Study of Solutions for a Degenerate Reaction Equation with a High Order Operator and Advection

José Luis Díaz Palencia, Julián Roa González and Almudena Sánchez Sánchez
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José Luis Díaz Palencia: Department of Education, Universidad a Distancia de Madrid, 28400 Madrid, Spain
Julián Roa González: Department of Education, Universidad a Distancia de Madrid, 28400 Madrid, Spain
Almudena Sánchez Sánchez: Department of Education, Universidad a Distancia de Madrid, 28400 Madrid, Spain

Mathematics, 2022, vol. 10, issue 10, 1-18

Abstract: The goal of the present study is to characterize solutions under a travelling wave formulation to a degenerate Fisher-KPP problem. With the degenerate problem, we refer to the following: a heterogeneous diffusion that is formulated with a high order operator; a non-linear advection and non-Lipstchitz spatially heterogeneous reaction. The paper examines the existence of solutions, uniqueness and travelling wave oscillatory properties (also called instabilities). Such oscillatory behaviour may lead to negative solutions in the proximity of zero. A numerical exploration is provided with the following main finding to declare: the solutions keeps oscillating in the proximity of the null stationary solution due to the high order operator, except if the reaction term is quasi-Lipschitz, in which it is possible to define a region where solutions are positive locally in time.

Keywords: homotopy; high order diffusion; Fisher-KPP; travelling waves; heterogeneous non-Lipschitz reaction (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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