\documentclass[twoside]{article}
\usepackage{amsfonts, amsmath} % used for R in Real numbers
\pagestyle{myheadings}

\markboth{ Local and global nonexistence }
{ Mohammed Guedda \&  Mokhtar  Kirane }

\begin{document}
\setcounter{page}{149}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
2002-Fez conference on Partial Differential Equations,\newline
Electronic Journal of Differential Equations,
Conference 09, 2002, pp 149--160. \newline
http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu (login: ftp)}
 \vspace{\bigskipamount} \\
%
  Local and global nonexistence of solutions to semilinear
  evolution equations
%
\thanks{ {\em Mathematics Subject Classifications:} 35K55, 35K65, 35L60.
\hfil\break\indent
{\em Key words:} Parabolic inequality, hyperbolic equation, fractional
power, Fujita-type result.
\hfil\break\indent
\copyright 2002 Southwest Texas State University. \hfil\break\indent
Published December 28, 2002.} }

\date{}
\author{Mohammed Guedda \&  Mokhtar  Kirane
\\ \quad\\
To Professor Bernard Risbourg, in memorium}
\maketitle


\begin{abstract}
 For a fixed $ p $ and $  \sigma > -1 $, such that
 $ p >\max\{1,\sigma+1\}$,
 one main concern of this paper  is to find sufficient  conditions
 for non solvability  of
 \[
 u_t = -(-\Delta)^{\frac{\beta}{2}}u - V(x)u + t^\sigma h(x)u^p + W(x,t),
 \]
 posed in $ S_T:=\mathbb{R}^N\times(0,T)$, where  $ 0 < T <+\infty$,
 $(-\Delta)^{\frac{\beta}{2}}$ with $ 0 < \beta \leq 2$ is  the
 $\beta/2$ fractional power of the $ -\Delta$, and
 $ W(x,t) = t^\gamma w(x) \geq 0$.  The potential  $ V  $ satisfies
 $ \limsup_{| x|\to +\infty }| V(x)| | x|^{a} < +\infty$, for some
positive $ a$.
 We shall see  that the existence of  solutions depends on
the behavior at infinity of both initial data and the function
 $h$ or of
both  $ w$ and $ h$.   The non-global existence  is also
 discussed. We
prove, among other things, that if  $ u_0(x) $ satisfies
\[
 \lim_{| x|\to+\infty}u_0^{p-1}(x) h(x)|
 x|^{(1+\sigma)\inf\{\beta,a\}} = +\infty,
\]
 any possible local solution blows up at a finite time for any locally
integrable function $W$.
 The situation is  then extended to nonlinear
hyperbolic equations.
\end{abstract}


\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\numberwithin{equation}{section}


\section{Introduction}

In this paper we consider the
problem
\begin{equation}\label{1.1}
\begin{gathered}
u_t =-(-\Delta)^{\frac{\beta}{2}}u- V(x)u+ t^\sigma h(x)u^p +
W(x,t),\quad
(x,t)\in \mathbb{R}^N\times(0,T),\\
u(x,0) = u_0(x)\geq 0,\quad x \in
\mathbb{R}^N,
\end{gathered}
\end{equation}
for some $ 0 < T \leq +\infty,
$
where $(-\Delta)^{\frac{\beta}{2}}$ with $0 < \beta \leq 2$ is  the
$
\beta/2 $ fractional power of
the $ -\Delta$, which stands for diffusion in
media with impurities,
$p>1$, $\sigma >-1$, the functions $h$ and $u_0$ are
nonnegative and
satisfy some growth conditions at infinity which will be
specified later.
 The function  $ W(x,t)\geq 0 $, which can be viewed as a
noise or as a
control, is locally integrable.  Even if we can handle
general $ W(x,t)$,
we will confine ourselves to the simple case where
$
W(x,t) = t^\gamma w(x)$, $\gamma > -1$. We assume that the potential
$ V $
satisfies
\begin{equation}\label{conditionV}
\limsup_{| x|\to+\infty}|
V(x)|| x |^{a}  < +\infty
,\end{equation}
for some $   a > 0$.

In the case
$ \beta = 2, \sigma = \gamma=0 $ and $  V = 0,  $  Pinsky
\cite{Pinsky2}
proved that all nontrivial nonnegative solutions blow up at
a finite time
if $ N \leq 2 $ or $ N \geq 3 $ and the
function $ w(x)| x|^{2-N} $ is not
integrable. It was also shown  that if
$h(x) \geq c| x|^m $ and $ w(x) \geq
c| x|^{-q}$,
$2 < q < N$, for large
$|x|$ there is no global solutions if
$
1 < p \leq 1 + \frac{2+m}{N-2}$.

