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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 20, pp. 1--6.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/20\hfil Lower bounds for the blowup time]
{Lower bounds for the blowup time of solutions to a nonlinear
parabolic problem}

\author[H. Li, W. Gao, Y. Han \hfil EJDE-2014/20\hfilneg]
{Haixia Li, Wenjie Gao, Yuzhu Han}  % in alphabetical order

\address{Haixia Li \newline
 School of Mathematics, Jilin University,
 Changchun 130012, China}
\email{lihaixia0611@126.com}

\address{Wenjie Gao \newline
 School of Mathematics, Jilin University,
 Changchun 130012, China}
\email{wjgao@jlu.edu.cn}

\address{Yuzhu Han \newline
 School of Mathematics, Jilin University,
 Changchun 130012, China}
\email{yzhan@jlu.edu.cn} 

\thanks{Submitted December 20, 2013. Published January 10, 2014.}
\subjclass[2000]{35K58, 35B44}
\keywords{Blow-up time; lower bounds; gradient nonlinearity}

\begin{abstract}
 In this short article, we study the blow-up properties of solutions
 to a parabolic problem with a gradient nonlinearity under homogeneous
 Dirichlet boundary conditions. By constructing an auxiliary function
 and by modifying the first order differential inequality technique
 introduced by Payne et al., we obtain a lower bound for the blow-up
 time of solutions in a bounded domain $\Omega\subset \mathbb{R}^n$
 for any $n\geq3$. This article generalizes a  result in \cite{Payne1}.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks


\section{Introduction}

When dealing with a parabolic problem there are several interesting features 
to analyze, one of which is the so called finite time blow-up. 
The question of blow-up of solutions to nonlinear parabolic equations and 
systems has received considerable attention since the elegant
work of Fujita \cite{Fujita66}. We refer to the interested readers the 
survey papers \cite{Bandle98,Galaktionov02,Levine90} and the
book \cite{Quittner07}.

In practical situations, one would like to know, among other things,
whether the solutions blow up, and if so, at what time $T$ blow-up occurs. 
However, when the solution does blow up at some finite $T$,
this time can seldom be determined explicitly, and much effort has been 
devoted to the calculation of bounds for $T$. Most of the methods used until 
recently can only yield upper bounds for $T$, which are
of little value in particular situations when blow-up has to be avoided. 
By using the first-order differential inequality technique, 
lower bounds for the blow-up time of solutions to semilinear
heat equations under different boundary conditions and suitable constraint 
on the data were obtained by Payne et al. \cite{Payne3,Payne06aa,Payne2,Payne1}. 
Thereafter, the differential inequality technique was successfully employed
to derive lower bounds for the blow-up time of solutions to other parabolic 
problems, see \cite{Baghaei13,Bao13,Ding13,Payne10,Payne08jpam}.

In this article, we shall study a parabolic problem with a gradient 
nonlinearity of the following form
\begin{equation}\label{1.1}
\begin{gathered}
u_t=\Delta u+u^p-|\nabla u|^q, \quad (x,t) \in\Omega\times(0,T),\\
u(x,t)=0, \quad (x,t)\in\partial\Omega\times(0,T),\\
u(x,0)=u_0(x)\geq0, \quad x\in\Omega,
\end{gathered}
\end{equation}
where $\Omega$ is a bounded domain in $\mathbb{R}^n$ with smooth boundary 
$\partial\Omega$, $\Delta$ and $\nabla$ are the
Laplace and gradient operator with respect to $x$, respectively,
$T$ is the possible blow-up time and $p, q>1$ are fixed (finite) parameters.
In \cite{Chipot,Kawohl}, conditions on $p$, $q$
and $u_0(x)$ were given for which the solutions to \eqref{1.1} would blow up
in finite time. In fact the restrictions on p and q were
$$
1<p<\frac{n+2}{n-2},\quad 1<q<\frac{2p}{p+1},\quad \text{for }n\geq2,
$$
or
$$
\text{$p$ is large enough and $q=\frac{2p}{p+1}$, for $n=1$}.
$$
In a recent paper Payne et al.  \cite{Payne1}
obtained lower bounds of the blow-up time of solutions to \eqref{1.1}
when $n=3$. Naturally, we hope to obtain the lower bounds for blow-up 
time of solutions to \eqref{1.1} with any smooth bounds
$\Omega\subset \mathbb{R}^n$ and any $n\geq3$. 
That is what we will do in this article.

