Solvability and bow-up of solutions of some nonlinear parabolic problems with different boundary conditions
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Date
2022
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Université de Larbi Ben M'hidi- Oum El Bouaghi
Abstract
The aim of this dissertation is to investigate a class of nonlinear parabolic problems with different boundary conditions (local, non-local and nonlinear conditions), where we began with a reminder of some basic preliminary concepts and tools required in this work as a first chapter.
The second chapter concerning a nonlinear parabolic problem with the classical Neumann boundary conditions; where we show the existence and the uniqueness of weak solution by using the energy inequality method and an iterative process based on a priori estimate. After that, we moved to dynamic issue, exactly we study the blow-up solution by the energy function method.
The third chapter devoted to study a nonlinear problem with nonlocal conditions of the second type; we present firstly the solvability of the associated linear problem where we separate it into two linear problems and showing their existence using the variable separation method and the energy inequality method. Then by using the Linearization method we prove the existence and the uniqueness of the weak solution of the nonlinear problem. We study also the finite time blow-up of the solutions .
Finally, in the last chapter, we examined the existence of weak solution of initial boundary problem for anonlinear parabolic equation with nonlinear boundary conditions by using Faedo-Galerkin method.
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Keywords
Nonlinear parabolic equation, Energy inequality, Faedo-Galerkin method