Semigroup theory and asymptotic profiles of solutions for a higher-order Fisher-KPP problem in R^N

Abstract

We study a reaction-diffusion problem formulated with a higher-order operator, a non-linear advection, and a Fisher-KPP reaction term depending on the spatial variable. The higher-order operator induces solutions to oscillate in the proximity of an equilibrium condition. Given this oscillatory character, solutions are studied in a set of bounded domains. We introduce a new extension operator, that allows us to study the solutions in the open domain RN, but departing from a sequence of bounded domains. The analysis about regularity of solutions is built based on semigroup theory. In this approach, the solutions are interpreted as an abstract evolution given by a bounded continuous operator. Afterward, asymptotic profiles of solutions are studied based on a Hamilton-Jacobi equation that is obtained with a single point exponential scaling. Finally, a numerical assessment, with the function bvp4c in Matlab, is introduced to discuss on the validity of the hypothesis.