% submitted to Jesus Ildefonso Diaz on January 10, 2018.
% 
\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2018 (2018), No. 48, pp. 1--12.\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-2018/48\hfil Blow-up of solutions]
{Blow-up of solutions to singular parabolic equations with nonlinear sources}

\author[N. T. Duy, A. N. Dao \hfil EJDE-2018/48\hfilneg]
{Nguyen Tuan Duy, Anh Nguyen Dao}

\address{Nguyen Tuan Duy \newline
Department of Fundamental Sciences,
University of Finance-Marketing,
2/4 Tran Xuan Soan St., Tan Thuan Tay Ward,
Dist. 7, HCM City, Vietnam}
\email{tuanduy2312@gmail.com}

\address{Anh Nguyen Dao (corresponding author)\newline
Applied Analysis Research Group,
Faculty of Mathematics and Statistics,
Ton Duc Thang University, Ho Chi Minh City, Vietnam}
\email{daonguyenanh@tdt.edu.vn}

\dedicatory{Communicated by Jesus Ildefonso Diaz}

\thanks{Submitted January 10, 2018. Published February 15, 2018.}
\subjclass[2010]{35K55, 35K67, 35K65}
\keywords{Nonlinear parabolic equations;  blow-up of solutions;
\hfill\break\indent  gradient estimates}

\begin{abstract}
 We prove the existence of a local weak solutions for semi-linear
 parabolic  equations with a strong singular absorption and a general source.
 Also, we investigate criteria for the solutions to blow up in finite time.
\end{abstract}

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

\section{Introduction}

In this article, we are interested in nonnegative solutions of the
 equation
\begin{equation}\label{plap1}
\begin{gathered}
\partial_{t}u-\Delta u+u^{-\beta}\chi_{\{u>0\}} = f(u, x, t)\quad
\text{in } \Omega\times(0, T),\\
u(x,t)=0 \quad \text{on }   \partial\Omega\times(0, T),\\
u(x,0)=u_{0}(x) \quad \text{in } \Omega,
\end{gathered}
\end{equation}
where  $\Omega$ is a bounded domain in $\mathbb{R}^N$,  $\beta\in(0,1)$,
and $\chi_{\{u>0\}}$ denotes the
characteristic function of the set of points $(x,t)$ where
$u(x,t)>0$, i.e:
\[
\chi_{\{u>0\}}= \begin{cases}
1, & \text{if } u>0, \\
0, & \text{if } u\leq 0.
\end{cases}
\]
Note that the absorption term $u^{-\beta}\chi_{\{u>0\}}$ becomes singular
when $u$ is near to $0$, and we impose
$u^{-\beta}\chi_{\{u>0\}}=0$ whenever $u=0$.  Through this paper,
$f: [0,\infty)\times \overline{\Omega}\times[0,\infty) \to \mathbb{R}$ will be
assumed a nonnegative function satisfying the  hypothesis
\begin{itemize}
\item[(H1)] $ f\in\mathcal{C}^1\big([0,\infty)\times \overline{\Omega}
\times[0,\infty)\big)$,  $f(0,x,t)=0$, for all $(x,t)\in \Omega\times(0, \infty)$,
 and $f(u,x,t)\leq h(u)$ for all $(x,t)\in \Omega\times(0, \infty)$,
where  $h$ is a locally Lipschitz function on $[0, \infty)$, and $h(0)=0$.
\end{itemize}
 In the sequel, we always consider nonnegative initial data  $u_0\not=0$.


Problem \eqref{plap1}  can be considered as a limit of mathematical  models
describing enzymatic kinetics (see \cite{Banks}),  or  the
Langmuir-Hinshelwood model of the heterogeneous chemical catalyst
(see, e.g. \cite[p. 68]{Strieder-Aris} and \cite{D Pitman,Phillips}).
This problem has been  studied by the authors in
\cite{AnhDiaz,Davila-M,Kawara,Kawohl,Levine, Phillips,
 Winkler}, and references therein.
These authors have considered the existence and uniqueness, and the qualitative
behavior of these solutions.  For example, when $f=0$,
Phillips \cite{Phillips}  proved the existence of solution for the Cauchy
problem associating to equation \eqref{plap1}.
A partial uniqueness of  solution of equation \eqref{plap1}  was proved by
 Davila and Montenegro  \cite{Davila-M},
 for a class of  solutions with initial data
$u_0(x)\geq C \operatorname{dist}(x,\partial\Omega)^\mu$, for
$\mu\in(1, \frac{2}{1+\beta})$ (see also \cite{AnhDiazPaul}
the uniqueness in a different class of solutions).
A beautiful result established by  Winkler, \cite{Winkler},
showed that the uniqueness of solution fails in general.
One of the interesting behaviors of solutions of  \eqref{plap1}
is the extinction  that any solution  vanishes after a finite time
even beginning with a positive initial data, see  \cite{Phillips,Kawara}
(see also \cite{AnhDiaz} for a quasilinear equation of this type).
 It is known that this phenomenon occurs according  to the presence of
the nonlinear singular absorption  $u^{-\beta}\chi_{\{u>0\}}$.

Equation \eqref{plap1} with source term $f(u)$ satisfying the sublinear condition,
 i.e:  $f(u)\leq C(u+1)$,  was considered by  Davila and Montenegro
 \cite{Davila-M}. The authors
    proved the existence of solution and showed that the measure of the set
$\{(x,t)\in\Omega\times(0,\infty):  u(x,t)=0\}$ is positive
(see also a more general statement in \cite{Diaz1}). In other words,
the solution may exhibit the quenching behavior. Still in the sublinear
case with source term  $\lambda f(u)$,
Montenegro \cite{Mon}  proved that there is  a real number $\lambda_0>0$
such that for any $\lambda\in(0,\lambda_0)$, there is $t_0>0$ such that
     \[
u(x,t_0)=0, \quad \forall x\in \Omega.
     \]
He called this phenomenon complete quenching.

