in \partial \Omega\times[0,\infty), \\
u(x,0)=u_0(x), \quad  x\in \Omega,
\end{gathered} \label{1.2}
\end{equation}
and proved that the critical exponent is $p_{\beta}=1+\frac{\beta}{\lambda_1}$,
where $\lambda_1 $ is the first Dirichlet eigenvalue of the Laplacian in $\Omega$.
Zhang and Wang \cite{zw2011} studied the nonlocal diffusion equations with
Dirichlet boundary condition
\begin{equation}
\begin{gathered}
u_t=\int_{\mathbb{R}^{N}}J(x-y)\left(u(y,t)-u(x,t)\right)\,\mathrm{d}y
+e^{\beta t}u^{p}(x,t), \quad (x,t)\in \Omega\times(0,\infty), \\
u(x,t)=0, \quad (x,t)\notin \Omega\times[0,\infty), \\
u(x,0)=u_0(x)\geq0, \quad  x\in \Omega,
\end{gathered}\label{1.3}
\end{equation}
where $p>1,\beta>0 $ and the kernel $J\in C^{1}(\mathbb{R}^{N}) $
satisfies $J\geq0 $ in $B_1$ (the unit ball); $J=0 $ in
$\mathbb{R}^{N}\setminus B_1 $ with $\int_{B_1}J(z)\,\mathrm{d}z=1$.
They showed that the critical exponent is coincident with that of \cite{m1990}.

Motivated by above research, we consider the critical exponent for 
problem \eqref{1.1}. We first study the local existence of the solution 
via Banach fixed point theorem. Then, we deal with the existence of the 
critical exponent of problem \eqref{1.1} by the method of supersolutions 
and subsolutions. Meanwhile, we prove that the nonnegative and nontrivial 
solution blows up in finite time and give the blow-up rate on $L^{\infty}$-norm. 
Finally, we take two examples to support our results.

This paper is organized as follows. In Section \ref{s2}, we consider the 
local existence of the solution. In Section \ref{s3}, we share many 
important properties of problem \eqref{1.1}, such as critical exponent 
or blow-up rate. In the end, we take two examples to check our results 
theoretically in Section \ref{s4}.

\section{Local existence of solution}\label{s2}

In this section we prove the existence of local a solution for problem \eqref{1.1} 
via Banach fixed point theorem.
We first define the following Banach space:
$$
X_{t_0}=\{u(x,t): u(x,t)\in C\left(V\times[0,t_0]\right); 
u(x,t)\equiv 0, \forall x\in \partial S\},
$$
with the norm
$$
\|u\|_{X_{t_0}}=\max_{t\in [0,t_0]}\max_{x\in S}|u(x,t)|, 
$$
where $t_0>0 $ is a fixed constant. And then, we consider the operator 
$D:X_{t_0}\to X_{t_0}$ for the fixed $v_0(x) $ which is defined on $V$
\[
D_{v_0}[v](x,t)=\begin{cases}
 v_0(x)+\int_0^{t}\Delta_\omega v(x,\tau)\,\mathrm{d}\tau
+\int_0^{t}e^{\beta t}v^{p}(x,\tau)\,\mathrm{d}\tau, & x\in S, \\
0, &  x\in \partial S.
\end{cases}
\]

In the following lemmas, we prove that the operator 
$D_{v_0}:X_{t_0}\to X_{t_0}$ is well defined and strictly contractive under 
some suitable conditions.

\begin{lemma}\label{lem1}
The operator $D_{v_0}$ is well defined, mapping $X_{t_0}$ to $X_{t_0}$. 
Moreover, let $u_0$, $v_0$ be defined on $V$ and $u, v\in X_{t_0}$, 
then, there exists a positive constant 
$C=C(p, \omega(x,y),\|u\|_{X_{t_0}},\|v\|_{X_{t_0}}$, $\beta,|V|)$ such that
\begin{equation}
\|D_{u_0}[u]-D_{v_0}[v]\|_{X_{t_0}}
\leq \max_{x\in V}|u_0(x)-v_0(x)|+Ct\|u-v\|_{X_{t_0}},
\label{2.1}
\end{equation}
where $|V| $ denotes the number of the nodes of the graph G.
\end{lemma}

\begin{proof} 
 We first show that the operator $D_{v_0}$ maps $X_{t_0}$ to $X_{t_0}$. 
On the one hand, for any $(x,t)\in S\times [0,t_0]$, we have
\begin{equation}
\begin{aligned}
|D_{v_0}[v](x,t)-D_{v_0}[v](x,0)|
&= \big|\int_0^{t}\Delta_{\omega}v(x,\tau)\,\mathrm{d}\tau
 +\int_0^{t}e^{\beta \tau}v^{p}(x,\tau)\,\mathrm{d}\tau\big| \\
&\leq \Big(2\max_{(x,y)\in E}|\omega(x,t)||V|\|v\|_{X_{t_0}}
+e^{\beta t_0}\|v\|_{X_{t_0}}^{p}\Big)t.
\end{aligned}\label{2.2}
\end{equation}
From this inequality, we know that the operator $D_{v_0}$ is continuous at $t=0$.
Similarly, for any $(x,t_1)$, $(x,t_2)\in S\times[0,t_0]$, we have
\begin{equation}
\begin{aligned}
&|D_{v_0}[v](x,t_1)-D_{v_0}[v](x,t_2)|\\
&= \big|\int_{t_1}^{t_2}\Delta_{\omega}v(x,\tau)\,\mathrm{d}\tau
 +\int_{t_1}^{t_2}e^{\beta \tau}v^{p}(x,\tau)\,\mathrm{d}\tau\big| \\
&\leq \Big(2\max_{(x,y)\in E}|\omega(x,t)||V|\|v\|_{X_{t_0}}
 +e^{\beta t_0}\|v\|_{X_{t_0}}^{p}\Big)|t_2-t_1|,
\end{aligned}\label{2.3}
\end{equation}
which shows that $D_{u_0} $ is continuous in time for any $t\in [0,t_0]$.
On the other hand, the convolution
$\sum_{y\in V}\omega(x,y)\left(u(y,t)-u(x,t)\right) $ is uniformly continuous.
So it is easy to see that the operator $D_{v_0}$ is continuous as the
function of $x$. Hence, the operator $D_{v_0}$ maps $X_{t_0}$ to $X_{t_0}$.