As indicated in \cite{Souplet} it is well known that if $p\leq q$ 
the solution will not blow up in finite time. Also it is well known that
if the initial data are small enough the solution will actually decay 
exponentially as $t\rightarrow \infty$ (see e.g.\cite{Payne2,Straughan}).
Since we are interested in a lower bound for the blow-up time $T$, 
only the case $p>q$ is considered.

\section{A lower bound for the blow-up time}

In this section we seek a lower bound for the blow-up time $T$ of solutions 
to \eqref{1.1} in some appropriate measure.
The idea of the proof of the following theorem is inspired by that 
in \cite{Baghaei13}.

\begin{theorem}\label{thm2.1}
Let $u(x,t)$ be the nonnegative classical solution of problem \eqref{1.1} 
for $p>q>1$ in a smooth bounded domain $\Omega\subset \mathbb{R}^n$ with $n\geq3$.
Define
$$
\varphi(t)=\int_{\Omega}u^k \mathrm{d}x,
$$
where $k$ is a parameter restricted by the condition
\begin{equation}\label{2.1}
k>\max\Big\{1,\frac{(7n-16)(p-1)}{2},(q-1)(3n-8)\Big\}.
\end{equation}
If $u(x,t)$ blows up in the measure $\varphi$ at the finite time $T$, 
then $T$ is bounded from below as
\begin{equation}\label{2.2}
T\geq\int_{\varphi(0)}^{+\infty}\frac{1}{C_1+C_2\xi^{\frac{3n-6}{3n-8}}}d\xi,
\end{equation}
where $C_1$ and $C_2$ are positive constants which will be determined 
in the proof.
\end{theorem}

