Problémes D'existence Globale De Solutions Pour Des Équations D'évolution Non Linéaires. Doctorat Thesis(2015), Université De Batna 2.

Résumé: The aim of this thesis is to study the problems of global existence of solutions for non linear evolution equations. We rst consider the coupled Gierer-Meinhardt systems with homogeneous Neumann boundary conditions. By using the technique of Lyapunov function we prove global existence of solutions. Under suitable conditions, we contribute to the study of the asymptotic behaviour of solutions. The basic idea of this result is a Lyapunov function which is non increasing function. These results are valid for any positive initial data in C( ), without any differentiability conditions. Moreover, we show that under reasonable conditions on the exponents of the non linear terms the solutions for considered system blow up in nite time. The second part is devoted to study the uniform boundedness and so global existence of solutions for a Gierer-Meinhardt model of three substances described by reaction-disusion equations with homogeneous Neumann boundary conditions. The proof of this result is based on a suitable Lyapunov functional and from which a result on the asymptotic behaviour of the solutions is established. In the third and the last part, we investigate the local existence and uniqueness of mild solution for some hyper-viscous Hamilton-Jacobi equations. Under suitable conditions, the local existence of weak and strong solution, and the uniqueness of strong solution are also studied for considered problem. Moreover, we show the blow up in nite time of weak solution for some fractional Hamilton Jacobi-type equations

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