Next, we prove \eqref{2.1}. For any $(x,t)\in S\times[0,t_0]$, we have
\begin{equation}
\begin{aligned}
& |D_{u_0}[u](x,t)-D_{v_0}[v](x,t)| \\
&\leq \max_{x\in S}|u_0(x)-v_0(x)|
 +\int_0^{t}|\Delta_{\omega}(u(x,\tau)-v(x,\tau))|\,\mathrm{d}\tau\\
 &\quad +\int_0^{t}e^{\beta \tau}|u^{p}(x,\tau)-v^{p}(x,\tau)|\,\mathrm{d}\tau \\
&\leq \max_{x\in S}|u_0(x)-v_0(x)|+2\max_{(x,y)\in E}|\omega(x,t)|\,
 |V|\|u-v\|_{X_{t_0}}t\\
&\quad +e^{\beta t_0}p\xi^{p-1}\|u-v\|_{X_{t_0}}t \\
&= \max_{x\in S}|u_0(x)-v_0(x)|+C\|u-v\|_{X_{t_0}}t,
\end{aligned}\label{2.4}
\end{equation}
where $\xi=\max \{\|u\|_{X_{t_0}},\|v\|_{X_{t_0}}\} $ and
\[
C=2\max_{\left(x,y\right)\in E}|\omega(x,t)||V|\|u-v\|_{X_{t_0}}
+e^{\beta t_0}p\xi^{p-1}.
\]
The arbitrariness of $(x,t)\in S\times[0,t_0] $ gives the desired
estimate \eqref{2.1}.
\end{proof}

\begin{lemma}\label{lem2}
Suppose $t_0$ to be small enough, then $D_{v_0}$ is strictly contractive in the ball $B(u_0$, $2\|u_0\|_{L^{\infty}{\left(V\right)}})$.
\end{lemma}

\begin{proof}
 Let $u_0=v_0$, \eqref{2.1} ensures that $D_{v_0}$ is a strict contraction in 
the ball $B\left(u_0, 2\|u_0\|_{L^{\infty}{(V)}}\right) $ provided that
 $t_0 $ is small enough. In fact, for any $u$,
 $v\in B\left(u_0, 2\|u_0\|_{L^{\infty}{(V)}}\right)$, we have 
$$
\|u\|_{X_{t_0}}\leq 3\|u_0\|_{L^{\infty}{(V)}},\quad 
\|v\|_{X_{t_0}}\leq 3\|u_0\|_{L^{\infty}{(V)}},
$$
and thus, we obtain 
$$
\|D_{u_0}[u](x,t)-D_{u_0}[v](x,t)\|_{X_{t_0}}\leq C_1t_0\|u-v\|_{X_{t_0}},
$$
where 
$$
C_1=e^{\beta t_0}p\left(1+3\|u_0\|_{L^{\infty}{(V)}}^{p-1}\right)
+2\max_{\left(x,y\right)\in E}|\omega(x,t)||V|.
$$
Therefore, if $t_0$ is small enough such that $C_1t_0<\frac{1}{2}$, 
we obtain that $D_{v_0}$ is a strict contraction in the ball 
$B\left(u_0, 2\|u_0\|_{L^{\infty}{\left(V\right)}}\right)$. 
We complete the proof.
\end{proof}

\begin{theorem}\label{thm20}
If $p>1$, then problem \eqref{1.1} has a unique solution in $[0,T)$ for some $T>0 $ to be sufficiently small.
\end{theorem}

\begin{proof}
 By the Banach fixed point theorem, Lemma \ref{lem1} and Lemma \ref{lem2}, 
we can easily obtain the existence and the uniqueness of solution to \eqref{1.1}
 in the time interval $[0, t_0]$. Thus, if $\|u\|_{X_{t_0}}<\infty $ and 
initial data is taken as $u(x, t_0)$, then, the solution can be extended 
to some interval $[0, t_1)$, where $t_1>t_0$. Therefore, there exists a $T>0$, 
which is sufficiently small, such that problem \eqref{1.1} has a unique 
solution in $[0,T)$. 
\end{proof}

\section{Global existence and blow-up phenomenon}\label{s3}

To study the global existence and blow-up properties,
 we have the following statements.