In a recent paper \cite{GuK2}, we studied
the criticality for some evolution
inequalities. It was shown, among other
results, that for   $ V \leq 0, w(x)
\geq 0 $ if $ h(x) $ behaves like
$ |
x|^\gamma $ at infinity and if $ 1 < p \leq 1 + \frac{\gamma
+
\beta(1+\sigma)}{N}=: p_c $ then  there is no global
nonnegative weak
solutions except the trivial one. In the case where
$ p >p_c $  solutions
may exist, at least locally.
 More recently the first named author proved
in \cite{Lamfa} a similar
result  for  $ \sigma = 0$, but
$  V_+(x) \leq
\frac{b}{1 +| x|^a}$, where
$ V_+ = \max\left\{ V,0\right\}$, $a
>\frac{N(p-1)}{p} > 0$,
$b > 0 $ and $ p $ small.

 In
\cite{Lamfa},\cite{GuK2}   the problem of nonexistence of global
weak
solution, with unsigned initial data and $ w = V = 0$, is
also
considered. The authors obtained the absence of global solution for
initial
data satisfying
\[ 0 < \int_{\mathbb{R}^N}u_0(x)dx \leq +\infty,
\]
and
under some conditions on $ p $ and on the behavior at infinity of $
h$.


In the  present paper we are interested in   conditions for local
and
global solvability  of \eqref{1.1} from a different angle.
We
investigate, for any fixed
 $\sigma > -1  $ and $ p > \max\{1,1+\sigma\}$,
in contrast to the
Fujita-type result, the effect of the behavior of
$
u_0$, $ h$ and $ w $ at infinity on the non existence of local and
global
weak solutions to \eqref{1.1}.

This work is motivated by the paper
\cite{BaKer} in which Baras and Kersner
showed that the
problem
\begin{equation}
u_t = \Delta u + h(x) u^p,\quad u(x,0) = u_0(x)
\geq 0,
\end{equation}
has no local weak  solution if the initial data
satisfies
\[ \lim_{| x| \to +\infty}u_0^{p-1}h(x) = +\infty,
\]
and any
possible local weak  solution blows up at a finite time if
\[\lim_{| x| \to
+\infty}u_0^{p-1}h(x)| x|^2 = +\infty.
\]
Here, we attempt to extend this
result to \eqref{1.1}. The methods used are some
modifications and
adaptations of ideas
from \cite{BaKer} and \cite{Lamfa}.

Set
\[ S_T :=
\mathbb{R}^N\times(0,T).\]

\paragraph{Definition} %\label{def1.1}
  We say
that   $ u  \geq 0  $ is a  local  weak solution to
(\ref{1.1}), defined in
$ S_T, 0 < T < +\infty$,  if it is a locally
integrable function such that
$u^ph \in
L^1_{loc}(S_T)$,
and
\begin{multline}\label{definition}
\int_{\mathbb{R}^N}u(x,0
)\zeta(x,0)dx  + \int_{S_T}t^\gamma w h
\zeta \,dx\,dt +\int_{S_T}t^\sigma
h  u^p\zeta \,dx\,dt \\
= \int_{S_T} u(-\Delta)^{\frac{\beta}{2}}\zeta
\,dx\,dt
-\int_{S_T}u\zeta_t  \,dx\,dt-\int_{S_T}uV(x)\zeta
\,dx\,dt,
\end{multline}
is satisfied for any  $ \zeta \in
C^\infty_0(\overline{ S_{T}}) $
which  vanishes for large $ | x| $ and at $
t =T$.


\paragraph{Definition}
  We say that   $ u  \geq 0  $ is a  global
weak solution to
(\ref{1.1}), if it is a local solution to (\ref{1.1})
defined in $ S_T $
 for any $ T > 0$. \smallskip


Throughout this paper we
may assume  that there exists $ R_0 > 0 $ such
that  $ w(x) $  are
nonnegative  for all
$| x|\geq R_0$ and condition (1.2) is
satisfied.

\begin{theorem} \label{thm1.1}
 Let $ \sigma > -1$, $p >
\max\left\{1, 1+\sigma\right\}$.
Assume that one of the following two
conditions
\begin{gather}\label{1.6}
 \lim_{| x| \to +\infty}u_0^{p-1}h(x)
= +\infty,\\
\label{1.6bis}
 \lim_{| x| \to +\infty}w^{p-1}h(x) = +\infty,
\end{gather}
is satisfied.
Then there is no $ T > 0 $ such that problem
\eqref{1.1} has a
solution defined in $ S_T$.
\end{theorem}

This result
shows in particular that any local solution to (\ref{1.1}) blows
up at $ t= 0$.
The proof of  Theorem \ref{thm1.1}  is based on an upper estimate of
the blowing up
time as it is shown in the following
theorem.

