\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 113, 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 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/113\hfil Lower bounds for the blow-up time]
{Lower bounds for the blow-up time of nonlinear parabolic problems with
Robin boundary conditions}

\author[K. Baghaei, M. Hessaraki \hfil EJDE-2014/113\hfilneg]
{Khadijeh Baghaei, Mahmoud Hesaaraki}  % in alphabetical order

\address{Khadijeh Baghaei \newline
 Department of mathematics,
 Iran University of Science and Technology, 
 Tehran, Iran}
\email{khbaghaei@iust.ac.ir}

\address{Mahmoud Hesaaraki \newline
 Department of mathematics,
 Sharif University of Technology,
 Tehran, Iran}
\email{hesaraki@sina.sharif.edu}

\thanks{Submitted June 6, 2013. Published April 16, 2014.}
\subjclass[2000]{35K55, 35B44}
\keywords{Parabolic equation; Robin boundary condition; blow-up; lower bound}

\begin{abstract}
In  this article, we find a lower bound for the blow-up time of
solutions to some nonlinear parabolic equations under Robin boundary
conditions in bounded domains of $\mathbb{R}^n$.
\end{abstract}

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

\section{Introduction}

In this article, we  consider  the nonlinear initial-boundary value problem
\begin{equation} \label{e1}
\begin{gathered}
(b(u))_{t}  =\nabla \cdot (g(u)\nabla u)+f(u), \quad x\in \Omega, \; t > 0 \\
\frac{\partial u}{\partial \nu}+\gamma u  =0,\quad x \in \partial \Omega, \; t > 0, \\
u(x,0) =u_0(x)\geq 0,  \quad  x \in \Omega
\end{gathered}
\end{equation}
where $ \Omega\subseteq \mathbb{R}^{n}, n\geq 3$, 
is a  bounded domain with smooth boundary, $\nu$ is the outward normal vector to 
$\partial\Omega$, $ \gamma$ is a positive constant and 
$u_0(x)\in C^{1}(\overline{\Omega})$ is the initial value.
We assume that $f $ is a nonnegative $C(\mathbb{R}^{+})$ function and  
the nonnegative functions $g$ and $b$ satisfy
\begin{equation}\label{g}
\begin{gathered}
 g\in C^{1}(\mathbb{R}^{+}), \quad  g(s) \geq g_{m}>0, \quad g'(s)   \leq 0, \quad 
 \forall s>0, \\
 b \in C^{2}(\mathbb{R}^{+}), \quad 0< b'(s) \leq b'_{M}, \quad  b''(s)\leq 0, 
\quad  \forall s > 0,
\end{gathered}
\end{equation}
where $ g_{m}$ and $ b'_{M}$  are positive constants.

The reader is referred to \cite{b1, e1,  p1, p2, p3, z1} for results on bounds 
for blow-up time  in nonlinear parabolic problems.
Ding \cite{d1} studied  problem \eqref{e1} under assumptions \eqref{g} and derived 
 conditions on the data which imply
blow-up or  the global existence of solutions.  
In addition, Ding obtained a lower bound for the
blow-up time when $\Omega\subseteq \mathbb{R}^{3}$ is a bounded convex domain.
Here  we obtain  a lower bound for the  blow-up time for \eqref{e1} 
in general  bounded  domains $\Omega\subseteq\mathbb{R}^{n}, n\geq3$. 

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

In this section we find a lower bound  for the  blow-up time $T$ in an 
appropriate measure.  The idea of the proof of the following theorem
comes from \cite{b1}.