\begin{definition} \label{def3.1} \rm
A nonnegative function $\overline{u}(x,t)\in C^{1}\left(V\times[0,T)\right) $ 
is a supersolution of problem \eqref{1.1} if it satisfies
\begin{equation}
\begin{gathered}
\overline{u}(x,t)\geq \sum_{y\in V}\omega(x,y)\left(\overline{u}(y,t)
 -\overline{u}(x,t)\right)+e^{\beta t}\overline{u}^{p}(x,t),\quad
  (x,t)\in S\times[0,T), \\
\overline{u}(x,t)\geq0,\quad  (x,t)\in \partial S\times[0,T), \\
\overline{u}(x,0)\geq u_0(x),\quad  x\in V.
\end{gathered} \label{3}
\end{equation}
Similarly, we can define the subsolution $\underline u(x,t)$ by reversing
the inequalities.
\end{definition}

Now, we introduce the comparison principle for the nonnegative solutions 
of  \eqref{1.1} which plays an important role in the proof of 
the existence of the critical exponent.

\begin{theorem}\label{thm4}
 Let $\overline{u}(x,t), \underline{u}(x,t)$, be supersolution and subsolution 
of \eqref{1.1}, respectively. Meanwhile, there exists a point $y\in S $ such that 
$\omega(x,y)\neq0 $ for any $x\in S$. Then, for any $(x,t)\in V\times[0,T)$, we have
$\overline{u}(x,t)\geq\underline{u}(x,t)$.
\end{theorem}

\begin{proof} Denote $v=\underline{u}-\overline{u}$, and choose $T_1<T$. 
In $V\times[0,T_1]$, we have
\begin{equation} \begin{aligned}
\frac{\partial v(x,t)}{\partial t}
&\leq \Delta_{\omega}v(x,t)+e^{\beta t}
 \left(\underline u^{p}(x,t)-\overline u^{p}(x,t)\right) \\
&= \Delta_{\omega}v(x,t)+pe^{\beta t}\xi^{p-1}(x,t)v(x,t),
\end{aligned} \label{3.2}
\end{equation}
where $\xi(x,t)$ is bounded in $V\times[0,T_1]$.

Let $v_{+}=\max \{v,0\}$. Multiplying both sides of \eqref{3.2} by $v_{+} $ 
and integrating on $V$, we have
\begin{equation}
\frac{1}{2}\Big(\int_{V}v^2_{+}(x,t)\Big)_t\leq \int_{V}
 \Delta_{\omega}v(x,t)v_{+}(x,t)+\int_{V}pe^{\beta t}\xi^{p-1}v^2_{+}(x,t),
\label{20.1}
\end{equation}
for all $(x,t)\in V\times[0,T_1]$, due to the fact that $v_{+}=0 $ on
 $\partial S$. Now, set $I(t)=\{x\in V: \underline u>\overline u\}$.
A direct use of definition of $\Delta_{w}u(x,t) $ yields
\begin{equation}
\begin{aligned}
\int_{V}\Delta_{\omega}v(x,t)v_{+}(x,t)
&= \int_{I(t)}\Delta_{\omega}v(x,t)v_{+}(x,t) \\
&= \sum_{x\in I(t)}\sum_{y\in V}\omega(x,y)\left(v(y,t)-v(x,t)\right)v_{+}(x,t) \\
&= \sum_{x\in I(t)}\sum_{y\in I(t)}\omega(x,y)\left(v(y,t)-v(x,t)\right)v_{+}(x,t) \\
&\quad +\sum_{x\in I(t)}\sum_{y\in V\setminus I(t)}\omega(x,y)
\left(v(y,t)-v(x,t)\right)v_{+}(x,t).
\end{aligned} \label{3.3}
\end{equation}
We first note that
\begin{equation}
\begin{aligned}
&\sum_{x\in I(t)}\sum_{y\in I(t)}\omega(x,y)\left(v(y,t)-v(x,t)\right)v_{+}(x,t) \\
&=\frac{1}{2}\sum_{x\in I(t)}\sum_{y\in I(t)}\omega(x,y)\left(v(y,t)-v(x,t)\right)v(x,t) \\
&\quad +\frac{1}{2}\sum_{x\in I(t)}\sum_{y\in I(t)}\omega(x,y)\left(v(x,t)-v(y,t)\right)v(y,t) \\
&=-\frac{1}{2}\sum_{x\in I(t)}\sum_{y\in I(t)}\omega(x,y)
\left(v(y,t)-v(x,t)\right)^2
\leq 0.
\end{aligned}\label{3.4}
\end{equation}
Then, notice that if $x\in I(t)=\{x\in V:\underline u> \overline u\}$ and
$y\in V\setminus I(t)=\{y\in V:\overline u\geq\underline u\}$,
then
$$
\underline u(x,t)>\overline u(x,t),\quad
 \overline u(y,t)\geq\underline u(y,t),
$$
so we have
$$
\underline u(y,t)-\underline u(x,t)\leq\overline u(y,t)-\overline u(x,t).$$
Then
\begin{equation}
\sum_{x\in I(t)}\sum_{y\in V\setminus I(t)}
\omega(x,y)\left(v(y,t)-v(x,t)\right)v_{+}(x,t)\leq0\,.
\end{equation}
Therefore,
\begin{equation}
\int_{V}\Delta_{\omega}v(x,t)v_{+}(x,t)\leq0
\label{20.2}
\end{equation}
and
\begin{equation}
\begin{aligned}
\int_{V}pe^{\beta t}\xi^{p-1}v^2_{+}(x,t)
&=\sum_{x\in V}pe^{\beta t}\xi^{p-1}v^2_{+}(x,t) \\
&\leq pe^{\beta T_1}\xi^{p-1}\sum_{x\in V}v^2_{+}(x,t)
=C\sum_{x\in V}v^2_{+}(x,t),
\end{aligned}\label{3.5}
\end{equation}
where $C=pe^{\beta T_1}\xi^{p-1}$.
From \eqref{20.1}, \eqref{20.2} and \eqref{3.5}, we have
$$
\Big(\int_{V}v^2_{+}(x,t)\Big)_t\leq 2C\int_{V}v^2_{+}(x,t).
$$
Since $v_{+}( \cdot, 0)=0$, we arrive at $v_{+}( x, t)=0$ in
$V\times [0,T_1]$. Due to the arbitrariness of $T_1$, we obtain
$$
\overline u(x,t)\geq\underline u(x,t),\quad \forall (x,t)\in V\times [0,T).
$$
The proof is complete.
\end{proof}