\begin{theorem} \label{thm1.2}
Let $ \sigma > -1$, $p >
\max\left\{1, 1+\sigma\right\}$.   There
exist  positive constants $ K_1,
K_2 $ such that if  problem
\eqref{1.1} has local solution defined in $
S_T, T < +\infty$, the
following two estimates
hold:
\begin{gather}
\liminf_{| x| \to +\infty}u_0^{p-1}h(x) \leq
K_1\frac{1}{T^{1+\sigma}},\\
\liminf_{| x| \to +\infty}w^{p-1}h(x)
\leq
K_2\frac{1}{T^{(1+\gamma)(p-1)}}.
\end{gather}
\end{theorem}

We can
deduce from the above result that if problem \eqref{1.1} has a
global
solution then the initial data and the function $ w $
must
satisfy
\[ \liminf_{| x| \to +\infty}u_0^{p-1}h(x) = \liminf_{| x|
\to
+\infty}w^{p-1}h(x)= 0.
\]
But those conditions are  not sufficient for the
global existence as it can
be seen from the following
statement.

\begin{theorem} \label{thm1.3}
Let $ \sigma > -1$, $p >
\max\left\{1, 1+\sigma\right\}$. Assume
that
\begin{equation}\label{1.7}
\lim_{| x| \to +\infty}u_0^{p-1}h(x)|
x|^{(1+\sigma)\inf\left\{\beta,a\right\}}=+\infty,
\end{equation}
or
\begin{equation}\label{1.8}
 \lim_{| x| \to
+\infty}w^{p-1}h(x)|
x|^{\inf\left\{\beta,a\right\}\left[\gamma(p-1)+p+\sigma\right]} =
+\infty.
\end{equation}
Then  problem \eqref{1.1} has no global
solution.
\end{theorem}

\begin{remark} \rm
It is interesting to note here,
that in some sense, there
is no effect of $ V $ on the global solvability
of \eqref{1.1}
if $ a \geq 2$. In case $ V\leq 0 $  assumptions
(1.9),(1.10) have to be
read with $ \beta $ instead of
$
\inf\{\beta,a\}$.
\end{remark}

The second part of our paper deals with
non existence results for the
hyperbolic
problem
\begin{equation}\label{lastinequality}
\begin{gathered}
 u_{tt} =
\Delta u +V(x)u + t^\sigma h(x) u^p+ t^\gamma w(x) ,\\
u(x,0) =
u_0(x),\quad u_t(x,0) = u_1(x).
\end{gathered}
\end{equation}
We use the
similar approach to  establish the

\begin{theorem} \label{thm1.4}
Let $ p
> \max\{1,1+\sigma\}$, $\sigma > -1$. Assume that
\[
\limsup_{| x| \to
\infty}| x|^{2a}|  V(x)|<+\infty
\]
and one of the following two
conditions
\begin{gather}
\lim_{| x| \to +\infty}
u_1h^{1/(p-1)}|
x|^{\inf\{1,a\}\frac{1+\sigma + p}{p-1}} = + \infty,
\\
\lim_{| x| \to +\infty} wh^{1/(p-1)}|
x|^{\inf\{1,a\}\left(1+\gamma +
\frac{p+1+\sigma}{p-1}\right)} = +
\infty,\end{gather}
holds.
Then {\rm
(\ref{lastinequality})} has no   global weak solution.
\end{theorem}


Observe that the conditions required does not involve the initial
position
$ u_0$.
The  method of the proofs is based on a judicious  choice
of the test
function in the form
\[ \zeta(x,t) = \eta(t/T)\Phi(x),
\]
where
$ \eta \in C_0^\infty([0,+\infty)) $ and $ \Phi
\in
C_0^\infty(\mathbb{R}^N)$. For Problem (1.11) we demand to $ \eta $
to
satisfy $ \eta'(0) = 0 $ therefore $ \zeta_t(x,0) = 0, $ and
this
condition eliminates in the definition of solution to (1.12)
the term
which contains $ u_0$.
The strong point in this result is to obtain
necessary conditions for non
local and non global existence of solutions
for any
local integrable initial data even if $ u_0 $ has a compact
support.  This
remark was first noticed by Pohozaev and Veron
\cite{PV}.

\section{Nonexistence of local solutions}

In this section, we
provide a necessary condition for the local solvability of
{\rm
(\ref{1.1})}. We first obtain estimates (1.7), (1.8) and then
prove
Theorem \ref{thm1.2}.
Without lost of generality we may assume that
for large $ | x|$,
\[
| V(x)| \leq | x|^{-a}, \quad a >
0.
\]