\begin{theorem}\label{thm1}
 Let $\Omega$ be a bounded domain in $\mathbb{R}^{n}, n\geq3$, and let
the functions   $f, g, b$ satisfy
\begin{equation}\label{e14}
0< f(s) \leq c g(s)\Big(\int_0^s \frac{b'(y)}{g(y)} dy \Big)^{p+1}, \quad s > 0,
\end{equation}
for some constants  $c>0$ and $p\geq1$. If $u(x,t) $ is a nonnegative classical 
solution to  problem \eqref{e1}, which becomes unbounded in the measure
\[
\Phi(t)= \int_{\Omega} \Big(\int_0^{u(x,t)}\frac{b'(y)}{g(y)}dy\Big)^{2k}dx,
\]
where  $k$ is a parameter restricted by the condition
\begin{equation}\label{k}
k > \max \big\{p(n-2), 1\big\},
\end{equation}
then $ T$ is bounded from below by
\begin{equation}\label{xxx}
\int_{\Phi(0)}^{+\infty} \frac{d\xi}{k_1+ k_2\xi ^{\frac{3(n-2)}{3n-8}}
  +  k_3 \xi^{\frac{2n-3}{2(n-2)}}},
\end{equation}
where $ k_1, k_2$ and $ k_3$ are  positive constants which will be determined
later in the proof.
\end{theorem}

\begin{proof}
To simplify our computations we define
\begin{equation}\label{e3}
v(s)=\int_0^{s}\frac{b'(y)}{g(y)}dy, \quad s>0.
\end{equation}
Hence,
\begin{align*}
\frac{d\Phi}{dt} 
&=\frac{d}{dt}\int_{\Omega} v^{2k} \,dx= 2k\int_{\Omega} v^{2k-1}\frac{b'(u)}{g(u)} u_{t} \,dx  \\
&=2k \int_{\Omega}  v^{2k-1} \frac{(b(u))_{t}}{g(u)} \, dx   \\
&=2k \int_{\Omega}  v^{2k-1} \frac{1}{g(u)}\Big[\nabla\cdot(g(u)\nabla u)
 + f(u)\Big]\, dx  \\
&= - 2k (2k-1)\int_{\Omega}v^{2k-2}v'(u) |\nabla u|^{2} \,dx 
+ 2k \int_{\Omega}v^{2k-1}\frac{g'(u)}{g(u)}|\nabla u|^{2} \, dx  \\
& \quad - 2k\gamma \int_{\partial\Omega} v^{2k-1} u \,ds 
 +2k \int_{\Omega}v^{2k-1}\frac{f(u)}{g(u)} \, dx  \\
& \leq - 2k (2k-1)\int_{\Omega}v^{2k-2} \frac{b'(u)}{g(u)} |\nabla u|^{2} \,dx 
 +2k \int_{\Omega}v^{2k-1}\frac{f(u)}{g(u)} \, dx,
\end{align*}
 where in the above inequality  we  used  $u\geq0$  and  $  g'(u) \leq 0$  
from \eqref{g}. From \eqref{e3}, we have
\begin{equation}\label{e4}
|\nabla u|^{2} = \Big(\frac{g(u)}{b'(u)}\Big)^{2}|\nabla v|^{2}.
\end{equation}
By \eqref{g}, \eqref{e4}, and \eqref{e14} we have
\begin{equation} \label{e5}