Then, we study the existence of the critical exponent for problem \eqref{1.1}.

\begin{definition} \rm
$p_{\beta} $ is called the critical exponent of problem \eqref{1.1}, 
if it satisfies:
\begin{itemize}
\item[(1)] when $p>p_{\beta}$, there is a nonnegative and nontrivial
 global solution $u$ of equation \eqref{1.1};

\item[(2)] when $1<p<p_{\beta}$, the nontrivial solution $u$ of\eqref{1.1} blows 
up in finite time.
\end{itemize}
\end{definition}

The following theorems imply that if $p>1$, problem \eqref{1.1} admits 
a critical exponent $p_{\beta}=1+\frac{\beta}{\lambda_1}$, where
 $\lambda_1$ is the principle eigenvalue of the  eigenvalue problem
\begin{equation}
\begin{gathered}
-\Delta_{\omega}\varphi(x)=\lambda_1\varphi(x),\quad x\in S, \\
\varphi(x)=0, \quad x\in \partial S.
\end{gathered}
\end{equation}

\begin{theorem}\label{thm7}
Suppose $p>p_{\beta}=1+\frac{\beta}{\lambda_1}$ and the initial value 
$u_0\leq z_0\phi_1(x)$, then the solution of  \eqref{1.1} is global 
and positive. Here $\phi_1(x)$ which corresponds to $\lambda_1 $ is the 
positive eigenfunction with $\|\phi_1(x)\|_{L^{\infty}}=1$ and $z_0$ 
is a constant satisfying 
$0<z_0<\big(\lambda_1-\frac{\beta}{p-1}\big)^{\frac{1}{p-1}}$.
\end{theorem}

\begin{proof} 
Let $v(x,t)=\phi_1(x)e^{-\lambda_1 t}$, then it is the solution of the 
 problem
\begin{equation}
\begin{gathered}
\frac{\partial v(x,t)}{\partial t}=\Delta_{\omega}v(x,t), \quad
(x,t)\in S\times(0,\infty), \\
v(x,t)=0, \quad (x,t)\in \partial S\times[0,\infty), \\
v(x,0)=\phi_1(x), \quad  x\in V.
\end{gathered} \label{3.6}
\end{equation}
In addition, let $z(t) $ be the solution of the initial-value problem
\begin{equation}
\begin{gathered}
\frac{\mathrm{d}z}{\mathrm{d}t}=e^{\beta t}\|v(\cdot,t)\|_{L^{\infty}}^{p-1}z^{p}(t), \\
z(0)=z_0>0,
\end{gathered}\label{3.7}
\end{equation}
where $z_0 $ is a constant satisfying
$z_0<\left(\lambda_1-\frac{\beta}{p-1}\right)^{\frac{1}{p-1}}$.
Solve the ODE problem of \eqref{3.7} with
$\|v(\cdot,t)\|_{L^{\infty}}^{p-1}=e^{-\lambda_1 (p-1)t}$, then we have
$$
z(t)=\Big(z_0^{1-p}-\frac{p-1}{(p-1)\lambda_1-\beta}
\big(1-e^{(\beta-(p-1)\lambda_1)t}\big)\Big)^{\frac{1}{1-p}},
$$
which is bounded uniformly for $t\in [0,\infty)$.

Now, let $\overline u(x,t)=z(t)v(x,t) $ for $(x,t)\in S\times (0,\infty)$, then
\begin{equation}
\begin{aligned}
\frac{\partial \overline u(x,t)}{\partial t}
&= \frac{\,\mathrm{d}z(t)}{\,\mathrm{d}t}v(x,t)+z(t)\frac{\partial v}{\partial t} \\
&= e^{\beta t}\|v(\cdot,t)\|_{L^{\infty}}^{p-1}z^{p}(t)v(x,t)+z(t)\Delta_{\omega}v(x,t) \\
&\geq \Delta_{\omega}\overline u(x,t)+e^{\beta t}v^{p}z^{p}(t) \\
&= \Delta_{\omega}\overline u(x,t)+e^{\beta t}\overline u^{p}(x,t),
\end{aligned}\label{3.8}
\end{equation}
and
\begin{equation}
\overline u(x,t)=0,\quad \forall(x,t)\in \partial S\times [0,\infty),
\label{20.8}
\end{equation}
furthermore,
\begin{equation}
\overline u(x,0)=z_0\phi_1(x)\geq u_0(x).
\label{3.9}
\end{equation}
From \eqref{3.8}, \eqref{20.8} and \eqref{3.9}, we obtain that
 $\overline u(x,t)$ is a supersolution of problem \eqref{1.1}.
By Theorem \ref{thm4}, we  conclude that  \eqref{1.1} admits a global
positive solution provided that the initial value $u_0$ is small.
\end{proof}