\paragraph{Proof of Theorem \ref{thm1.2}}
 Suppose that $ u $ is a
local solution to {\rm (\ref{1.1})} defined in
$S_T$, $0 < T <
+\infty$.
Let  $ \zeta $  be a test function which is nonnegative.
According to
(\ref{definition}) we
have
\begin{multline}\label{defbis}
\int_{\mathbb{R}^N}u(x,0)\zeta(x,0)dx
+\int_{S_T}t^\gamma w \zeta
\,dx\,dt + \int_{S_T}t^\sigma h
 u^p\zeta
\,dx\,dt  \\
\leq \int_{S_T}  u  ((-\Delta)^{\frac{\beta}{2}}\zeta)_+
\,dx\,dt
+\int_{S_T} u |\zeta_t| \,dx\,dt
  +\int_{S_T} u| V|\zeta
\,dx\,dt,
\end{multline}
where $[.]_+ = \max\{.,0\}$.
By considering the
Young inequality, with $ p' = p/(p-1)$,  we
obtain
\begin{multline*}
\int_{S_T} u | \zeta_t| \,dx\,dt \\
\leq
\frac{1}{3}\int_{S_T} t^\sigma
u^p  h \zeta \,dx\,dt
+
(p-1)3^{1/(p-1)}p^{-p/(p-1)}\int_{S_T}|\zeta_t|^{p'}(t^\sigma
h\zeta)^{1-p'} \
,dx\,dt,
\end{multline*}
\begin{multline*}
 \int_{S_T}
u((-\Delta)^{\frac{\beta}{2}}\zeta)_+ \,dx\,dt
 \leq \frac{1}{3}\int_{S_T}
t^\sigma u^p
  h\zeta \,dx\,dt +
(p-1)3^{1/(p-1)}p^{-p/(p-1)}\\
\times
\int_{S_T}((-\Delta)^{\frac{\beta}{2}}\zeta)_+^{p'}(t^\sigma
h\zeta)^{1-p'}\,dx\,dt,
\end{multline*}
and
\begin{multline*}
 \int_{S_T} u | V(x)|\zeta \,dx\,dt\\
 \leq \frac{1}{3}\int_{S_T} t^\sigma u^p h\zeta \,dx\,dt +
(p-1)3^{1/(p-1)}p^{-p/(p-1)}\int_{S_T}| V|^{p'}\zeta(t^\sigma
h)^{1-p'} \,dx\,dt.
\end{multline*}
 Using the above estimates in  (\ref{defbis}), we obtain
\begin{align*}
\int_{\mathbb{R}^N} &u(x,0) \zeta(x,0)dx +\int_{S_T}t^\gamma w \zeta
\,dx\,dt \\
\leq&(p-1)3^{1/(p-1)}p^{-p/(p-1)}
\Big[\int_{S_T}|\zeta_t|^{p'}(t^\sigma h\zeta)^{1-p'}\,dx\,dt\\
&+\int_{S_T}((-\Delta)^{\frac{\beta}{2}}\zeta)_+^{p'}(t^\sigma
h\zeta)^{1-p'}
\,dx\,dt+\int_{S_T}| V|^{p'}\zeta(t^\sigma h)^{1-p'}
\,dx\,dt\Big].
\end{align*}
At this stage, let
\[\zeta (x,t) = \left(\eta(t/T)\right)^{p'}\Phi(x),
\]
where  $ \Phi \in C^\infty_0(\mathbb{R}^N)$, $\Phi \geq 0 $ and
$\eta \in C^\infty_0(\mathbb{\mathbb{R}_+})$, $0 \leq \eta \leq 1$,
satisfying
\[
\eta(r)=\begin{cases}
    1 &\mbox{if } r \leq \frac{1}{2}, \\
    0 &\mbox{if } r \geq 1.
\end{cases}
\]
With the above choice of $ \zeta$, we obtain
\begin{equation}
\begin{aligned}
&\frac{1}{1+\gamma}\big(\frac{T}{2}\big)^{1+\gamma}\int_{\mathbb{R}^N}\Phi
w dx+\int_{\mathbb{R}^N}\Phi u_0 dx \\
 &\leq
(p-1)3^{1/(p-1)}p^{-p/(p-1)}C_\star^{p'-1}\Big\{(p')^{p'}T^{(1+\sigma)(1-p')}\in
t_{\mathbb{R}^N}\Phi
h^{1-p'}  \,dx\,dt\\
&\quad +  T^{1+\sigma(1-p')}\int_{\mathbb{R}^N}
((-\Delta)^{\frac{\beta}{2}}\Phi)_+^{p'}(\Phi h)^{1-p'}dx
+ T^{1+\sigma(1-p')}\int_{\mathbb{R}^N}| V|^{p'}\Phi h^{1-p'}dx \Big\},
\end{aligned}
\end{equation}
where
\[ C_\star^{p'-1} =
 \frac{\max\left\{1,\Vert \eta'\Vert_{\infty}^{p'}\right\}}{ 1+
\sigma(1-p')}.\]
Next,  we consider $\Phi(x) = \varphi(x/R)$, $R > 0$,
where
\[ \varphi \in C^\infty_0(\mathbb{R}^N), \quad 0 \leq \varphi \leq 1,\quad
{\rm supp }\ \varphi
\subset \left\{ 1 < | x| < 2\right\},\quad
((-\Delta)^{\frac{\beta}{2}})\varphi)_+ \leq \varphi.
\]
Accordingly,  via (2.2) we find
\begin{equation}\label{via1} \inf_{ | x| > R}\left(u_0(x)h^{p' -
1}\right)\int_{\mathbb{R}^N}\Phi
h^{1-p'} dx \leq
(p-1)3^{1/(p-1)}p^{-p/(p-1)}C_\star^{p'-1} I(R),\end{equation}
and
\begin{multline}\label{via2}
\inf_{ | x| > R}\left(w(x)h^{p' -1}\right)\int_{\mathbb{R}^N}\Phi
h^{1-p'} dx \\
\leq (\gamma+1)2^{1+\gamma}T^{-1-\gamma}
(p-1)3^{1/(p-1)}p^{-p/(p-1)}C_\star^{p'-1} I(R),
\end{multline}
for $ R > R_0$, where
\[ I(R)
:=
\Big[\big(\frac{p}{p-1}\big)^{p'}T^{(1-p')(1+\sigma)}
+
T^{1+\sigma(1-p')}\Big\{ \frac{1}{R^{\beta p'}} +
\frac{1}{R^{a
p'}}\Big\}
\Big]\int_{\mathbb{R}^N}\Phi h^{1-p'}dx.
\]
Then estimates
(1.8), (1.9), with
$ K_1 = \frac{3C_\star}{p-1}, K_2 =
(\gamma+1)^{p-1}2^{1+\gamma}(p-1)K_1, $
are easily obtained by
dividing
(\ref{via1}) and (\ref{via2}) by $ \int_{\mathbb{R}^N}\Phi
h^{1-p'}dx  $
and letting $ R \to +\infty$. This completes the
proof.
\hfill$\square$\smallskip