 From our knowledge, equation \eqref{plap1} with a general source term
$f(u,x,t)$ has not been studied completely. Thus, we would like to investigate
first  the existence of solutions to equation \eqref{plap1}.
Furthermore,  it is well known that nonlinear parabolic equations with general
source $f(u,x,t)$ may cause the finite time blow-up. As mentioned above,
the nonlinear absorption $u^{-\beta}\chi_{\{u>0\}}$ causes the complete
quenching phenomenon.  Thus, it is interesting to see when the complete
 quenching prevails the blow-up, and conversely. We also note that the
above qualitative behavior of solutions  were studied by the authors
in \cite{AnhDiaz1,AnhDiaz2} for the $p$-Laplacian equation in one-dimension
of this type. In this paper, we only consider the blowing-up solutions  of
\eqref{plap1}.  Before giving  our results,
 it is necessary to introduce a notion of weak solution  of equation
\eqref{plap1}.

 \begin{definition} \rm
 Let $u_{0}\in L^\infty(\Omega)$. A nonnegative function $u(x,t)$ is called a
weak solution of equation
 \eqref{plap1} if $u^{-\beta}\chi_{\{u>0\}}\in L^{1}(\Omega\times(0, T))$,
and $u\in L^{2}(0, T; W_{0}^{1,2}(\Omega))\cap
 L^{\infty}(\Omega\times(0, T))\cap\mathcal{C}([0, T
 );L^{1}(\Omega))$ satisfies equation
 \eqref{plap1} in
 the sense of distributions $\mathcal{D'}(\Omega\times(0, T))$, i.e.
 \begin{equation}\label{plapdef}
 \int_{0}^{T}\int_{\Omega}\left(-u\phi_{t}+\nabla u \cdot \nabla \phi+u^{-\beta}\chi
 _{\{u>0\}}\phi  - f(u,x,t)\phi \right)    \,dx\,dt=0,
\end{equation}
for all $\phi\in  \mathcal{C}_{c}^{\infty}(\Omega\times(0, T))$.
 \end{definition}

Our first result is  the  existence of a local solution to \eqref{plap1}.

\begin{theorem}\label{thelocalexist}
 Let $u_{0}\in L^\infty(\Omega)$, and let $f$ satisfy $(H1)$.
Then, there exists a finite time $T=T(u_0)>0$ such that equation
\eqref{plap1} has a  maximal weak solution $u$ in $\Omega\times(0, T)$,
i.e: for any weak solution $v$ in $\Omega\times(0, T)$, we have
 \[
 v\leq u, \quad \text{in } \Omega\times(0, T).
 \]
  Moreover,  there is a positive constant $C=C( f, \|u_0\|_\infty)$ such
 that
 \begin{equation}\label{plap2}
 |\nabla u(x,\tau)|^2 \leq C u^{1-\beta} \left(\tau^{-1}+1\right), \quad
\text{for a.e. } (x,\tau) \in \Omega\times(0, T).
 \end{equation}
 Besides, if   $\nabla (u_0^{1/\gamma})\in L^\infty(\Omega)$, with
$\gamma=\frac{2}{1+\beta}$, then there is a positive constant $C=C(f,  u_0)$
 such that
 \begin{equation}\label{plap2a}
 |\nabla u(x,\tau)|^2\leq C  u^{1-\beta} (x, \tau), \quad \text{for a.e. }
(x,\tau) \in \Omega\times(0, T).
 \end{equation}
 \end{theorem}

\begin{remark} \label{rmk3}\rm
 Theorem \ref{thelocalexist}  implies that  $u$ is continuous up to the boundary.
Furthermore,  $u$ is continuous up to  $t=0$  provided
 $\nabla (u^{1/\gamma}_0) \in L^\infty(\Omega)$
(see for example \cite{AnhDiaz1,AnhDiaz,AnhDiaz2,Phillips}).
 \end{remark}

 \begin{remark} \label{rmk4}\rm
 Similarly as in the case of $p$-Laplacian of the equation of this type
(see \cite{AnhDiaz1}), when $f(u,x,t)=f(u)$,  the results of Theorem
\ref{thelocalexist} still hold s for $f$ a locally Lipschitz function on
$[0,\infty)$,  instead of $f\in C^2([0,\infty))$, required in the previous
works (see for example \cite{Davila-M,Mon}).
  For example, our existence result can take into account the function
$f(u)=(u-1)^+u$.
 \end{remark}

 After that,  we  study  the global nonexistence  of solutions of 
\eqref{plap1}, the so called finite time blowing-up solution.
In this paper,  we point out some criteria on initial data $u_0$
 to guarantee  the blow-up  of solution in a finite time.
For simplicity, we consider $f(u,x,t)=f(u)$. We will give the first
result of blow-up for the  superlinear case, i.e. $f(u)=u^p$, for $p>1$.
Then,  it is convenient to introduce the energy functional
\begin{equation}\label{plapenergy}
E(t)=\int_\Omega \Big(\frac{1}{2}| \nabla u(t)|^2+\frac{1}{1-\beta}u^{1-\beta}(t)
-\frac{1}{p+1} u^{p+1}(t) \Big) dx,
\end{equation}

 Our first criterion considers  $E(0)$ negative.

\begin{theorem}\label{theblow-up1}
 Let $u_0\in L^\infty(\Omega)\cap H^{1}_0(\Omega)$.
Suppose that  $f(u)=u^p$, for $p>1$, and $E(0)\leq 0$.
 Let $u$ be a solution of equation \eqref{plap1}.
Then, $u$ blows up in a finite time.
\end{theorem}

