No Cover Image

E-Thesis 43 views 20 downloads

Self-similar fast reaction limit of reaction diffusion systems with nonlinear diffusion / YINI DU

Swansea University Author: YINI DU

DOI (Published version): 10.23889/SUthesis.60766

Abstract

In this thesis, we present an approach to characterising fast-reaction lim-its of systems with nonlinear diffusion, when there are either two reaction-diffusion equations, or one reaction-diffusion equation and one ordinary dif-ferential equation on unbounded domains. Here, we replace the terms of the...

Full description

Published: Swansea 2022
Institution: Swansea University
Degree level: Doctoral
Degree name: Ph.D
Supervisor: Crooks, Elaine ; Mercuri, Carlo
URI: https://cronfa.swan.ac.uk/Record/cronfa60766
Tags: Add Tag
No Tags, Be the first to tag this record!
Abstract: In this thesis, we present an approach to characterising fast-reaction lim-its of systems with nonlinear diffusion, when there are either two reaction-diffusion equations, or one reaction-diffusion equation and one ordinary dif-ferential equation on unbounded domains. Here, we replace the terms of the form uxx in usual reaction-diffusion equation, which represent linear diffusion, by terms of form φ(u)xx, representing nonlinear diffusion. For appropriate initial data, in the fast-reaction limit k → ∞, spatial segregation results in the two components of the original systems each converge to the positive and negative points of a self-similar limit profile f(η), where η = √xt , that satisfies one of four ordinary differential systems. The existence of these self-similar solutions of the k → ∞ limit problems is proved by using shooting methods which focus on a, the position of the free boundary which separates the regions where the solution is positive and where it is negative, and γ, the derivative of −φ(f) at η = a. The position of the free boundary gives us intuition how one substance penetrates into the other, so for specific forms of nonlinear diffusion, the relationship between the given form of the nonlinear diffusion and the position of the free boundary is also studied.
Item Description: ORCiD identifier: https://orcid.org/0000-0002-9765-1314
Keywords: Nonlinear diffusion; Reaction diffusion problem; Fast reaction; Free boundary; Self-similar solution
College: College of Science