Note that assumption (1.2) are only used
to eliminate the second term
of  $ I(R) $ when $ R $ tends to
infinity. It
is obvious  that the conclusions of Theorems 1.1 and 1.2
remain true if we
assume
\[ \limsup_{| x|\to+\infty} | x|^a V_+(x) < +\infty,
\]
or
\[
\lim_{R\to+\infty}\ \ \max_{\left\{ R < | x| < 2R\right\}}
V(x)_+ =
0,\]
instead of (1.2).

\begin{remark} \label{rmk1.2} \rm
Following the
above proof, the condition $ \sigma > -1 $
is not used. It is easily
verified that estimates (1.7), (1.8)  are
satisfied for any $ \sigma$.
This leads in particular to
\[ \liminf_{| x| \to +\infty}u_0^{p-1}h(x) =
0,\]
if $ \sigma < -1, $ or if $ \sigma = -1 $  the limit is finite.
Therefore
there is no local solution if
$ \liminf_{| x| \to
+\infty}u_0^{p-1}h(x)  > 0 $ and $ \sigma  <
-1$.  For the case $ \sigma =
-1 $ there is no local solution  if
\[ \liminf_{| x| \to
+\infty}u_0^{p-1}h(x) > K_1.
\]
\end{remark}

\section{Necessary conditions
for  global solvability }

In this section, we discuss conditions for the
non existence of global
solution to \eqref{1.1}.

\begin{proposition}
\label{prop3.1}
Let $ p > \max\{1,1+\sigma\}$. Assume that {\rm
\eqref{1.1}} has a global
solution. Then the following two limits are
finite:
\begin{gather}\label{3.1}
 \liminf_{| x|\to+\infty}w^{p-1}h(x)
|x|^{\inf\left\{\beta,a\right\}(1+\sigma)},\\
\label{3.2}
\liminf_{|
x|\to+\infty}u_0^{p-1}h(x)
|x|^{\inf\left\{\beta,a\right\}(\gamma(p-1)+p+\sigma)
}
\end{gather}
\end{proposition}

\paragraph{Proof}
Assume that
\eqref{1.1}--(1.2) has a global weak solution. According
to the  proof of
Theorem \ref{thm1.2} we have, for any $ T >
0$,
\begin{multline}
\int_{\Omega_R}\Phi u_0 dx
\leq
C_1\Big\{\big(\frac{p}{p-1}\big)^{p'}T^{(1+\sigma)(1-p')}+
2R^{-\inf\{\beta,a\}p
'}T^{1+\sigma(1-p')}\Big\} \\
\times \int_{\Omega_R}\Phi
h^{1-p'}\,dx\,dt
\end{multline}
and
\begin{multline}
\int_{\Omega_R}\Phi w \,dx
\leq
C_2\Big\{\big(\frac{p}{p-1}\big)^{p'}T^{\sigma(1-p')-p'-\gamma}+
2R^{-\inf\{\beta,a\}p'}T^{\sigma(1-p')-\gamma}\Big\}\\
\times \int_{\Omega_R}\Phi
h^{1-p'}\,dx\,dt,
\end{multline}
where $\Omega_R =\{R < | x| < 2R\}$,
$C_1
:=(p-1)3^{1/(p-1)}p^{-p/(p-1)}C_\star^{p'-1}$,
$ C_2
=(\gamma+1)2^{\gamma+1}C_1$
and $\Phi(x) = \varphi(x/R)$, with $ \varphi
\in C^\infty_0(\mathbb{R}^N) $
nonnegative satisfying $((-\Delta)^{\beta/2}
\varphi)_+ \leq \varphi $
and  $\mathop{\rm supp}\varphi\subset \{ 1 < | x
| < 2 \}$.