It is interesting to find out an optimal condition  of  nonlinear
source $f(u)$ such that the explosion of solution holds.
Let us remind   a necessary and sufficient condition for blow-up of
solutions of equation \eqref{plap1} without the singular absorption
$u^{-\beta}\chi_{\{u>0\}}$,
\begin{equation}\label{plap1a}
\begin{gathered}
\partial_{t}u-\Delta u = f(u)\quad \text{in }\Omega\times(0, T),\\
u(x,t)=0 \quad \text{on } \partial\Omega\times(0, T),\\
u(x,0)=u_{0}(x) \quad \text{in } \Omega,
\end{gathered}
\end{equation}
It is known that if $f$ is a convex function on $(0,\infty)$, and
\begin{equation}\label{condblowup}
\int_{a}^{\infty}\frac{1}{f(s)}ds< +\infty,
\end{equation}
for some $a>0$, then the solution $u$ of \eqref{plap1a} must blow up
in a finite time provided that $u_0$ is large enough (see also
\cite{GaVaz1,GaVaz2}, necessary and sufficient conditions for blow-up of
solution of  the porous medium equation).
One can take for instance a  typical weak superlinear $f(u)=(1+u)\log^p(1+u)$,
which is convex and  satisfies \eqref{condblowup},  with $u\geq 0$, $p>1$.
We also note that  only condition \eqref{condblowup} is not sufficient to
guarantee the explosion of $u$ in a finite time if lacking of the convexity
of $f$, see \cite[Theorem 19.15]{QuiSou}.
Here, we will demonstrate that  the explosion of solution of \eqref{plap1}
 occurs with $f$ as above.

\begin{theorem}\label{theblow-up2}
 Let $f(u,x,t)=f(u)$ be a locally  Lipschitz function on $[0,\infty)$.
Suppose that $f(u)$  is a convex function on $(0,\infty)$, and
$f$ satisfies \eqref{condblowup} for some $a>0$.
Then, the solution $u$ of \eqref{plap1} blows up in a finite time
if $u_0\in\mathcal{C}_b(\Omega)$ is large enough.
\end{theorem}

Our proof of Theorem \ref{theblow-up2} is based on the first eigenvalue
method introduced by Kaplan \cite{Kaplan}.
Note that our equation contains the singular term $u^{-\beta}\chi_{\{u>0\}}$,
which causes a difficulty in estimating this solution. To overcome this obstacle,
we show that if $u_0$ is positive inside of $\Omega$ and  large enough,
then $u(t)$ is also positive inside of $\Omega$ for a certain
large time interval. Note that the concave method used  by the authors
in \cite{AnhDiaz1}  to prove the explosion of solutions for $p$-Laplacian
equation  in one dimension of this type cannot be applied to this situation.
Finally,  one can find a rich source of  topic of explosive solutions
in \cite{Gal-Vaz,Lev,QuiSou, Weissler}, and references therein.

This article is organized as follows:
In the next section, we prove the  existence  of a local solution to \eqref{plap1}.
To do that, we prove  some gradient estimates for the approximating solutions.
The last section is devoted to study of blowing-up of solutions.

The notation that will be used in this paper is  the following:
we denote by $C$ a general positive constant, possibly varying from line to line.
Furthermore, the constants which depend on parameters will be emphasized by
using parentheses. For example, $C=C(p,\beta,\tau)$ means that $C$
depends on $p,\beta,\tau$.


\section{Existence of a local solution}

In this section, we consider a regularized equation of \eqref{plap1}:
\begin{equation} \label{ePen}
\begin{gathered}
\partial_{t}u_{\varepsilon}-\Delta u_{\varepsilon}+g_{\varepsilon
}(u_{\varepsilon})= f(u_\varepsilon,x,t) \quad \text{in } \Omega\times(0,\infty),\\
u_{\varepsilon}=\eta \quad \text{on } \partial\Omega\times(0,\infty),\\
u_{\varepsilon}(0)=u_{0}+\eta \quad \text{on } \Omega
\end{gathered}
\end{equation}
for any   $0<\eta<\varepsilon$,  with
$g_{\varepsilon}(s)= \psi_{\varepsilon}(s)s^{-\beta}$,
$\psi_{\varepsilon}(s)=\psi({\frac{s}{\varepsilon}})$, and
$\psi \in{\mathcal{C}^{\infty}(\mathbb{R})}$ is a non-decreasing function on
$\mathbb{R}$ such that $\psi(s)=0$ for $s\leq1$, and $\psi(s)=1$ for $s\geq2$.
Note  that $g_{\varepsilon}$
is a globally Lipschitz function for any $\varepsilon>0$.
We will show that    solution $u_{\varepsilon,\eta}$ of  equation \eqref{ePen}
tends to a solution of equation \eqref{plap1} as
$\eta, \varepsilon\to 0$.
In  passing to the limit, we need to  derive some gradient estimates  for
solution $u_{\varepsilon,\eta}$, see also
\cite{AnhDiazPaul,Davila-M,Phillips}.
Then, we have  the following result.

\begin{lemma}\label{lemgradient}
Let $ u_0\in\mathcal{C}^\infty_c(\Omega)$, $u_0\not=0$.
 There exists a classical unique solution $u_{\varepsilon,\eta}$ of
\eqref{ePen} in $\Omega\times(0, T)$.
\begin{itemize}
\item[(i)] There is  a constant $C>0$ only depending on
$\beta, T, f, \|u_0\|_\infty$ such that
\begin{equation}\label{gradient1}
|\nabla u_{\varepsilon,\eta}(x,\tau)|^2
\leq C u_{\varepsilon,\eta}^{1-\beta} (x,\tau) \big(\tau^{-1} +1\big),
\quad\text{for any } (x,\tau)\in \Omega\times(0, T),
\end{equation}