\begin{proof}
Applying the divergence theorem to the first equation in \eqref{1.1},
 we have
\begin{equation} \label{2.3}
\begin{aligned}
\frac{\mathrm{d}\varphi}{\mathrm{d}t}
&= k\int_{\Omega}u^{k-1}u_t\mathrm{d}x \\
&= k\int_{\Omega}u^{k-1}(\triangle u+u^p-|\nabla u|^q)\mathrm{d}x \\
&= k\int_{\Omega}u^{k-1}\triangle u\mathrm{d}x+k\int_{\Omega}u^{k+p-1}
 \mathrm{d}x-k\int_{\Omega}u^{k-1}|\nabla u|^2\mathrm{d}x \\
&= -\frac{4(k-1)}{k}\int_{\Omega}|\nabla u^{k/2}|^2\mathrm{d}x
 +k\int_{\Omega}u^{k+p-1}\mathrm{d}x\\
&\quad -\frac{kq^q}{(k+q-1)^q}\int_{\Omega}|\nabla u^{\frac{k+q-1}{q}}|^q\mathrm{d}x.
\end{aligned}
\end{equation}
Moreover,  from \cite[(2.10)]{Payne3} it follows that
\begin{equation} \label{2.4}
\int_{\Omega}|\nabla u^{\frac{k+q-1}{q}}|^q\mathrm{d}x
\geq(\frac{2\sqrt{\lambda}}{q})^q\int_{\Omega}u^{k+q-1}\mathrm{d}x,
\end{equation}
where the positive constant $\lambda$ is the first eigenvalue of the
 problem
\begin{equation}\label{2.5}
\begin{gathered}
\triangle w+\lambda w=0 \quad\text{in }\Omega,\\
w=0 \quad\text{on } \partial\Omega.
\end{gathered}
\end{equation}
Thus by combining \eqref{2.3} with \eqref{2.4}  we obtain
\begin{equation} \label{2.6}
\frac{\mathrm{d}\varphi}{\mathrm{d}t}
\leq-\frac{4(k-1)}{k}\int_{\Omega}|\nabla u^{k/2}|^2\mathrm{d}x
+k\int_{\Omega}u^{k+p-1}\mathrm{d}x
-\frac{k(2\sqrt{\lambda})^q}{(k+q-1)^q}\int_{\Omega} u^{k+q-1}\mathrm{d}x.
\end{equation}
Noticing \eqref{2.1}, we can apply first H\"older's inequality
and then Young's inequality to the second
term on the right hand side of \eqref{2.3} to obtain
\begin{equation} \label{2.7}
\begin{aligned}
\int_{\Omega}u^{k+p-1}\mathrm{d}x
&\leq |\Omega|^{m_1}\Big(\int_{\Omega}u^{\frac{k(7n-14)}{7n-16}}
 \mathrm{d}x\Big)^{m_2} \\
&\leq m_1|\Omega|+m_2\int_{\Omega}u^{\frac{k(7n-14)}{7n-16}}\mathrm{d}x,
\end{aligned}
\end{equation}
where
$$
m_1=1-\frac{(k+p-1)(7n-16)}{k(7n-14)}\in(0,1),\quad
m_2=\frac{(k+p-1)(7n-16)}{k(7n-14)}\in(0,1).
$$
Combining \eqref{2.7} and \eqref{2.6} yields
\begin{equation} \label{2.8}
\begin{aligned}
\frac{\mathrm{d}\varphi}{\mathrm{d}t}
&\leq-\frac{4(k-1)}{k}\int_{\Omega}|\nabla u^{k/2}|^2\mathrm{d}x
 +km_1|\Omega|+km_2\int_{\Omega}u^{\frac{k(7n-14)}{7n-16}}\mathrm{d}x \\
&\quad -\frac{k(2\sqrt{\lambda})^q}{(k+q-1)^q}\int_{\Omega} u^{k+p-1}\mathrm{d}x.
\end{aligned}
\end{equation}
We now  use  H\"{o}lder's inequality in the third term on the right hand side
of \eqref{2.8}:
\begin{equation}\label{2.9}
\int_{\Omega}u^{\frac{k(7n-14)}{7n-16}}\mathrm{d}x
\leq\Big(\int_{\Omega}u^k\mathrm{d}x\Big)^{\alpha}
\Big(\int_{\Omega}u^{\frac{k}{2}\frac{2n}{n-2}}\mathrm{d}x\Big)^{1-\alpha},
\end{equation}
where $0<\alpha=\frac{2(3n-7)}{7n-16}<1$. Next, using the Sobolev inequality
for $W_0^{1,2}\hookrightarrow L^{\frac{2n}{n-2}}$ ($n\geq3$)  \cite{Talenti}),
we obtain
\begin{equation}\label{2.10}
\|u^{k/2}\|_{L^{\frac{2n}{n-2}}}^{\frac{2n(1-\alpha)}{n-2}}
\leq C_s^{\frac{2n(1-\alpha)}{n-2}}\|\nabla u^{k/2}
\|_{L^2}^{\frac{2n(1-\alpha)}{n-2}},
\end{equation}
where $C_s=\big(\frac{1}{n(n-2)\pi}\big)^{1/2}
\big(\frac{n!}{2\Gamma(\frac{n}{2}+1)}\big)^{1/n}$ is the best imbedding
 constant (see \cite[Chap. 7]{Gilbarg}).
By substituting \eqref{2.10} into \eqref{2.9}, we arrive at
\begin{equation} \label{2.11}
\int_{\Omega}u^{\frac{k(7n-14)}{7n-16}}\mathrm{d}x
\leq C_s^{\frac{2n(1-\alpha)}{n-2}}\Big(\int_{\Omega}u^k\mathrm{d}x\Big)^{\alpha}
\Big(\int_{\Omega}|\nabla u^{k/2}|^2\mathrm{d}x\Big)^{\frac{n(1-\alpha)}{n-2}},
\end{equation}
which, with the help of Young's inequality, gives
\begin{equation} \label{2.12}
\int_{\Omega}u^{\frac{k(7n-14)}{7n-16}}\mathrm{d}x
\leq\frac{C_s^{\frac{n}{3n-8}}(6n-16)}{(7n-16)\varepsilon_1^{\frac{n}{2(3n-8)}}}