\begin{theorem}\label{thm10}
Suppose that $u_0(x) $ is nonnegative and nontrivial. 
If $1<p<p_{\beta}=1+\frac{\beta}{\lambda_1}$, then the corresponding solution 
to \eqref{1.1} on graph $G$ blows up in the sense of 
$\lim_{t\to T^{*-}}\sum_{x\in S}u(x,t)\varphi(x)=+\infty $ 
and the blow-up time $T^{*} $ satisfies
$$
T^{*}=\frac{\ln \big(1-\frac{\beta-(p-1)\lambda_1}{1-p}G^{1-p}_0\big)}
{\beta-(p-1)\lambda_1}.
$$
\end{theorem}

\begin{proof} Consider the eigenvalue problem
\begin{equation}
\begin{gathered}
-\Delta_{\omega}\varphi(x)=\lambda_1\varphi(x),\quad x\in S, \\
\varphi(x)=0, \quad x\in \partial S, \\
\|\varphi(x)\|_{L^{\infty}}=1,
\end{gathered}\label{3.16}
\end{equation}
where $\lambda_1 $ is the principle eigenvalue of the eigenvalue problem.
Multiplying $\varphi(x) $ on the both sides of \eqref{1.1} and summing on $S$,
we have
\begin{equation}
\sum_{x\in S}u_t(x,t)\varphi(x)-\sum_{x\in S}\Delta_{\omega}u(x,t)\varphi(x)
=\sum_{x\in S}e^{\beta t}u^{p}(x,t)\varphi(x). \label{100}
\end{equation}
Let $G(t)=\sum_{x\in S}u(x,t)\varphi(x)$, then
$G'(t)=\sum_{x\in S}u_t(x,t)\varphi(x)$,
and
\begin{equation}
\begin{aligned}
&\sum_{x\in S}\Delta_{\omega}u(x,t)\varphi(x) \\
&= \sum_{x\in S}\sum_{y\in V}\omega(x,y)\left(u(y,t)-u(x,t)\right)\varphi(x) \\
&= \sum_{x\in S}\sum_{y\in S}\omega(x,y)\left(u(y,t)-u(x,t)\right)\varphi(x)-\sum_{x\in S}\sum_{y\in \partial S}\omega(x,y)u(x,t)\varphi(x) \\
&= -\frac{1}{2}\sum_{x\in S}\sum_{y\in S}\omega(x,y)\left(u(y,t)-u(x,t)\right)
\left(\varphi(y)-\varphi(x)\right) \\
&\quad -\sum_{x\in S}\sum_{y\in \partial S}\omega(x,y)u(x,t)\varphi(x) \\
&= \sum_{x\in S}\sum_{y\in S}\omega(x,y)\left(\varphi(y)-\varphi(x)\right)u(x,t)-\sum_{x\in S}\sum_{y\in \partial S}\omega(x,y)u(x,t)\varphi(x) \\
&= \sum_{x\in S}\sum_{y\in V}\omega(x,y)\left(\varphi(y)-\varphi(x)\right)u(x,t) \\
&= \sum_{x\in S}\Delta_{\omega}\varphi(x)u(x,t)
= -\lambda_1G(t).
\end{aligned}\label{10.11}
\end{equation}
Moreover, applying Jensen's inequality, we obtain
\begin{equation}
\sum_{x\in S}e^{\beta t}u^{p}(x,t)\varphi(x)
\geq e^{\beta t}\Big(\sum_{x\in S}u(x,t)\varphi(x)\Big)^{p}
=e^{\beta t}G^{p}(t).
\label{10.22}
\end{equation}
Substituting \eqref{10.11} and \eqref{10.22} into \eqref{100}, we have
\begin{equation}
G'(t)\geq-\lambda_1G(t)+e^{\beta t}G^{p}(t).
\label{10.23}
\end{equation}
Thus, using the comparison for the linear ODE, we have
\begin{equation}
G^{1-p}(t)\leq\Big(G^{1-p}_0-\frac{1-p}{\beta-\lambda_1 (p-1)}
+\frac{1-p}{\beta-\lambda_1 (p-1)}e^{\big(\beta-(p-1)\lambda_1\big)t}\Big)
e^{(p-1)\lambda_1t},
\label{10.24}
\end{equation}
and then
\begin{equation}
G^{p-1}(t)\geq\frac{1}{\big(G^{1-p}_0-\frac{1-p}{\beta-\lambda_1 (p-1)}
+\frac{1-p}{\beta-\lambda_1 (p-1)}e^{\left(\beta-(p-1)\lambda_1\right)t}\big)
e^{(p-1)\lambda_1t}}.
\label{10.25}
\end{equation}
Since $1<p<p_{\beta}=1+\frac{\beta}{\lambda_1}$,
 we have $\frac{1-p}{\beta-(p-1)\lambda_1}<0 $ and
$G^{1-p}(0)-\frac{1-p}{\beta-(p-1)\lambda_1}>0$.
Thus, $G(t)=\sum_{x\in S}u(x,t)\varphi(x) $ cannot be global.