\item[(ii)] If  $\nabla (u_0^{1/\gamma})\in L^\infty(\Omega)$, then we obtain
\begin{equation}\label{uniplapgradient}
|\nabla u_{\varepsilon,\eta}(x,\tau)|^2\leq C  u_{\varepsilon,\eta}^{1-\beta}
(x, \tau), \quad\text{for any } (x,\tau)\in \Omega\times(0, T),
  \end{equation}
  with $C>0$ merely depends on $\beta, T, f, \|u_0\|_\infty,
\|\nabla (u^{1/\gamma}_0)\|_\infty$.
\end{itemize}
\end{lemma}

\begin{proof}
(1) Fix $\varepsilon\in (0,\|u_0\|_\infty)$. For any $\eta\in(0,\varepsilon)$,
the existence and uniqueness of a classical solution
$u_{\varepsilon,\eta}$ of problem \eqref{ePen} is well-known (see \cite{LaSoU}).
 We denote by $u=u_{\varepsilon,\eta}$ for short.
Let $\Gamma(t)$ be the flat solution of the ODE:
 \begin{equation}\label{plap5b}
\begin{gathered}
 \partial_t \Gamma = h(\Gamma),  \quad \text{in } [0,T'], \\
 \Gamma(0)=2\|u_0\|_{\infty},
 \end{gathered}
 \end{equation}
 where $h$ is the function in  (H1) above, and $T'$ is the maximal existence
time of $\Gamma(t)$. Note that $T'$ depends merely on $\|u_0\|_{\infty}$,
see \cite[Chapter 1]{Cod-Lev}.
 It follows from the comparison principle that
 \[
 \eta\leq u\leq \Gamma(t),\quad\forall t\in [0, T'].
 \]
Let us put  $u=\phi(v)=v^\gamma$, with $\gamma=2/(1+\beta)$.
Then
\begin{equation}\label{2.3b}
v_t-\Delta v=\frac{\phi''}{\phi'}|\nabla v|^2
-\frac{1}{\phi'}\big(g_\varepsilon(\phi(v))-f(\phi(v),x,t) \big).
\end{equation}
 For any $\tau\in(0, T'/3)$, let us consider a cut-off function
$\xi(t)\in\mathcal{C}^\infty(0,\infty)$, $0\leq \xi(t)\leq 1$, such that
\begin{equation*}
 \xi(t)= \begin{cases}
1, &\text{on } [\tau,\frac{T'}{3}], \\
 0,& \text{outside } (\frac{\tau}{2}, \frac{T'}{3}+\frac{\tau}{2}),
 \end{cases}
\end{equation*}
and $|\xi_t|\leq \frac{c_0}{\tau}$, for some constant $c_0>0$.

Then, we set  $w=\xi(t)|\nabla v|^2$.
 If $\max_{\Omega\times[0,T]} w=0$, then
$\nabla v(\tau)=0$, so estimate \eqref{gradient1} is trivial.

If not, there is a point $(x_0, t_0)\in \Omega\times(0,2T'/3)$ such that
 $\max_{\Omega\times[0,T']} w= w(x_0, t_0)$.
Thus, we have at $(x_0, t_0)$:
 \begin{equation}\label{2.3c}
 w_t=0,\quad \nabla w=0, \quad \Delta w\leq 0.
 \end{equation}
 This implies
 \[
 0\leq w_t-\Delta w= \xi_t |\nabla v|^2+2\xi(t) \big(\nabla v.\nabla v_t
- \nabla v.\nabla(\Delta v)\big)-2\xi(t)|D^2 v|^2,
 \]
or
\begin{equation}\label{2.4b}
 0\leq \xi_t |\nabla v|^2+2\xi(t)\nabla v\cdot\nabla(v_t  -\Delta v).
 \end{equation}
A combination of \eqref{2.3b} and \eqref{2.4b} provides us with
 \begin{equation*}
  0\leq \xi_t |\nabla v|^2+2\xi(t)\nabla v\cdot\nabla
 \big( \frac{\phi''}{\phi'}|\nabla v|^2
-\frac{g_\varepsilon(\phi(v))-f(\phi(v),x,t)}{\phi'} \big).
  \end{equation*}
Since $\xi(t_0)>0$, we obtain
   \begin{equation}\label{2.4ab}
    0\leq \frac{1}{2}\xi^{-1}  \xi_t |\nabla v|^2+ \nabla v\cdot\nabla
\big( \frac{\phi''}{\phi'}|\nabla v|^2
-\frac{g_\varepsilon(\phi(v))-f(\phi(v),x,t)}{\phi'} \big).
    \end{equation}
 At the moment, we estimate the terms on the right hand side of \eqref{2.4ab}.
 First of all, we have from \eqref{2.3c} that $\nabla (|\nabla v(x_0, t_0)|^2)=0$,
so
 \begin{equation}\label{2.3cc}
 \nabla v \cdot\nabla \big(\frac{\phi''}{\phi'
  }|\nabla v|^2\big)= \nabla v \cdot\nabla \big(\frac{\phi''}{\phi'
    }\big)|\nabla v|^2=
  (\gamma-1)(2\gamma-3)v^{-2} |\nabla v|^4.
 \end{equation}
 Next, we have
 \begin{equation}\label{2.3d}
 \begin{split}
& \nabla v.\nabla\Big( \frac{f(\phi, x_0,t_0)}{\phi'} \Big)\\
&= \frac{D_x f(\phi, x_0, t_0)}{\phi'} \nabla v
+ D_u f (\phi, x_0, t_0) |\nabla v|^2  - f(\phi,x_0,t_0)
\frac{\phi''}{\phi^{\prime2}} |\nabla v|^2  \\
&=\frac{1}{\gamma} D_x f(\phi,x_0,t_0) v^{1-\gamma} \nabla v+ D_u f(\phi,x_0,t_0)
|\nabla v|^2 \\
&\quad   - (\frac{\gamma-1}{\gamma} ) f(\phi,x_0,t_0)   v^{-\gamma} |\nabla v|^2.
 \end{split}
 \end{equation}
Since $f\geq 0$, and $\gamma>1$,  it follows from \eqref{2.3d} that
 \begin{equation}\label{2.3e}
  \begin{split}
  \nabla v \cdot\nabla\Big( \frac{f(\phi, x_0,t_0)}{\phi'} \Big)
   \leq
   \frac{1}{\gamma} |D_x f(\phi,x_0,t_0)| v^{1-\gamma}|\nabla v|
+ |D_u f(\phi,x_0,t_0)| |\nabla v|^2.
  \end{split}
  \end{equation}
  Concerning the last term, we have
 \[
 \nabla v  \cdot\nabla\Big(  \frac{g_\varepsilon(\phi)}{\phi'}\Big)
=(g_{\varepsilon }'-g_{\varepsilon}\frac{\phi''}{\phi'^{2}
 })|\nabla v|^2
=\Big(  \psi_{\varepsilon}'(\phi)v^{-\beta}
 -(\beta+\frac{\gamma-1}{\gamma})\psi_{\varepsilon}(\phi)v^{-(1+\beta
 )\gamma}\Big)  |\nabla v|^{2}.
 \]
 Since $\psi'_\varepsilon\geq 0$, and $0\leq \psi_\varepsilon\leq 1$, we obtain
 \begin{equation}\label{2.3f}
 -\nabla v \cdot\nabla \left(  \frac{g(\phi)}{\phi'}\right)\leq (\beta
 +\frac{\gamma-1}{\gamma})v^{-(1+\beta)\gamma}  |\nabla v|^{2}.
 \end{equation}
By inserting   \eqref{2.3cc}, \eqref{2.3e} and \eqref{2.3f} into  \eqref{2.4ab},
 we obtain
\begin{equation}\label{2.7b}
\begin{split}
(\gamma-1)v^{-2}|\nabla v|^4
&\leq \frac{1}{2}\xi^{-1} \xi_t |\nabla v|^2
+ (\beta +1 -\frac{1}{\gamma})v^{-(1+\beta)\gamma}|\nabla v|^2
\\
&\quad +|D_u f||\nabla v|^2+\frac{1}{\gamma}v^{1-\gamma}|D_x f||\nabla v|.
\end{split}
\end{equation}
Now, we multiply both sides of \eqref{2.7b} by $v^2$ to get
\begin{equation}\label{2.8b}
\begin{split}
(\gamma-1)|\nabla v|^4
&\leq \frac{1}{2}\xi^{-1}|\xi_t| v^2 |\nabla v|^2
+ (\beta +1 -\frac{1}{\gamma})|\nabla v|^2
+ v^2 |D_u f||\nabla v|^2 \\
&\quad +\frac{1}{\gamma}v^{3-\gamma}|D_x f||\nabla v|.
\end{split}
\end{equation}
If $|\nabla v(x_0,t_0)|\leq 1$, then  $w(x_0, t_0)\leq 1$.
This leads  to $w(x,\tau)\leq 1$, thereby proves
\[
|\nabla u(x, \tau)|^2\leq \frac{4}{(1+\beta)^2} u^{1-\beta}(x, \tau).
\]
Then, estimate \eqref{gradient1} follows immediately.

