Estudio Detallado de Problemas Parabólicos Locales. Una Introducción a Problemas no Locales

Abstract

A partial differential equation (PDE) is an equation that involves an unknown function of two or more variables and some of its partial derivatives. This helps to describe a wide variety of physical situations such as: sound, heat, elasticity, dynamic fluids, etc. This work focuses on the elliptic and parabolic equations. For the local case, the evolutionary equations of parabolic type are elaborated detailing in a detailed manner all the present proofs in (Evans, 2010) regarding the existence, uniqueness and regularity of weak solutions using the Galerkin approximation technique, to then enter the study of nonlocal equations: elliptical case and parabolic case. The latter topic includes nonlocal vector calculation results and the study of fractional Sobolev spaces, with the aim of demonstrating, in a specific case, the equivalence that exists between these spaces and the non-local spaces of energy which are detailed in due time. Finally, the variational formulations of both problems are carried out under Dirichlet and Neumann volume conditions, and the existence and uniqueness of the weak solutions for the non-local elliptic case under both volume conditions is studied.