Study of Local and Nonlocal Elliptic and Parabolic Problems
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Date
2024
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Publisher
Faculty of Sciences
Abstract
In this work, we focus on partial differential equations of the elliptic and parabolic types.
The aim of this thesis is to find weak solutions for these equations, Which includes both the
classical and fractional Laplace operator.
We first present the basic concepts and general results necessary for the study. We then address
the existence and uniqueness of solutions for elliptic equations with homogeneous Dirichlet
boundary conditions, using methods from convex analysis and critical point theory.
The study is then extended to fractional Laplace operator with the same boundary condition.
After , we study the existence of non-trivial solutions for equations driven by a non-local inte-
grodifferential operator with semi-linear term and homogeneous Dirichlet boundary conditions.
employing variational methods : the Mountain Pass theorem and the Linking theorem.
Finally, we discuss a model problem of parabolic partial differential equations, in both linear
and nonlinear cases. For linear case, we apply the Faedo-Galerkin method, while for nonlinear
case, we use the Schauder Fixed Point theorem.