If not, we have $|\nabla v(x_0, t_0)|>1$,  it follows then from \eqref{2.8b}
\begin{align*}
(\gamma-1)|\nabla v|^4
&\leq \frac{1}{2}\xi^{-1}|\xi_t| v^2 |\nabla v|^2 + (\beta +1
-\frac{1}{\gamma})|\nabla v|^2
+ v^2 |D_u f||\nabla v|^2 \\
&\quad + \frac{1}{\gamma}v^{3-\gamma}|D_x f||\nabla v|^2.
\end{align*}
By simplifying the term $|\nabla v|^2$ both sides of the last inequality, we obtain
\[
(\gamma-1)|\nabla v|^2\leq \frac{1}{2}\xi^{-1}|\xi_t| v^2
 + (\beta +1 -\frac{1}{\gamma}) + v^2 |D_u f|+
\frac{1}{\gamma}v^{3-\gamma}|D_x f|.
\]
Multiplying both sides of the above inequality by $\xi(t_0)$ yields
\begin{equation}\label{2.9b}
(\gamma-1)\xi(t_0)|\nabla v|^2\
leq \frac{1}{2}|\xi_t| v^2 + \xi(t_0)\Big((\beta +1 -\frac{1}{\gamma})
+ v^2 |D_u f|+ \frac{1}{\gamma}v^{3-\gamma}|D_x f|\Big).
\end{equation}
Recall that $w(x_0, t_0)=\xi(t_0)|\nabla v(x_0, t_0)|^2$, $0\leq \xi(t)\leq 1$,
and  $|\xi_t|\leq \tau^{-1}$. It follows from \eqref{2.9b}
that there is a constant $C=C(\beta)>0$ such that
\[
w(x_0, t_0)\leq C\big(\tau^{-1}v^2   + v^2 |D_u f|+
v^{3-\gamma}|D_x f|+1\big).
\]
Since $w(x_0, t_0)\geq w(x, \tau)=|\nabla v(x,\tau)|^2$, we obtain
\[
|\nabla v(x,\tau)|^2\leq C\big(\tau^{-1}v^2   + v^2 |D_u f|+
v^{3-\gamma}|D_x f|+1\big)
\]
Moreover, we have
\[
v^\gamma(x,t)=u(x,t)\leq \Gamma(T'), \quad\text{for any }
 (x,t)\in \Omega\times[0, T'].\]
Then
\begin{align*}
|\nabla v(x,\tau)|^2
&\leq C\Big(\tau^{-1} \Gamma^{1+\beta}(T')
+ \Gamma^{1+\beta}(T')   \Theta(D_u f,\Gamma(T')) \\
&\quad + \Gamma^{\frac{1+3\beta}{2}}(T')  \Theta(D_x f,\Gamma(T'))+1\Big),
\end{align*}
with $\Theta(g,r)=\max_{0\leq s\leq r}\{|g(s)|\}$,
or
\begin{align*}
|\nabla u(x,\tau)|^2
&\leq C_1 u^{1-\beta} \Big(\tau^{-1} \Gamma^{1+\beta}(T')
+ \Gamma^{1+\beta}(T')   \Theta(D_u f,\Gamma(T'))\\
&\quad + \Gamma^{\frac{1+3\beta}{2}}(T')  \Theta(D_x f,\Gamma(T'))+1\Big).
\end{align*}
Thus, (i) follows by choosing $T=T'/3$.
\smallskip