From the right side of \eqref{10.25}, we have the blow-up time is
$$
T^{*}=\frac{\ln \big(1-\frac{\beta-(p-1)\lambda_1}{1-p}G^{1-p}_0\big)}
{\beta-(p-1)\lambda_1}.
$$
\end{proof}

Next we have a blow-up rate in the  $L^{\infty}$-norm.

\begin{theorem}\label{thm9}
 Let $1<p<p_{\beta} $ and $u(x,t) $ be a solution of problem \eqref{1.1} 
which blows up at time $T$, then
\begin{equation}
\lim_{t\to T^{-}}(T-t)^{\frac{1}{p-1}}\max_{x\in V}u(x,t)
=\Big(\frac{e^{-\beta T}}{p-1}\Big)^{\frac{1}{p-1}}.
\label{3.74}
\end{equation}
\end{theorem}

\begin{proof}
 Let $U(t)=u\big(x(t),t\big)=\max_{x\in V}u(x,t)$. On the one hand,
\begin{equation}
U'(t)= \sum_{y\in V}\omega(x,y)\left(U(y,t)-U(x,t)\right)+e^{\beta t}U^{p}(x,t).
\label{3.23}
\end{equation}
Integrating  over $(t,T)$, we obtain
$$
\frac{1}{1-p}\left(U^{1-p}(T)-U^{1-p}(t)\right)
\leq\frac{1}{\beta}(e^{\beta T}-e^{\beta t}),
$$
then
$$
\max_{x\in V}u(x,t)=U(t)\geq\big(\frac{\beta}{p-1}\big)^{\frac{1}{p-1}}
(e^{\beta T}-e^{\beta t})^{-\frac{1}{p-1}},
$$
so
\begin{equation}
\lim_{t\to T^{-}}(T-t)^{\frac{1}{p-1}}\max_{x\in V}u(x,t)\geq\lim_{t\to T^{-}}
\Big(\frac{(T-t)\frac{\beta}{p-1}}{e^{\beta T}-e^{\beta t}}\Big)^{\frac{1}{p-1}}
=\Big(\frac{e^{-\beta T}}{p-1}\Big)^{\frac{1}{p-1}}.
\label{3.24}
\end{equation}
On the other hand,
\begin{equation}
\begin{aligned}
u_t(x,t)
&= \sum_{y\in V}\omega(x,y)\left(u(y,t)-u(x,t)\right)+e^{\beta t}u^{p}(x,t) \\
&\geq -ku(x,t)+e^{\beta t}u^{p}(x,t) \\
&= u^{p}(x,t)\left(e^{\beta t}-ku^{1-p}(x,t)\right),
\end{aligned}\label{3.25}
\end{equation}
where $k=\max_{x\in S}\sum_{y\in V}\omega(x,y)$.
In particular, we have
\begin{equation}
\begin{aligned}
U'(t)
&\geq U^{p}(x,t)\big(e^{\beta t}-kU^{1-p}(x,t)\big) \\
&\geq U^{p}(x,t)\Big(e^{\beta t}-k\frac{p-1}{\beta}(e^{\beta T}-e^{\beta t})\Big).
\end{aligned}\label{3.26}
\end{equation}
Integrating as before over $(t,T)$, we have
$$
U^{1-p}(t)\geq\frac{(p-1)(\beta+k(p-1))}{\beta^{2}}
(e^{\beta T}-e^{\beta t})-\frac{k(p-1)^{2}}{\beta}(T-t)e^{\beta T},
$$
then
$$
U(t)\leq\Big(\frac{(p-1)\left(\beta+k(p-1)\right)}{\beta^{2}}
(e^{\beta T}-e^{\beta t})-\frac{k(p-1)^{2}}{\beta}(T-t)e^{\beta T}
\Big)^{\frac{1}{1-p}},
$$
so
\begin{equation}
\lim_{t\to T^{-}}(T-t)^{\frac{1}{p-1}}\max_{x\in V}u(x,t)
\leq\Big(\frac{e^{-\beta T}}{p-1}\Big)^{\frac{1}{p-1}}.
\label{3.27}
\end{equation}
From \eqref{3.24} and \eqref{3.27}, we complete the proof.
\end{proof}

\section{Examples and numerical experiments}\label{s4}

In this section, we give two examples to illustrate our results from section \ref{s3}.
First, we consider the special graph $G_1 $ for problem \eqref{1.1}. 
The graph $G_1 $ has two nodes $x_1 $ and $x_2$, where $x_1 $ is
boundary and $x_2 $ is interior.
Then, problem \eqref{1.1} can be rewritten as
\begin{equation}
\begin{gathered}
u_t(x_2,t)=-\omega u(x_2,t)+e^{\beta t}u^{p}(x_2,t), \quad t>0, \\
u(x_2,0)=u_0>0,
\end{gathered}\label{4.1}
\end{equation}
where $\omega$, $\beta>0 $ and $p>1 $ are real constants.

