Electron. J. Differential Equations, Vol. 2023 (2023), No. 04, pp. 1-17.

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

Jose Luis Diaz Palencia

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.

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.
DOI: https://doi.org/10.58997/ejde.2023.04

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José Luis Díaz Palencia
Department of Mathematics and Education
Universidad a Distancia de Madrid
28400, Madrid, Spain
email: joseluis.diaz.p@udima.es

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