(ii) The proof of estimate \eqref{uniplapgradient} is similar  to the
one of estimate \eqref{gradient1}. We just make a slight change by
 considering a cut-off function
$\overline{\xi}(t)\in \mathcal{C}^\infty(\mathbb{R})$ (instead of $\xi(t)$ above),
such that $0\leq \overline{\xi}(t)\leq 1$,  $\overline{\xi}_t(t)\leq 0$, and
\[
\overline{\xi}(t)=\begin{cases}
1, & \text{if } t\leq T'/3,\\
0, & \text{if } t\geq 2T'/3.
\end{cases}
\]
 Then, we observe that either $w(x,t)$ attains  its maximum  at the initial data, i.e.
\[
\max_{(x,t)\in I\times[0,2T_0]} w(x,t)=w(x_0, 0)
=  \overline{\xi}(0) |\nabla v(x_0, 0)|^2\leq \|\nabla(u^{1/\gamma}_0)\|^2_\infty,
\]
for some $x_0\in \Omega$,
which implies
\begin{equation}\label{plap15}
|\nabla u(x,\tau)|^2\leq \gamma^2  \|\nabla (u^{1/\gamma}_0)\|^2_\infty
 u^{1-\beta}(x,\tau), \quad \text{for all } x\in \Omega.
\end{equation}
Thus, we obtain estimate \eqref{uniplapgradient} immediately;
or there is a point  $(x_0, t_0)\in \Omega\times(0, 2T'/3)$ such that
\[
\max_{(x,t)\in \Omega\times[0,T']} w(x,t)=w(x_0, t_0)
\]
Then, we repeat the proof of (i) for this case until \eqref{2.7b} to get
\begin{align*}
(\gamma-1)v^{-2}|\nabla v|^4
&\leq \frac{1}{2}\overline{\xi}^{-1} \overline{\xi}_t |\nabla v|^2
 + (\beta +1 -\frac{1}{\gamma})v^{-(1+\beta)\gamma}|\nabla v|^2  \\
&\quad +|D_u f||\nabla v|^2+\frac{1}{\gamma}v^{1-\gamma}|D_x f||\nabla v|.
\end{align*}
Since $\overline{\xi}_t(t)\leq 0$, from the above inequality we have
\[
\begin{split}
(\gamma-1)v^{-2}|\nabla v|^4\leq (\beta +1 -\frac{1}{\gamma})v^{-(1+\beta)\gamma}
|\nabla v|^2
+|D_u f||\nabla v|^2+\frac{1}{\gamma}v^{1-\gamma}|D_x f||\nabla v|.
\end{split}
\]
By repeating  the proof of (i) after this inequality, we obtain
\begin{equation}\label{plap15a}
\begin{aligned}
|\nabla u(x,\tau)|^2
&\leq C  u^{1-\beta} (x, \tau)  \big(
  \Gamma^{1+\beta}(T')  \Theta(D_u f, \Gamma(T')) \\
&\quad + \Gamma^{\frac{1+3\beta}{2}} (T')  \Theta( D_x f, \Gamma(T'))
+ 1 \big),
\end{aligned}
\end{equation}
with  $C=C(\beta)>0$.
Combining \eqref{plap15} and \eqref{plap15a} yields estimate
\eqref{uniplapgradient}, and completes the proof.
\end{proof}


The proof of Theorem \ref{thelocalexist} is similar to the one in \cite{AnhDiaz}
(see also \cite{AnhDiazPaul}).  It applies Lemma \ref{lemgradient}
to pass to the limit as $\eta\to0$ and $\varepsilon\to 0$.
We let the reader to do it.


\section{Non-global existence of solutions}

In this section, we study the  non-global existence of solutions to
equation \eqref{plap1}.

\begin{proof}[Proof of Theorem \ref{theblow-up1}]
By multiplying by $u$ (resp. $u_t$) in equation \eqref{plap1},
we have the  integral equations
\begin{equation}\label{6.1}
\frac{1}{2}\frac{d}{dt} \int_{\Omega} u^2(x,t) dx
= -\int_{\Omega} \big(|\nabla u(x,t)|^2  + u^{1-\beta}(x,t) -u^{q+1}(x,t) \big)dx,
\end{equation}
and
\begin{equation}\label{6.2}
\begin{split}
&\int_0^t\int_{\Omega} |u_t|^2 dxds+ \int_{\Omega}\big( \frac{1}{2}|\nabla u(t)|^2
+ \frac{1}{1-\beta}u^{1-\beta}(t) - \frac{1}{q+1} u^{q+1}(t) \big) dx \\
&= \int_{\Omega}\big( \frac{1}{2}|\nabla u_0|^2
 + \frac{1}{1-\beta}u^{1-\beta}_0 - \frac{1}{q+1} u^{q+1}_0 \big) dx,
\end{split}
\end{equation}
 see \cite{QuiSou}.
By combining \eqref{6.1} and \eqref{6.2}, we obtain
\[
\frac{1}{2}\frac{d}{dt} \int_{\Omega} u^2(x,t) dx
= -2E(t)+\frac{1+\beta}{1-\beta} \int_{\Omega} u^{1-\beta} (x,t)dx
+\frac{q-1}{q+1}\int_\Omega u^{q+1} (x,t) dx.
\]
Since $E(0)\leq 0$, \eqref{6.2} implies $E(t)\leq 0$, for any $t>0$.
It follows then from the last inequality that
\begin{equation}\label{6.3}
\frac{1}{2}\frac{d}{dt} \int_{\Omega} u^2(x,t) dx
\geq \frac{q-1}{q+1}\int_\Omega u^{q+1} dx.
\end{equation}
By Holder's inequality, 
\begin{equation}\label{6.4}
\int_\Omega u^{2} dx
\leq \Big(\int_\Omega u^{q+1} dx\Big)^\frac{2}{q+1} |\Omega|^\frac{q-1}{q+1} .
\end{equation}
From  \eqref{6.3} and \eqref{6.4}, we obtain
$y'(t) \geq C y^{\frac{q+1}{2}} (t)$,
with
\[ 
y(t)=\int_{\Omega} u^2(x,t) dx, \quad
C=\frac{2(q-1)}{(q+1)|\Omega|^{\frac{q-1}{2}}}.
\]
This inequality implies that $y(t)\to +\infty$ as $t\to T^-_0$,
with $T_0=\frac{4\|u_0\|^{1-q}_{L^2(\Omega)}}{(q+1)|\Omega|^{\frac{q-1}{2}}} $.
\end{proof}