\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig1}  % G1.eps
\caption{Graph $G_1$} \label{fig1}
\end{center}
\end{figure}

Equation \eqref{4.1} is the well-known Bernoulli equation. 
The explicit solution of \eqref{4.1} is 
\begin{equation}
u(x_2,t)
=\Big(e^{\omega (p-1)t}\Big(u_0^{1-p}-\frac{1-p}{\beta-(p-1)\omega}
+\frac{1-p}{\beta-(p-1)\omega}e^{\left(\beta-\omega(p-1)\right)t}\Big)
\Big)^{\frac{1}{1-p}}.
\label{4.2}
\end{equation}
Consider $1<p<p_{\beta} $ and $\omega=\lambda_1$, then
$\frac{1-p}{\beta-(p-1)\omega}<0$, so we can get that
$$
u_0^{1-p}-\frac{1-p}{\beta-(p-1)\omega}
+\frac{1-p}{\beta-(p-1)\omega}e^{\left(\beta -\omega (p-1)\right )t}
$$
is decreasing on $t$.
Notice that $u_0>0 $ and $\frac{1-p}{\beta-(p-1)\omega}<0$, then the solution
 $u(t) $ will blow up in finite time. The blow-up time on $L^{\infty}$-norm
to problem \eqref{4.1} is
$$
T=\frac{\ln \big(1-\frac{\left(\beta-\omega(p-1)\right)u_0^{1-p}}{1-p}\big)}
{\beta-\omega(p-1)}.
$$
Then we have the limit,
$$
\lim_{t\to T^{-}}(T-t)^{\frac{1}{p-1}}u(x_2,t)
=\left(\frac{e^{-\beta T}}{p-1}\right)^{\frac{1}{p-1}}.
$$

We remark that if $\frac{1-p}{\beta-(p-1)\omega}>0 $ and 
$u_0^{1-p}-\frac{1-p}{\beta-(p-1)\omega}>0$, then the only solution \eqref{4.2} 
to problem \eqref{4.1} is global.

\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig2} % G2.eps
\caption{Graph $G_2$} \label{fig2}
\end{center}
\end{figure}

Now, we consider a complicated graph $G_2 $ which has six nodes, where
 $x_1, x_{4} $ and $x_{6} $ are boundary and $x_2, x_3 $ and $x_5 $
are interior. Moreover, we only consider the weight function $\omega$=1.
Thus, the problem can be rewritten as
\begin{equation}
\begin{gathered}
u_t(x_2,t)=u(x_3,t)+u(x_5,t)-3u(x_2,t)+e^{\beta t}u^{p}(x_2,t), \\
u_t(x_3,t)=u(x_2,t)+u(x_5,t)-3u(x_3,t)+e^{\beta t}u^{p}(x_3,t), \\
u_t(x_5,t)=u(x_2,t)+u(x_3,t)-3u(x_5,t)+e^{\beta t}u^{p}(x_5,t), \\
u(x_2,0)=\alpha>0, \\
u(x_3,0)=\zeta>0, \\
u(x_5,0)=\gamma>0.
\end{gathered} \label{4.3}
\end{equation}
Let $U=\left(u(x_2,t),u(x_3,t), u(x_5,t)\right)^{\rm T}$,
and the coefficient matrix is
\[
A=\begin{pmatrix}
-3 & 1 & 1 \\
1 & -3 & 1 \\
1 & 1 & -3
\end{pmatrix}.
\]
Thus, \eqref{4.3} can be rewritten as
\begin{equation}
\begin{gathered}
U_t=A\ast U+e^{\beta t}U^{p}, \\
U_0=(\alpha, \zeta, \gamma )^{\rm T}.
\end{gathered} \label{4.4}
\end{equation}
Because of nonlinearity, it is hard to handle system \eqref{4.4}
by exact analysis technique. Instead, we calculate the solution by
difference method. Then the explicit scheme is
\begin{equation}
\begin{gathered}
\frac{U^{n+1}-U^{n}}{\Delta t}=A\ast U^{n}+(U^{n})^{p}, \\
U^{0}=(\alpha, \zeta, \gamma )^{\rm T},
\end{gathered}\label{4.5}
\end{equation}
where $U^{n}=\left(u(x_2,n\Delta t),u(x_3,n\Delta t),u(x_5,n\Delta t)\right)^{\rm T}$
and $\Delta t$ is the time step. Moreover, we set
$\alpha=0.4, \zeta=0.6, \gamma=0.7, \beta=3 $ and $p=2$, respectively.
The numerical experiment result is shown in Figure \ref{fig3}. We observe that
the solution blows up in finite time.

\begin{figure}[htb]
\begin{center}
 \includegraphics[width=0.6\textwidth]{fig3} % G3.eps
\caption{Blow-up phenomenon for the equation \eqref{4.3}} \label{fig3}
\end{center}
\end{figure}


\subsection*{Acknowledgments}
This work was partly supported by the National Natural Science Foundation of 
China (Grant No. 11301277, 41475091) and the Qing Lan Project of Jiangsu Province.


\begin{thebibliography}{99}

\bibitem{cb2005} S.-Y. Chung,  C. A. Berenstein;
\emph{$\omega$-harmonic functions and inverse conductivity problems on networks}, 
SIAM J. Appl. Math. \textbf{65} (2005), no.~4, 1200--1226.


\bibitem{cck2007} S.-Y. Chung, Y.-S. Chung, J.-H. Kim;
\emph{Diffusion and elastic equations on networks}, 
Publ. Res. Inst. Math. Sci. \textbf{43} (2007), no.~3, 699--726.


