Journal of Applied Analysis 9 (2003), No. 1, 103--121
Copyright Heldermann Verlag 2003
N. Alaa
Faculte des Sciences et Techniques, Universite Cadi Ayyad, BP 618 Marrakech, Marocco
M. Guedda
LAMFA, CNRS UMR 6140, Université de Picardie Jules Verne, Faculté de Mathématiques et d'Informatique, 33 Rue Saint-Leu, 80039 Amiens, France
We consider the problem $$ u_{tt}+\delta u_{t} +\varepsilon a \Delta u+\varphi(\int_{\Omega}|\nabla u|^2dx) \Delta u \geq f(x,t), $$ posed in $\Omega\times(0,+\infty)$. Here $\Omega \subset {\mathbb R}^N$ is a an open smooth bounded domain and $\varphi $ is like $\varphi(s) = bs^\gamma$, $\gamma> 0$, $a > 0 $ and $ \varepsilon = \pm 1$. We prove, in certain conditions on $f $ and $\varphi $ that there is absence of global solutions. The method of proof relies on a simple analysis of the ordinary inequality of the type $$w^{\prime\prime} + \delta w^\prime \geq \alpha w+ \beta w^p.$$ It is also shown that a global positive solution, when it exists, must decay at least exponentially.
Keywords: Nonlinear differential inequalities, hyperbolic and elliptic problems, blow-up, asymptotic behavior of solutions.
MSC 2000: 35L70, 35J65, 35B40.
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