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Self-similar fast reaction limit of reaction diffusion systems with nonlinear diffusion / YINI DU
Swansea University Author: YINI DU
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Copyright: The author, Yini Du, 2022.Download (1.48MB)
DOI (Published version): 10.23889/SUthesis.60766
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...
|Supervisor:||Crooks, Elaine ; Mercuri, Carlo|
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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.
ORCiD identifier: https://orcid.org/0000-0002-9765-1314
Nonlinear diffusion; Reaction diffusion problem; Fast reaction; Free boundary; Self-similar solution
College of Science