\bibitem{clc2011} Y.-S. Chung, Y.-S. Lee, S.-Y. Chung;
\emph{Extinction and positivity of the solutions of the heat equations 
with absorption on networks}, J. Math. Anal. Appl. \textbf{380} (2011),
 no.~2, 642--652.

\bibitem{pmd2013} L. M. Del Pezzo, C. A. Mosquera, J. D. Rossi;
\emph{Existence, uniqueness and decay rates for evolution equations on trees},
 Port. Math. \textbf{71} (2014), no.~1, 63--77.

\bibitem{f2011} R. Ferreira;
\emph{Quenching phenomena for a non-local diffusion equation with a 
singular absorption}, Israel J. Math. \textbf{184} (2011), 387--402.

\bibitem{fl2012} R. Ferreira,  M. P\'erez-Llanos;
\emph{Blow-up for the non-local $p$-Laplacian equation with a reaction term},
 Nonlinear Anal. \textbf{75} (2012), no.~14, 5499--5522.

\bibitem{kwz2012} L. H. Kong, J. H. Wang, S. L. Zheng;
\emph{Uniform blow-up profiles for heat equations with coupling nonlocal
 sources of asymmetric mixed type nonlinearities}, 
Acta Math. Sci. Ser. B Engl. Ed. \textbf{32} (2012), no.~3, 1126--1140

\bibitem{lc2012} Y.-S. Lee, S.-Y. Chung;
\emph{Extinction and positivity of solutions of the $p$-Laplacian evolution 
equation on networks}, J. Math. Anal. Appl. \textbf{386} (2012), no.~2, 581--592.

\bibitem{l1973} H. A. Levine;
\emph{Some nonexistence and instability theorems for solutions of formally 
parabolic equations of the form $Pu_t=-Au+F(u)$}, 
Arch. Rational Mech. Anal. \textbf{51} (1973), 371--386.


\bibitem{m1990} P. Meier;
\emph{On the critical exponent for reaction-diffusion equations},
 Arch. Rational Mech. Anal. \textbf{109} (1990), no.~1, 63--71.


\bibitem{pkc2009} J.-H. Park, J.-H. Kim, S.-Y. Chung;
\emph{The $p$-Schr\"odinger equations on finite networks}, 
Publ. Res. Inst. Math. Sci. \textbf{45} (2009), no.~2, 363--381.

\bibitem{lr2009} M. P\'erez-Llanos,  J. D. Rossi;
\emph{Blow-up for a non-local diffusion problem with Neumann boundary 
conditions and a reaction term}, Nonlinear Anal. \textbf{70} (2009), 
no.~4, 1629--1640.

\bibitem{t2013} F. Tahamtani, M. Shahrouzi;
\emph{Asymptotic stability and blow up of solutions for a Petrovsky 
inverse source problem with dissipative boundary condition},
 Math. Methods Appl. Sci. \textbf{36} (2013), no.~7, 829--839.

\bibitem{t2003} H. Takeuchi;
\emph{The spectrum of the $p$-Laplacian and $p$-harmonic morphisms on graphs},
 Illinois J. Math. \textbf{47} (2003), no.~3, 939--955.

\bibitem{w2011} M. X. Wang;
\emph{Blow-up rate estimates for semilinear parabolic systems}, 
J. Differential Equations \textbf{170} (2001), no.~2, 317--324.

\bibitem{wcz2013} Y. Wang, J. Chen, B. Zhou;
\emph{Blowup analysis for a nonlocal reaction diffusion equation with potential}, 
WSEAS Trans. Math. \textbf{12} (2013), no.~7, 2224--2880.

\bibitem{w2013} S.-T. Wu;
\emph{Blow-up of solutions for a system of nonlinear parabolic equations}, 
Electron. J. Differential Equations \textbf{2013}, No. 216, 9 pp.

\bibitem{xml20114} Q. Xin, C. Mu,  D. Liu;
\emph{Extinction and positivity of the solutions for a $p$-Laplacian equation 
with absorption on graphs}, J. Appl. Math. \textbf{2011}, Art. ID 937079, 12 pp.

\bibitem{xxm2013} Q. Xin, L. Xu, C. Mu;
\emph{Blow-up for the $\omega$-heat equation with Dirichlet boundary conditions 
and a reaction term on graphs}, Appl. Anal.   \textbf{93} (2014), no.~8,  1691--1701.

\bibitem{zw2011} G. Zhang, Y. Wang;
\emph{Critical exponent for nonlocal diffusion equations with Dirichlet boundary 
condition}, Math. Comput. Modelling \textbf{54} (2011), no.~1-2, 203--209.

\bibitem{z2012} J. P. Zhao;
\emph{Blow-up of solutions for a class of reaction-diffusion equations with
 a gradient term under nonlinear boundary condition},
 Z. Naturforsch. 67a  (2012), 479--482.

\bibitem{zmmz2013} S. Zhou, C.  Mu, Y. Mi, F. Zhang, Fuchen;
\emph{Blow-up for a non-local diffusion equation with exponential 
reaction term and Neumann boundary condition}, 
Commun. Pure Appl. Anal. \textbf{12} (2013), no.~6, 2935--2946.

\end{thebibliography}
\end{document}