Dinâmica de EDP parabólicas locais ou não locais: existência, unicidade e comportamento assintótico de solução

Abstract

This work is dedicated to the study of dynamical properties of some partial differential equations (PDE, for short) of parabolic type, local or nonlocal ones. We prove existence, uniqueness and establish the asymptotic behavior, for large times, for two classes of PDE. Namely, a (nonlocal) parabolic equation with a Kirchhoff term and flux boundary condition and two parabolic PDE driven by the p-Laplacian with logistic terms involving potentials and weights which may be indefi nite and unbounded. First we prove existence and continuous dependence on data for the PDE with Kirchhoff term using Faedo-Galerkin method and a suitable change of variables. Some sufficient conditions are given to ensure uniqueness of solution. Concerning asymptotic behavior, we show that the omega limit set of each (semi) orbit contains, at least, one stationary solution. We then study stability of local minima of the associated energy functional showing first a result on asymptotic stability of a global minimum for the energy. We also prove a sufficient condition for the existence of isolated local minimum of the energy functional, which is proved to be an asymptotically stable stationary solution in a suitable neighborhood. Finally we determine the dynamics of positive solutions with bounded initial datum for two classes of parabolic PDE driven by the p-Laplacian with indefi nite and unbounded potentials and logistic sources having weights which are also indefi nite and unbounded. The boundary conditions appearing are also of flux type (linear or nonlinear). The asymptotic stability properties of the stationary solutions are described using principal eigenvalues of some elliptic eigenvalue problems involving a parameter of the original PDE. Those eigenvalue problems are also studied herein in order to be used as a tool for obtaining our results due to the low regularity assumptions on their coeficients from the viewpoint of the literature.