Análise da equação de difusão não linear com potências fracionárias do laplaciano via grupo de renormalização

Abstract

The main objective of this study is to describe and apply the techniques of the Renormalization Group (RG) in the study of the asymptotic behavior of the solution to the following initial value problem: "Formula disponível no texto completo" where f is the initial data, ρ, η E [0, 1] and M is an operator in the Fourier space defined by M —(—A)β with3 4 < β < 1. The method of the Renormalization Group (RG) emerged in the late 1950s in Quantum Field Theory [10] and later it was used to study Critical Phenomena in Statistical Mechanics. In the early 1990s, it was applied to the asymptotic analysis of solutions of differential equations [3], through the use of concepts such as scales invariance and universality, in the search for sets of initial data and perturbations of equations whose solutions presented the same asymptotic behavior. This method involves a multi-scale problem, whose idea is to look for a solution that is scales invariant, and this solution then appears as a fixed point of an operator. For a better understanding of the method, this study was divided in three cases: linear case (η = ρ = 0), linear case with dispersive term (ρ = 0) and nonlinear case (η = 0 e ρ = 0). In all the cases, we see that the solutions behave as follows: "Formula disponível no texto completo" where and are critical exponents, A is a pre-factor that has the information of the initial data and the nonlinear term upux and f is the prole function. The study developed in this work was based on Article [1]. "Formula disponível no texto completo"