Electron. J. Differential Equations, Vol. 2023 (2023), No. 04, pp. 117.
Semigroup theory and asymptotic profiles of solutions for a higherorder
FisherKPP problem in R^N
Jose Luis Diaz Palencia
Abstract:
We study a reactiondiffusion problem formulated with a higherorder operator,
a nonlinear advection, and a FisherKPP reaction term depending on the spatial variable.
The higherorder 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^{N}, 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 HamiltonJacobi 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; FisherKPP equation; HamiltonJacobi equation.
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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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