A simple minimization of the right hand side of (3.3) with
respect
to $ T > 0 $ yields
 \[\int_{\Omega_R}\Phi u_0 dx \leq
A_\star^{p'-1} R^{-\inf\{\beta,a\}\frac{1+\sigma}{p-1}}
\int_{\Omega_R}\Phi h^{1-p'}dx,
\]
where $ A_\star = A_\star(p,\sigma) $ is a positive
constant.
This  leads to the  estimate
\begin{multline*}
\inf_{| x| > R}
\left(u_0(x)
h^{p'-1}(x)|
x|^{\inf\{\beta,a\}\frac{1+\sigma}{p-1}}\right)\int_{\Omega_R}\Phi
h^{1-p'}| x|^{-\inf\{\beta,a\}\frac{1+\sigma}{p-1}}dx \\
\leq
A_\star^{p'-1}\int_{\Omega_R}\Phi
|x|^{{-\inf\{\beta,a\}
\frac{1+\sigma}{p-1}}}h^{1-p'}dx.
\end{multline*}
Thus
\[
\liminf_{| x| \to + \infty}\left(u_0(x)
h^{p'-1}(x)|
x|^{\inf\{\beta,a\}\frac{1+\sigma}{p-1}}\right) \leq
A_\star^{p'-1}.
\]
To confirm (3.2) we use (3.4 ), with $ T =
R^{\inf\{\beta,a\}} $, to deduce
\[
\int_{\Omega_R}\Phi w dx
\leq
B_\star^{p'-1} R^{-\inf\{\beta,a\}(\gamma + p'
+\sigma(p'-1))}
\int_{\Omega_R}\Phi h^{1-p'}dx.
\]
The rest of the proof as
above.
\hfill$\square$

\begin{remark} \label{rmk31.} \rm
As in section 2,
condition (1.2) can be relaxed to
\[ \limsup_{| x|\to \infty}V_+(x)| x|^a <
+\infty,
\]
where $ V_+ = \max\{V,0\}$.
For equation \eqref{1.1} with $ W =
0$, i.e, equation
\begin{equation}\label{eq:3.4}
 u_t =
-(-\Delta)^{\frac{\beta}{2}}u - V(x) u + t^\sigma  u^p,
\end{equation}
we
have  no global  solution whenever
\begin{equation}
\lim_{| x| \to
\infty}u_0^{p-1}(x)|x|^{\inf\{\beta,a\}(1+\sigma)}>A_\star.
\end{equation}
Now i
f we keep the function
$K_2^{1/(p-1)}T_0^{-\gamma-1}t^\gamma\frac{| x|}{1+|
x|},T_0 > 0$,
 in \eqref{1.1}, any local solution
ceases to exist before
$T_0$.
\end{remark}

\begin{remark} \label{rmk3.2} \rm
In
\cite{Pinsky2,Zh1,Zh2}  a crucial role is played by some estimate of
the
heat kernel associated to the linear operator involved in the
considered
equations.  The methods
used in this paper seem to be more efficient
because there are not
based on a knowledge of the kernel of the involved
operators. The methods
have a remarkable degree of simplicity and
versality. For instance,
equations
with nonlinear diffusion can be handled
by the methods presented  in here
as we can see below while the methods
adopted in \cite{Pinsky2,Zh1,Zh2}
are clearly inoperative. For example,
we can consider the
\[ u_t \geq \Delta (a(x,t) u^m) + t^\sigma h(x)
u^p,
\]
where $ a \geq 0 $  in $  L^\infty(\mathbb{R}^N\times(0,+\infty)) $
and
$ 0< m < p$.
These methods can also be used to derive  a  non global
existence
of weak solutions to
\begin{equation}
 u_t = -|
x|^\alpha(-\Delta)^{\beta/2}u - V(x)u + t^\sigma h(x)
u^p+ t^\gamma w(x),
u(x,0) = u_0(x),
\end{equation}
where $ 0 < \alpha < N$. We note that this
equation has a
diffusion that vanishes at the origin $ x = 0 $
\cite{GuK3}.
Concerning the nonexistence of global solutions to the last
problem,
the  following result can be  established without any major
difficulty.
\end{remark}

\begin{theorem} \label{thm3.1}
 Let $ \sigma >
-1, p > \max\left\{1, 1+\sigma\right\}. $ Assume
that
\[
 \lim_{| x| \to
+\infty}u_0^{p-1}h(x)|
x|^{-\alpha
p+(1+\sigma)\inf\left\{\beta,a+\alpha\right\}}
=+\infty,
\]
or
\[\lim_{| x| \to +\infty}w^{p-1}h(x)|
x|^{\inf\left\{\beta,a\right\}\left[\gamma(p-1)+\sigma\right]-\alpha
p}= +\infty.\]
Then  problem {\rm (3.7)} has no global weak
solution.
\end{theorem}