\begin{aligned}
\frac{d\Phi}{dt}
&\leq  - 2k (2k-1)\int_{\Omega}v^{2k-2} \frac{g(u)}{b'(u)} |\nabla v|^{2} \, dx
 + 2k\int_{\Omega}v^{2k-1}\frac{f(u)}{g(u)} \, dx  \\
& \leq - \frac{2(2k-1)g_{m}}{k b'_{M}} \int_{\Omega}  |\nabla v ^{k}|^{2} \, dx
  + 2kc \int_{\Omega}v^{2k+p}\, dx.
\end{aligned}
\end{equation}
From  \eqref{k},  H\"{o}lder, and Young inequalities, we infer
\begin{equation} \label{e6}
\begin{aligned}
\int_{\Omega}v^{2k+p}\, dx
&\leq |\Omega|^{m_1}\Big( \int_{\Omega}v^{\frac{k(2n-3)}{n-2}}\, dx \Big)^{m_2}   \\
& \leq m_1|\Omega| + m_2 \int_{\Omega}v^{\frac{k(2n-3)}{n-2}}\, dx,
\end{aligned}
\end{equation}
where
\[
m_1=\frac{k(2n-3)-(n-2)(2k+p)}{k(2n-3)}, \quad m_2=\frac{(n-2)(2k+p)}{k(2n-3)}.
\]
From \eqref{e6} and the Cauchy-Schwartz inequality we have:
\begin{equation} \label{e7}
\begin{aligned}
\int_{\Omega}v^{\frac{k(2n-3)}{n-2}}\,dx
&\leq  \Big(\int_{\Omega} v^{2k}\,dx \Big)^{1/2}
\Big(\int_{\Omega} v^{\frac{2k(n-1)}{n-2}}\,dx \Big)^{1/2}   \\
&\leq\Big(\int_{\Omega} v^{2k}\,dx \Big)^{\frac{3}{4}}
\Big(\int_{\Omega} (v^{k})^{\frac{2n}{n-2}}\,dx \Big)^{1/4}.
\end{aligned}
\end{equation}
Applying  the Sobolev inequality (see \cite{t1}) to the last term in \eqref{e7},
for $n>3$, we obtain
\begin{equation} \label{e8}
\begin{aligned}
\|v^{k}\|_{L^{\frac{2n}{n-2}}(\Omega)}^{\frac{n}{2(n-2)}}
&\leq(c_{s})^{\frac{n}{2(n-2)}}\|v^{k}\|_{W^{1,2}(\Omega)}^{\frac{n}{2(n-2)}}  \\
&\leq (c_{s})^{\frac{n}{2(n-2)}}
\Big( \|\nabla v^{k}\|_{L^{2}(\Omega)}^{\frac{n}{2(n-2)}}
 +  \| v^{k}\|_{L^{2}(\Omega)}^{\frac{n}{2(n-2)}} \Big)
\end{aligned}
\end{equation}
In the case, $n=3$, we have
\begin{equation}
\begin{aligned} \label{e15}
\|v^{k}\|_{L^{\frac{2n}{n-2}}(\Omega)}^{\frac{n}{2(n-2)}}
&\leq (c_{s})^{\frac{n}{2(n-2)}}\|v^{k}\|_{W^{1,2}(\Omega)}^{\frac{n}{2(n-2)}}   \\
&\leq 2^{\frac{4-n}{2(n-2)}}(c_{s})^{\frac{n}{2(n-2)}}
 \Big( \|\nabla v^{k}\|_{L^{2}(\Omega)}^{\frac{n}{2(n-2)}}
 +  \| v^{k}\|_{L^{2}(\Omega)}^{\frac{n}{2(n-2)}}  \Big).
\end{aligned}
\end{equation}
Here, $c_{s}$ is the best constant in the Sobolev inequality.

