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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...

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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
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first_indexed 2022-08-05T15:29:17Z
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spelling v2 60766 2022-08-05 Self-similar fast reaction limit of reaction diffusion systems with nonlinear diffusion 2bfcd9725f897025ac9db0056fd3dc64 YINI DU YINI DU true false 2022-08-05 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. E-Thesis Swansea Nonlinear diffusion; Reaction diffusion problem; Fast reaction; Free boundary; Self-similar solution 29 7 2022 2022-07-29 10.23889/SUthesis.60766 ORCiD identifier: https://orcid.org/0000-0002-9765-1314 COLLEGE NANME COLLEGE CODE Swansea University Crooks, Elaine ; Mercuri, Carlo Doctoral Ph.D 2022-08-05T16:41:34.6879800 2022-08-05T16:26:43.5715149 College of Science Mathematics YINI DU 1 60766__24888__d2f295452d9d4f66ab553c2c096ebeea.pdf Du_Yini_PhD_Thesis_Final_Redacted_Signature.pdf 2022-08-05T16:34:59.6957234 Output 1552277 application/pdf E-Thesis – open access true Copyright: The author, Yini Du, 2022. true eng
title Self-similar fast reaction limit of reaction diffusion systems with nonlinear diffusion
spellingShingle Self-similar fast reaction limit of reaction diffusion systems with nonlinear diffusion
YINI DU
title_short Self-similar fast reaction limit of reaction diffusion systems with nonlinear diffusion
title_full Self-similar fast reaction limit of reaction diffusion systems with nonlinear diffusion
title_fullStr Self-similar fast reaction limit of reaction diffusion systems with nonlinear diffusion
title_full_unstemmed Self-similar fast reaction limit of reaction diffusion systems with nonlinear diffusion
title_sort Self-similar fast reaction limit of reaction diffusion systems with nonlinear diffusion
author_id_str_mv 2bfcd9725f897025ac9db0056fd3dc64
author_id_fullname_str_mv 2bfcd9725f897025ac9db0056fd3dc64_***_YINI DU
author YINI DU
author2 YINI DU
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department_str Mathematics{{{_:::_}}}College of Science{{{_:::_}}}Mathematics
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description 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.
published_date 2022-07-29T16:41:32Z
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score 10.898149