Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 04, pp. 1-17.

Title: Semigroup theory and asymptotic profiles of solutions for a higher-order 
Fisher-KPP problem in R^N
       
Author: Jose Luis Diaz Palencia (Univ. a Distancia de Madrid, Spain)
         
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 R<sup>N</sup>, 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.

Submitted June 15, 2022. Published January 16, 2023.
Math Subject Classifications: 35K92, 35K91, 35K55.
Key Words: Higher order diffusion; semigroup theory; Fisher-KPP equation; Hamilton-Jacobi equation.