2002-Fez conference on Partial Differental Equations,
Electron. J. Diff. Eqns., Conf. 09, 2002, pp. 149-160.

Local and global nonexistence of solutions to semilinear evolution equations

Mohammed Guedda & Mokhtar Kirane

Abstract:
For a fixed $ p $ and $ \sigma greater than -1 $, such that $ p greater than \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 less than \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.
Subject classfications: 35K55, 35K65, 35L60.
Key words: Parabolic inequality, hyperbolic equation, fractional power, Fujita-type result.

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Mohammed Guedda
Universite de Picardie Jules Verne
Faculte de Mathematiques et d'Informatique
33, rue Saint-Leu 80039 Amiens, France
e-mail: Guedda@u-picardie.fr
Mokthar Kirane
Laboratoire de Mathematiques,
Pole Sciences et Technologies,
Universite de la Rochelle, Av. M. Crepeau,
17042 La Rochelle Cedex, France
e-mail: mokhtar.kirane@univ-lr.fr

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