Observe that we can  also consider  the
equation
\begin{equation}
u_t = \Delta u + t^\sigma
h(x)(1+u)\log(1+u)^p,
\end{equation}
with an initial data $ u_0 \geq0$. We
refer the reader to \cite{sgkm} for
the case $ \sigma = 0 $ and $ h = 1$.
Equation (3.8)
can be written
\[
v_t = \Delta v + t^\sigma h(x)  v^p + |
\nabla v|^2 \geq \Delta
v + t^\sigma h(x)  v^p,
\]
via the transformation $
v = \log(u+1)$.
According to the previous results,  if
\[ \liminf_{| x|\to
+\infty}(\log(1+u_0))h(x)^{1/(p-1)}|
x|^{2(1+\sigma)/(p-1)}>
C_0,\]
for
some positive constant $ C_0$, where
$p > \max\{1,1+\sigma\}$, Equation
(3.8) with initial value $ u_0 $
does not possess global
solution.


\section{Nonexistence results for nonlinear hyperbolic
equations}

This short section deals with the equivalent of Theorems 1.1
and 1.3 for
nonlinear hyperbolic equations  of the
form
\begin{equation}
u_{tt} = \Delta u - V(x) u + t^\sigma h(x)
u^p+t^\gamma w(x),
\end{equation}
for $ x \in \mathbb{R}^N$ and $t \in
(0,T)$ subject to the conditions
\begin{equation}
u(x,0)= u_0(x),\quad
u_t(x,0) = u_1(x),\quad x\in \mathbb{R}^N.
\end{equation}
The extension of
the results  of Section 2 to equations of
type  (4.1) presents no
conceptual difficulty. We will  follow
 the routine calculations, except
that we choose
\begin{equation}
\zeta(x,t)
=\big(\eta(\frac{t^2}{T^2})\big)^{2p'}\Phi(x)
\end{equation}
as a test
function.  Without loss of generality, we assume that the
potential $ V $
satisfies
\begin{equation}
 | V(x) | \leq | x|^{-2a},\quad  a >0,
\end{equation}
for large $ | x|$.

\paragraph{Proof of Theorem
\ref{thm1.4}}
Since the proof is similar to the one  in the preceding
sections, we only
give here a sketch of the proof. First we assume
on the
contrary that problem (4.1)--(4.2) has a global solution, say $ u$.
Let $
\zeta $ be a test function defined by (4.3)
where  $ \eta $ and $ \Phi $
are defined  in Sections 2 and 3. Observe that
$\zeta_t(x,0) = 0$ for all $
x $ in $ \mathbb{R}^N$. Therefore,
 we have the
estimate
\begin{multline*}
\int_{\mathbb{R}^N}u_1(x)\zeta(x,0)dx  +
\int_{S_T}t^\gamma w \zeta
\,dx\,dt+ \int_{S_T}t^\sigma h  u^p\zeta\,dx\,dt
\\
\leq \int_{S_T}  u (-\Delta\zeta)_+  \,dx\,dt +\int_{S_T} u
|
\zeta_{tt}| \,dx\,dt +\int_{S_T} u | V(x)| \zeta
\,dx\,dt,
\end{multline*}
which leads to
\begin{multline}
\label{estimateT}
\int_{\mathbb{R}^N}u_1(x)\Phi(x)dx \\
\leq
\big[C_1T^{-p'+(1+\sigma)(1-p')}
+C_2R^{-\inf\{2,2a\}p'}
T^{1+\sigma(1-p')}\big]
\int_{\mathbb{R}^N}\frac{\Phi(x)
}{h^{p'-1}}dx,
\end{multline}
for some positive constants $ C_1, C_2$, and
for any $ T > 0$.  Therefore,
by a minimization argument, we deduce
that
\[
\int_{\mathbb{R}^N}u_1(x)\Phi(x)dx
\leq
K_1R^{-\inf\{1,a\}(p+1+\sigma)(p'-1)}\int_{\mathbb{R}^N}
\frac{\Phi(x)}{h^{
p'-1}}dx.
\]
Hence, as in section 2,
\[
\liminf_{| x| \to
+\infty}u_1^{p-1}h(x)|x|^{\inf\{1,a\}(p+1+\sigma)}
< +\infty,
\]
which is
impossible. The rest of the proof is similar to that  in Section 3
and is
hence left to the reader.
\hfill$\square$

\begin{remark} \label{rmk4.1}
\rm
An immediate necessary conditions for the local existence
can be
obtained  from (4.5) which  leads to
\[
\liminf_{| x| \to
+\infty}u_1h^{1/(p-1)}
\leq \frac{K_1}{T^{1+\sigma + 2p}}.
\]
Concerning
the function $ w $ we have also a necessary condition for the
local
existence,
\[ \liminf_{| x| \to +\infty}wh^{1/(p-1)}\leq
\frac{K_2}{T^{\gamma
(p-1)+\sigma + 2p}}.\]
\end{remark}

