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Exact solutions and critical behaviour for a linear growth-diffusion equation on a time-dependent domain

Jane Allwright Allwright

Proceedings of the Edinburgh Mathematical Society, Volume: 65, Issue: 1, Pages: 53 - 79

Swansea University Author: Jane Allwright Allwright

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DOI (Published version): 10.1017/s0013091521000754

Abstract

A linear growth-diffusion equation is studied in a time-dependent interval whose location and length both vary. We prove conditions on the boundary motion for which the solution can be found in exact form and derive the explicit expression in each case. Next, we prove the precise behaviour near the...

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Published in: Proceedings of the Edinburgh Mathematical Society
ISSN: 0013-0915 1464-3839
Published: Cambridge University Press (CUP) 2022
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URI: https://cronfa.swan.ac.uk/Record/cronfa59142
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first_indexed 2022-01-10T16:24:01Z
last_indexed 2023-01-11T14:40:12Z
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spelling 2022-07-12T10:36:37.0406227 v2 59142 2022-01-10 Exact solutions and critical behaviour for a linear growth-diffusion equation on a time-dependent domain cb793783c92e676896135595f2f736f1 Jane Allwright Allwright Jane Allwright Allwright true false 2022-01-10 SMA A linear growth-diffusion equation is studied in a time-dependent interval whose location and length both vary. We prove conditions on the boundary motion for which the solution can be found in exact form and derive the explicit expression in each case. Next, we prove the precise behaviour near the boundary in a ‘critical’ case: when the endpoints of the interval move in such a way that near the boundary there is neither exponential growth nor decay, but the solution behaves like a power law with respect to time. The proof uses a subsolution based on the Airy function with argument depending on both space and time. Interesting links are observed between this result and Bramson's logarithmic term in the nonlinear FKPP equation on the real line. Each of the main theorems is extended to higher dimensions, with a corresponding result on a ball with a time-dependent radius. Journal Article Proceedings of the Edinburgh Mathematical Society 65 1 53 79 Cambridge University Press (CUP) 0013-0915 1464-3839 reaction-diffusion equation; time-dependent domain 1 2 2022 2022-02-01 10.1017/s0013091521000754 COLLEGE NANME Mathematics COLLEGE CODE SMA Swansea University SU Library paid the OA fee (TA Institutional Deal) EPSRC-funded studentship (project reference 2227486) 2022-07-12T10:36:37.0406227 2022-01-10T16:21:28.1782649 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics Jane Allwright Allwright 1 59142__23771__0570b875ebcd460082379864068dc8af.pdf 59142.pdf 2022-04-04T13:56:48.0640633 Output 549772 application/pdf Version of Record true © The Author(s), 2021. This is an Open Access article, distributed under the terms of the Creative Commons Attribution-NonCommercial licence true eng https://creativecommons.org/licenses/by-nc/4.0/
title Exact solutions and critical behaviour for a linear growth-diffusion equation on a time-dependent domain
spellingShingle Exact solutions and critical behaviour for a linear growth-diffusion equation on a time-dependent domain
Jane Allwright Allwright
title_short Exact solutions and critical behaviour for a linear growth-diffusion equation on a time-dependent domain
title_full Exact solutions and critical behaviour for a linear growth-diffusion equation on a time-dependent domain
title_fullStr Exact solutions and critical behaviour for a linear growth-diffusion equation on a time-dependent domain
title_full_unstemmed Exact solutions and critical behaviour for a linear growth-diffusion equation on a time-dependent domain
title_sort Exact solutions and critical behaviour for a linear growth-diffusion equation on a time-dependent domain
author_id_str_mv cb793783c92e676896135595f2f736f1
author_id_fullname_str_mv cb793783c92e676896135595f2f736f1_***_Jane Allwright Allwright
author Jane Allwright Allwright
author2 Jane Allwright Allwright
format Journal article
container_title Proceedings of the Edinburgh Mathematical Society
container_volume 65
container_issue 1
container_start_page 53
publishDate 2022
institution Swansea University
issn 0013-0915
1464-3839
doi_str_mv 10.1017/s0013091521000754
publisher Cambridge University Press (CUP)
college_str Faculty of Science and Engineering
hierarchytype
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 A linear growth-diffusion equation is studied in a time-dependent interval whose location and length both vary. We prove conditions on the boundary motion for which the solution can be found in exact form and derive the explicit expression in each case. Next, we prove the precise behaviour near the boundary in a ‘critical’ case: when the endpoints of the interval move in such a way that near the boundary there is neither exponential growth nor decay, but the solution behaves like a power law with respect to time. The proof uses a subsolution based on the Airy function with argument depending on both space and time. Interesting links are observed between this result and Bramson's logarithmic term in the nonlinear FKPP equation on the real line. Each of the main theorems is extended to higher dimensions, with a corresponding result on a ball with a time-dependent radius.
published_date 2022-02-01T04:16:13Z
_version_ 1763754087264813056
score 11.014067

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