\Big(\int_{\Omega}u^k\mathrm{d}x\Big)^\frac{3n-7}{3n-8}
+\frac{n(1-\alpha)\varepsilon_1}{n-2}\int_{\Omega}|\nabla u^{k/2}|^2\mathrm{d}x.
\end{equation}
Here $\varepsilon_1$ is a positive constant to be determined later.
By H\"older's inequality, we have
\begin{eqnarray}\label{2.13}
\int_{\Omega}u^{q+k-1}\mathrm{d}x
\geq |\Omega|^{-\frac{q-1}{k}}\Big(\int_{\Omega}u^k\mathrm{d}x
\Big)^{1+\frac{q-1}{k}}.
\end{eqnarray}
Combining \eqref{2.12} and \eqref{2.13} with \eqref{2.8} gives
\begin{equation} \label{2.14}
\begin{aligned}
\frac{\mathrm{d}\varphi}{\mathrm{d}t}
&\leq km_1|\Omega|+\big[\frac{n(1-\alpha)\varepsilon_1km_2}{n-2}
 -\frac{4(k-1)}{k}\big]\int_{\Omega}|\nabla u^{k/2}|^2\mathrm{d}x \\
&\quad +\frac{km_2C_s^{\frac{n}{3n-8}}(6n-16)}{(7n-16)
 \varepsilon_1^{\frac{n}{2(3n-8)}}}\varphi^{\frac{3n-7}{3n-8}}
 -\frac{k(2\sqrt{\lambda})^q}{(k+q-1)^q}|\Omega|^{-\frac{q-1}{k}}
 \varphi^{1+\frac{q-1}{k}}.
\end{aligned}
\end{equation}
Next, we  apply Young's inequality to the third term on the right-hand
side of \eqref{2.14} to conclude that
\begin{eqnarray}\label{2.15}
\varphi^{\frac{3n-7}{3n-8}}
\leq\frac{\varepsilon_2}{m_3}\varphi^{1+\frac{q-1}{k}}
+\frac{1}{m_4}\varepsilon_2^{-\frac{m_4}{m_3}}\varphi^{\frac{3n-6}{3n-8}},
\end{eqnarray}
where
$$
m_3=\frac{2k-(q-1)(3n-8)}{k},\quad
m_4=\frac{2k-(q-1)(3n-8)}{k-(q-1)(3n-8)},
$$
and $\varepsilon_2$ is a positive constant to be fixed.
Combining \eqref{2.15} and \eqref{2.14}, we obtain
\begin{equation} \label{2.16}
\begin{aligned}
\frac{\mathrm{d}\varphi}{\mathrm{d}t}
&\leq C_1+\big[\frac{n(1-\alpha)\varepsilon_1km_2}{n-2}
 -\frac{4(k-1)}{k}\big]\int_{\Omega}|\nabla u^{k/2}|^2\mathrm{d}x
 +C_2\varphi^{\frac{3n-6}{3n-8}} \\
&\quad +\big[\frac{\varepsilon_2km_2C_s^{\frac{n}{3n-8}}(6n-16)}{(7n-16)
 \varepsilon_1^{\frac{n}{2(3n-8)}}m_3}
-\frac{k(2\sqrt{\lambda})^q|\Omega|^{-\frac{q-1}{k}}}{(k+q-1)^q}\big]
\varphi^{1+\frac{q-1}{k}},
\end{aligned}
\end{equation}
where
$$
C_1=km_1|\Omega|,\quad
C_2=\frac{km_2C_s^{\frac{n}{3n-8}}(6n-16)
 \varepsilon_2^{-\frac{m_4}{m_3}}}{(7n-16)\varepsilon_1^{\frac{n}{2(3n-8)}}m_4}.
$$
Therefore, by choosing 
$$
\varepsilon_1=\frac{4(k-1)(n-2)}{nk^2m_2(1-\alpha)}
$$
first and 
$$
\varepsilon_2=\frac{(7n-16)m_3k(2\sqrt{\lambda})^q
|\Omega|^{-\frac{q-1}{k}}\varepsilon_1^\frac{n}{2(3n-8)}}
{km_2(6n-16)C_s^{\frac{n}{3n-8}}(k+q-1)^q}
$$ 
next, we obtain the differential inequality
\begin{equation} \label{2.17}
\frac{\mathrm{d}\varphi}{\mathrm{d}t}\leq C_1+C_2\varphi^{\frac{3n-6}{3n-8}},
\end{equation}
or equivalently
\begin{equation} \label{2.18}
\frac{\mathrm{d}\varphi}{C_1+C_2\varphi^{\frac{3n-6}{3n-8}}}\leq \mathrm{d}t.
\end{equation}
Integrating of the differential inequality \eqref{2.18} from 0 to $t$ leads to
\begin{eqnarray}\label{2.19}
\int_{\varphi(0)}^{\varphi(t)}\frac{1}{C_1+C_2\xi^{\frac{3n-6}{3n-8}}}d\xi\leq t.
\end{eqnarray}
Passing to the limit as $t\rightarrow T^-$, we obtain
\begin{eqnarray}\label{2.20}
\int_{\varphi(0)}^{+\infty}\frac{1}{C_1+C_2\xi^{\frac{3n-6}{3n-8}}}d\xi\leq T.
\end{eqnarray}
Thus, the proof is complete.
\end{proof}

\begin{remark} \rm
It is easy to see that when $n=3$, the lower bound for the blow-up time 
derived here is consistent with the one obtained by Payne et al. \cite{Payne1}.
\end{remark}


\subsection*{Acknowledgments}
This research was supported by NSFC (11271154), by Key Lab of Symbolic
Computation and Knowledge Engineering of Ministry of Education and
by the 985 program of Jilin  University.
The authors would like to thank the anonymous referees for their valuable
comments and suggestions which improve the original manuscript.

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\end{document}