We illustrate our results with the  example
\[
u_{tt} = \Delta u + t^\sigma| u|^p + (1 + | x|^2)^{-q},
u_t(x,0) = A(1+| x|^2)^{-(k+1)/2}.
\]
If $ k < \frac{2+\sigma}{p-1} $ or $ q < \frac{1+\sigma + p}{2(p-1)} $ the
problem has no global weak solution even if $ u_0 $ has a  compact support
or if $ u_0\equiv 0$. Now if $ k = \frac{2+\sigma}{p-1} $  and
$ q \geq \frac{1+\sigma+ p}{2(p-1)}$ the problem has no global weak
solution if $ A $ is large enough.

\begin{thebibliography}{99} \frenchspacing

\bibitem{BaKer}
{\sc P. Baras } and {\sc R.  Kersner}, { \it  Local and global solvability
of a class of semilinear parabolic equations,} J. Diff.
Eqns, 68, no. 2, (1987), pp. 238--252.

\bibitem{bp}
{\sc P.  Baras } and  {\sc M. Pierre,} { \it
Crit\`ere d'existence de solutions positives pour des \' equations
semi-lin\'eaires non monotones, }
Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, { 2}, no 3, (1985), pp.
185--212.


\bibitem{Lamfa}
{\sc M. Guedda,} { \it Non existence of solutions to some evolution
inequalities,} Coll. Math., to appear.


\bibitem{GuK}
{\sc M. Guedda} and {\sc M. Kirane,} {\it A Note on nonexistence of
global solutions to a nonlinear
integral equation}, Bull. Belg.
Math. Soc., { 6}, (1999), pp. 491--497.

\bibitem{GuK3}
{\sc M. Guedda} and {\sc M.  Kirane,} { \it Non existence of global
solutions to nonlinear hyperbolic
inequalities with a singularity,} LAMFA, { 16}, Universit\'e de
Picardie Jules Verne, 2000.

\bibitem{GuK2}
{\sc M. Guedda} and {\sc M. Kirane,} { \it Criticality for some evolution
equations,} LAMFA, { 11}, Differential Equations (Russian), Vol. 37, N 4,
(2001), pp. 511-520.

\bibitem{Pinsky2}
{\sc R. G. Pinsky,}  { \it  Finite time blow-up for the inhomogeneous
equation $u\sb t=\Delta u+a(x)u\sp p+\lambda\phi$ in
$\mathbb{R}\sp d$,} Proc. Amer. Math. Soc.,  127, no. 11,  (1999), pp.
3319--3327.

\bibitem{PV}
  {\sc S. Pohozaev}  and  {\sc L. Veron,} { \it Blow-up results for
nonlinear hyperbolic inequalities,}  Ann. Scuola Norm. Sup. Pisa
Cl. Sci., (4), 29,  no. 2, (2000), pp. 393--420.

\bibitem{sgkm}
{\sc A. Samarskii},  {\sc V. Galaktionov,}  {\sc S. Kurdyumov}
and {\sc A.  Mikhailov,} Blow-up in
quasilinear parabolic equations,   de Gruyter Expositions in Mathematics,
19. Walter de Gruyter and Co., Berlin, 1995.

\bibitem{Zh1}
 {\sc Qi S. Zhang,}{ \it  Semilinear parabolic problems on manifolds and
applications to the non-compact Yamabe problem,}  Electron.
J. Diff.  Eqns, No. 46,   (2000) ,  pp. 1--30.

\bibitem{Zh2}
{\sc  Qi S. Zhang,}{ \it  The quantizing effect of potentials on the
critical number of reaction--diffusion equations,} J. Diff.
Eqns, 170, (2001), pp. 188--241.

\end{thebibliography}


\noindent\textsc{Mohammed Guedda } \\
 Lamfa,  CNRS UMR 6140,
Universit\'e de Picardie Jules Verne,\\
Facult\'e  de  Math\'ematiques et d'Informatique, 33,\\
 rue Saint-Leu 80039 Amiens, France \\
e-mail: Guedda@u-picardie.fr \smallskip

\noindent\textsc{Mokhtar Kirane}\\
Laboratoire de Math\'ematiques,
P\^{o}le Sciences et Technologies,\\
Universit\'e   de La Rochelle, \\
Avenue Michel Cr\'epeau,
17042 La Rochelle Cedex, France\\
e-mail: mokhtar.kirane@univ-lr.fr

\end{document}