 By inserting \eqref{e8} in \eqref{e7} for $n>3$ and \eqref{e15} in \eqref{e7} for 
$n=3$, we have 
\begin{equation} \label{e9}
\begin{aligned}
&\int_{\Omega}v^{\frac{k(2n-3)}{n-2}}\, dx \\
&\leq c_0\Big(\int_{\Omega} v^{2k}\, dx \Big)^{\frac{3}{4}}
\Big( \|\nabla v^{k}\|_{L^{2}(\Omega)}^{\frac{n}{2(n-2)}}
  +  \| v^{k}\|_{L^{2}(\Omega)}^{\frac{n}{2(n-2)}} \Big)  \\
&= c_0\Big(\int_{\Omega} v^{2k}\, dx \Big)^{\frac{3}{4}}
 \Big(\int_{\Omega} |\nabla v^{k}|^{2}\, dx \Big)^{\frac{n}{4(n-2)}}
 +  c_0 \Big(\int_{\Omega} v^{2k}\, dx \Big)^{\frac{2n-3}{2(n-2)}},
\end{aligned}
\end{equation}
where
\[
c_0= \begin{cases}
2^{\frac{4-n}{2(n-2)}}(c_{s})^{\frac{n}{2(n-2)}}, & \text{for }  n=3,  \\[4pt]
(c_{s})^{\frac{n}{2(n-2)}}, & \text{for } n>3.
\end{cases}
\]
Now, using Young's inequality we obtain
\begin{equation}\label{e16}
\begin{aligned}
&\int_{\Omega}v^{\frac{k(2n-3)}{n-2}}\, dx  \\
& \leq \frac{c_0^{\frac{4(n-2)}{3n-8}}(3n-8)}{4(n-2)\epsilon^{\frac{n}{3n-8}}}
 \Phi^{\frac{3(n-2)}{3n-8}}
 + \frac{n\epsilon}{4(n-2)}\int_{\Omega} |\nabla v^{k}|^{2}\, dx
  +   c_0 \Phi^{\frac{2n-3}{2(n-2)}},
\end{aligned}
\end{equation}
where $ \epsilon$ is a positive constant to be determined later.
Substituting  \eqref{e16} into \eqref{e6} yields
\begin{align*}
2kc \int_{\Omega}v^{2k+p}\, dx 
&\leq  2kc m_2 \Big\{ \frac{(3n-8)}{4(n-2)\epsilon^{\frac{n}{3n-8}}}
 c_0^{\frac{4(n-2)}{3n-8}}\Phi^{\frac{3(n-2)}{3n-8}} 
 +  \frac{n\epsilon}{4(n-2)}\int_{\Omega} |\nabla v^{k}|^{2}\, dx\\
&\quad+   c_0 \Phi^{\frac{2n-3}{2(n-2)}}\Big\}
 +2kc m_1|\Omega|.
\end{align*}
 By inserting  the last inequality in \eqref{e5}, we have
\[
\frac{d\Phi}{dt} \leq \Big(- \frac{2(2k-1)g_{m}}{k b'_{M}} + \frac{nkc m _2\epsilon}{2(n-2)}\Big) \int_{\Omega}  |\nabla v ^{k}|^{2} \, dx
  +k_1 +k_2\Phi^{\frac{3(n-2)}{3n-8}}
  +  k_3\Phi^{\frac{2n-3}{2(n-2)}},
\]
where
\[
k_1=2kcm_1|\Omega|, \quad
k_2=\frac{2kcm_2(c_0)^{\frac{4(n-2)}{3n-8}}(3n-8)}{4(n-2)\epsilon^{\frac{n}{3n-8}}},
\quad k_3=2kcc_0m_2.
\]
For
\[
\epsilon=\frac{4(n-2)(2k-1)g_{m}}{nk^{2}c m_2b'_{M}},
\]
the above inequality becomes
\[
\frac{d\Phi}{dt}\leq k_1+ k_2\Phi ^{\frac{3(n-2)}{3n-8}}
+ k_3 \Phi ^{\frac{2n-3}{2(n-2)}}.
\]
Thus,
\begin{equation}\label{e111}
\frac{d\Phi}{k_1+ k_2\Phi ^{\frac{3(n-2)}{3n-8}}
  +  k_3 \Phi ^{\frac{2n-3}{2(n-2)}}}\leq d t.
\end{equation}
We integrate from $0$ to $t$  to  obtain
\[
\int_{\Phi(0)}^{\Phi(t)} \frac{d\xi}{k_1+ k_2\xi ^{\frac{3(n-2)}{3n-8}}
  + k_3 \xi^{\frac{2n-3}{2(n-2)}}}\leq t,
\]
where
\[
\Phi(0)=\int_{\Omega}\Big(\int_0^{u_0(x)}\frac{b'(y)}{g(y)}dy\Big)^{2k}dx.
\]
Passing to the limit as $ t\rightarrow T^{-}$, we conclude that
\[
\int_{\Phi(0)}^{+\infty} \frac{d\xi}{k_1+ k_2\xi ^{\frac{3(n-2)}{3n-8}}
  +  k_3 \xi^{\frac{2n-3}{2(n-2)}}}\leq T.
\]
The proof is complete.
\end{proof}

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\end{document}