Electronic Journal of Differential Equations,
Conference 09 (2002), pp. 149-160.

Title: Local and global nonexistence of solutions to semilinear
  evolution equations.

Authors: Mohammed Guedda (Univ. de Picardie Jules Verne, France) 
  Mokhtar  Kirane (Univ. de  La Rochelle Cedex, France)
 
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.

Published December 28, 2002.
Math Subject Classifications: 35K55, 35K65, 35L60.
Key Words: Parabolic inequality; hyperbolic equation; fractional power; Fujita-type result.