# Contributions aux problèmes d'évolution

Abstract : In the first chapter, the large time behavior of non-negative solutions to the reaction-diffusion equation $\partial_t u=-(-\Delta)^{\alpha/2}u - u^p,$ $(\alpha\in(0,2], \;p>1)$ posed on $\mathbb{R}^N$ and supplemented with an integrable initial condition is studied. We show that the anomalous diffusion term determines the large time asymptotics for $p>1+{\alpha}/{N},$ while nonlinear effects win if $p\leq1+{\alpha}/{N}.$\\ In chapter two, we present first a new technique to prove, in a general case, the recent result of Cazenave, Dickstein and Weissler on the blowing-up solutions to a temporally nonlocal nonlinear parabolic equation.\\ Then, we study the blow-up rate and the global existence in time of the solutions. Furthermore, we establish necessary conditions for global existence.\\ In the chapter three, we investigate the local existence and the finite-time blow-up of solutions of a semilinear parabolic system with nonlocal in time nonlinearities.\\ Moreover, we investigate the blow-up rate and the necessary conditions for local and global existence.\\ Finally, we study the local existence solutions of a hyperbolic equation with a nonlocal in time nonlinearity. Moreover, we give blow-up theorem for the solution under some conditions on the initial data and the exponents of the nonlinear term.
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https://tel.archives-ouvertes.fr/tel-00437141
Submitted on : Monday, February 22, 2010 - 7:00:03 AM
Last modification on : Thursday, October 8, 2020 - 1:00:03 PM
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• HAL Id : tel-00437141, version 1

### Citation

Ahmad Fino. Contributions aux problèmes d'évolution. Mathématiques [math]. Université de La Rochelle, 2010. Français. ⟨tel-00437141⟩

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