Next, we prove Theorem \ref{theblow-up2}. Since our proof below is
just a local argument, it suffices to consider initial data
$u_0(x)=c\Phi(x)$, with $c>0$, and $\Phi$ is  the first eigenfunction  of
the Dirichlet problem
 \begin{equation}\label{plapeigenvalue}
 \begin{gathered}
 -\Delta \Phi = \lambda_1 \Phi \quad \text{in }  \Omega, \\
 \Phi (x)=0,\quad \text{on }  \partial\Omega\,.
 \end{gathered}
\end{equation}
We have the following result.

 \begin{theorem}\label{theblow-up3}
Let $f(u,x,t)=f(u)$ be a locally  Lipschitz function on $[0,\infty)$ such that
$f(0)=0$. Suppose that $f(u)$  is a convex function on $(0,\infty)$, and
$f$ satisfies \eqref{condblowup} for some $a>0$. Let $u_0(x)=c\Phi(x)$,
where $c>0$ is large enough. Then, solution $u$ must blow up in a finite time.
 \end{theorem}

We first modify a result by Davila and Montenegro \cite{Davila-M}
to show that $u(t)$ is positive inside of $\Omega$ for a certain
large time interval $(0,T)$.

\begin{lemma}\label{lempositivity}
Suppose that $u_0(x)=C \Phi^\mu (x)$, for $C>1$, and for some
$\mu\in(1,\frac{2}{1+\beta})$. Then, we have
\begin{equation}\label{8.2}
u(x,t)\geq  C e^{-At}\Phi^\mu (x), \quad\forall (x,t)\in\Omega\times(0, T_{A,C}),
\end{equation}
where   $A>0$ is chosen later, and $T_{A,C}=\log (C)/A$.
\end{lemma}

\begin{proof}
For any  $\varepsilon>0$,  let $u_\varepsilon$ be a unique solution of the equation
\begin{equation} \label{ePe}
\begin{gathered}
\partial_{t}u-\Delta u+ g_\varepsilon(u)= f(u, x, t)\quad 
\text{in } \Omega\times(0, T), \\
u(x,t)=0 \quad \text{on } \partial\Omega\times(0, T),\\
u(x,0)=u_{0}(x) \quad \text{in } \Omega,
\end{gathered}
\end{equation}
 obtained by passing to the limit as $\eta\to 0$ in \eqref{plap1}. 
Note that $u_\varepsilon$ converges to  $u$, uniformly on any compact set 
in $\Omega\times(0,T)$, see  \cite{AnhDiaz1}. Thus,
it suffices to prove that for any $\varepsilon>0$,
\[
u_\varepsilon(x,t)\geq C e^{-At}\Phi^\mu (x),
\quad\forall (x,t)\in\Omega\times(0, T_{A,C}).
\]
Put $w=  C e^{-At}\Phi^\mu (x)$. We show that $w$ is a sub-solution of 
\eqref{ePe} for  $A>0$ large enough.
In fact, we have
\begin{align*}
&\partial_t w -\Delta w + g_\varepsilon(v)-f(w)\leq \partial_t w -\Delta w
+ w^{-\beta}\chi_{\{w>0\}}\\
&=-C A e^{-At} \Phi^\mu  -C \mu e^{-At}\Phi^{\mu-1} \Delta \Phi
 -C\mu(\mu-1)e^{-At}\Phi^{\mu-2}|\nabla \Phi|^2 \\
&\quad + C^{-\beta}e^{A\beta t}\Phi^{-\beta \mu} \chi_{\{\Phi>0\}} \\
&= C(-A   +\lambda_1\mu ) e^{-At}\Phi^{\mu} +C e^{-At}\Phi^{-\beta\mu}
\Big( - \mu(\mu-1)\Phi^{\mu(\beta+1)-2}|\nabla \Phi|^2 \\
&\quad + C^{-\beta-1}e^{A\beta t+At}\chi_{\{\Phi>0\}}\Big).
\end{align*}
Note that for any $t\in (0,T_{A,C})$, we obtain
$C^{-\beta-1}e^{A\beta t+At}\leq 1$.
This leads to
\begin{equation}\label{8.7}
\begin{aligned}
&\partial_t w -\Delta w + g_\varepsilon(w)-f(w) \\
&\leq  C e^{-At} \Big((-A   +\lambda_1\mu ) \Phi^{\mu}
  + \Phi^{-\beta\mu}\big( - \mu(\mu-1)\Phi^{\mu(\beta+1)-2}|\nabla \Phi|^2+1\big)
\Big).
\end{aligned}
\end{equation}
It is clear that   $(-A   +\lambda_1\mu ) \Phi^{\mu}\leq 0$ in
$\Omega\times(0, T_{A,C})$,  if  $A>2\lambda_1$.

Let $\omega_\delta=\{x\in\Omega : \operatorname{dist}(x, \partial\Omega)<\delta \}$,
for any $\delta>0$. Obviously,  we have
\begin{equation}\label{8.8}
\big( -\mu(\mu-1)\Phi^{\mu(\beta+1)-2}|\nabla \Phi|^2+ 1\big)<0,
\quad\text{for  any } x\in\omega_\delta,
\end{equation}
if $\delta>0$ is small enough because of $\mu(1+\beta)-2<0$.

Fix $\delta>0$ such that \eqref{8.8} holds.  On the set
$\Omega\backslash \omega_\delta$, we choose $A>0$ large enough such that
\begin{equation}\label{8.9}
(-A   +\lambda_1\mu ) \Phi^{\mu} +\Phi^{-\beta\mu}
\big( -\mu(\mu-1)\Phi^{\mu(\beta+1)-2}|\nabla \Phi|^2+ 1\big)<0.
\end{equation}
A combination of \eqref{8.7}, \eqref{8.8}, and \eqref{8.9}
 implies that $w$ is a sub-solution of equation
\eqref{ePe}; thereby it proves
\[
w\leq u_\varepsilon, \quad \text{in }\Omega\times(0,T_{A,C}).
\]
which completes the proof.
\end{proof}


\begin{remark}\label{remT_C} \rm
Note that $A$ is chosen independently of $C$, see \eqref{8.9} again.
If we  fix $A>0$ such that \eqref{8.9} holds, then  $T_{A,C}=T_C$ is as large as
$\log C$.
\end{remark}

Now we have sufficient information to complete the proof of the above theorem.

\begin{proof}[Proof of theorem \ref{theblow-up3}]
Fix $\mu\in(1, \frac{2}{1+\beta})$. Since $\Phi$ is continuous
on $\overline{\Omega}$, we have
\[
u_0(x)=c \Phi(x)\geq c  \eta_0 \Phi^\mu(x),\quad \text{in } \Omega,
\]
with   $\eta_0=\big(\max_{x\in\Omega} \{\Phi(x)\}\big)^{1-\mu}>0$.
 By applying Lemma \ref{lempositivity}, we obtain
\[
u(x,t)\geq C_0 e^{-At} \Phi^\mu (x), \quad
\forall (x,t)\in \Omega\times(0,T_{C_0}),
\]
with $C_0=c\eta_0$, and $T_{C_0}=\log(C_0)/A$.
Multiply both sides  of  \eqref{plap1} by $\Phi$ yields
\begin{align*}
\frac{d}{dt}\int_{\Omega}u(x,t)\Phi (x) dx
& =  \int_{\Omega} f(u(x,t))\Phi(x) dx
 -\lambda_1 \int_{\Omega}u(x,t)\Phi (x) dx \\
&\quad -\int_{\Omega}u^{-\beta}\chi_{\{u>0\}}(x,t)\Phi (x) dx.
\end{align*}
Thanks to Lemma \ref{lempositivity}, we obtain that for any $t\in(0, T_{C_0})$,
\begin{equation}\label{8.3}
\begin{aligned}
\frac{d}{dt}\int_{\Omega}u(x,t)\Phi (x) dx
&\geq \int_{\Omega} f(u)\Phi(x) dx -\lambda_1 \int_{\Omega}u\Phi(x) dx \\
&\quad - C^{-\beta}_0e^{A\beta t} \int_{\Omega}\Phi^{1-\mu\beta} dx.
\end{aligned}
\end{equation}
Note that  $ C^{-\beta}_0e^{A\beta t}\leq 1$, for any $t\in(0, T_{C_0})$.
By the convexity of $f$ and  \eqref{8.3}, we obtain
 \begin{equation}\label{8.4}
  z'(t)\geq f(z(t))-\lambda_1 z(t)-\int_{\Omega}\Phi^{1-\mu\beta} dx, \quad
\text{for } t\in(0, T_{C_0}),
 \end{equation}
with $z(t)=\int_{\Omega}u(x,t)\Phi (x) dx$.
Since $f$ is a convex function, it follows  from \eqref{condblowup}
 that $\frac{f(s)}{s}\to +\infty$ as $s\to +\infty$.
Thus, there is a constant $s_0>0$ such that
 \begin{equation}\label{8.5}
 \frac{1}{2}f(s)\geq  \lambda_1 s+ \int_{\Omega}\Phi^{1-\mu\beta} dx, \quad
\forall s>s_0.
 \end{equation}
Since $c$ is sufficiently large,  we have
$z(0)=c \int_{\Omega} \Phi^2(x) dx> \max\{a, s_0\}$.
It follows  from \eqref{8.4} and \eqref{8.5} that $z(t)\geq z(0)$ and
 \[
 z'(t)\geq \frac{1}{2} f(z(t)), \quad \text{for any } t\in(0, T_{C_0}).
 \]
Therefore,
\[
\frac{1}{2}t\leq \int_{0}^{t} \frac{z'(t)dt}{f(z(t))}
=\int_{z(0)}^{z(t)} \frac{dz}{f(z)}
\leq\int^{+\infty}_{a} \frac{dz}{f(z)},\quad \text{for any } t\in (0, T_{C_0}).
\]
 This  implies
\begin{equation}\label{8.6}
\frac{T_{C_0}}{2}\leq\int^{+\infty}_{a} \frac{dz}{f(z)}\,.
\end{equation}
The right-hand side of \eqref{8.6} is bounded by a constant,
while $T_{C_0}$ is as large as $\log C_0=\log (c\eta_0)$
(see Remark \ref{remT_C}).
Then, we obtain a contradiction if  $c$ is large enough.
This completes the proof.
 \end{proof}

\begin{remark} \rm
It is not difficult to show that the blow-up result in
Theorem \ref{theblow-up2} still holds if $u_0$ is assumed to be
positive and large enough in a ball $B(x_0, r_0)\Subset\Omega$.

Note that the result in Theorem \ref{theblow-up3} still holds if  $f$
is only assumed  to be a convex function on $(a,\infty)$, for some $a>0$.
\end{remark}


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\end{document}