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Analysis of Reaction-Diffusion Equations on a Time-Dependent Domain / JANE ALLWRIGHT

Swansea University Author: JANE ALLWRIGHT

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DOI (Published version): 10.23889/SUthesis.61569

Abstract

We consider non-negative solutions to reaction-diffusion equations on time-dependent domains with zero Dirichlet boundary conditions. The reaction term is either linear or else monostable of the KPP type. For a range of different forms of the boundary motion, we use changes of variables, exact solutio...

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Published: Swansea 2022
Institution: Swansea University
Degree level: Doctoral
Degree name: Ph.D
Supervisor: Crooks, Elaine
URI: https://cronfa.swan.ac.uk/Record/cronfa61569
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first_indexed 2022-10-17T09:33:08Z
last_indexed 2023-01-13T19:22:24Z
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spelling 2022-10-17T11:42:32.2827840 v2 61569 2022-10-17 Analysis of Reaction-Diffusion Equations on a Time-Dependent Domain ed7b154c961013554bb98ae338ce827e JANE ALLWRIGHT JANE ALLWRIGHT true false 2022-10-17 We consider non-negative solutions to reaction-diffusion equations on time-dependent domains with zero Dirichlet boundary conditions. The reaction term is either linear or else monostable of the KPP type. For a range of different forms of the boundary motion, we use changes of variables, exact solutions, supersolutions and subsolutions to study the long-time behaviour.For a linear equation on (A(t), A(t) + L(t)), we prove that the solution can be found exactly by a separation of variables method when ¨LL3 and ¨AL3 are constants. In these cases L(t) has the form L(t) = √at2 + 2bt + l2. We also analyse the linear problem near the boundary, deriving conditions on L(t) such that the gradient at the boundary remains bounded above and below away from zero. Interesting links are observed between this ‘critical’ boundary motion and Bramson’s logarithmic term for the nonlinear KPP equation.The exact solutions and investigation of behaviour near the boundary are also extended to a ball with time-dependent radius, and a time-dependent box.We then consider time-periodic bounded domains Ω(t). The long-time be-haviour is determined by a principal periodic eigenvalue µ, for which we derive several bounds and also consider the large and small frequency limits. For the nonlinear problem, we prove that the solution converges to either zero or a unique positive periodic solution u∗.The nonlinear problem is also studied on a bounded domain in RN moving at constant velocity c, and we derive several properties of the positive stationary limit Uc.Results describing long-time behaviour for the nonlinear equation are then extended to certain other types of time-dependent domain that have non-constant velocity and non-constant length, and to time-dependent cylinders. E-Thesis Swansea Time-dependent domain; moving boundary; reaction-diffusion equations; separation of variables 11 10 2022 2022-10-11 10.23889/SUthesis.61569 COLLEGE NANME COLLEGE CODE Swansea University Crooks, Elaine Doctoral Ph.D EPSRC: EP/R51312X/1 2022-10-17T11:42:32.2827840 2022-10-17T10:21:57.4990948 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics JANE ALLWRIGHT 1 61569__25477__0cf86d49a9a84d54b11a4f522627902a.pdf Allwright_Jane_PhD_Thesis_Final_Redacted_Signature_Redacted.pdf 2022-10-17T11:36:44.1842232 Output 1719426 application/pdf E-Thesis – open access true Copyright: The author, Jane Allwright, 2022. true eng
title Analysis of Reaction-Diffusion Equations on a Time-Dependent Domain
spellingShingle Analysis of Reaction-Diffusion Equations on a Time-Dependent Domain
JANE ALLWRIGHT
title_short Analysis of Reaction-Diffusion Equations on a Time-Dependent Domain
title_full Analysis of Reaction-Diffusion Equations on a Time-Dependent Domain
title_fullStr Analysis of Reaction-Diffusion Equations on a Time-Dependent Domain
title_full_unstemmed Analysis of Reaction-Diffusion Equations on a Time-Dependent Domain
title_sort Analysis of Reaction-Diffusion Equations on a Time-Dependent Domain
author_id_str_mv ed7b154c961013554bb98ae338ce827e
author_id_fullname_str_mv ed7b154c961013554bb98ae338ce827e_***_JANE ALLWRIGHT
author JANE ALLWRIGHT
author2 JANE ALLWRIGHT
format E-Thesis
publishDate 2022
institution Swansea University
doi_str_mv 10.23889/SUthesis.61569
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 We consider non-negative solutions to reaction-diffusion equations on time-dependent domains with zero Dirichlet boundary conditions. The reaction term is either linear or else monostable of the KPP type. For a range of different forms of the boundary motion, we use changes of variables, exact solutions, supersolutions and subsolutions to study the long-time behaviour.For a linear equation on (A(t), A(t) + L(t)), we prove that the solution can be found exactly by a separation of variables method when ¨LL3 and ¨AL3 are constants. In these cases L(t) has the form L(t) = √at2 + 2bt + l2. We also analyse the linear problem near the boundary, deriving conditions on L(t) such that the gradient at the boundary remains bounded above and below away from zero. Interesting links are observed between this ‘critical’ boundary motion and Bramson’s logarithmic term for the nonlinear KPP equation.The exact solutions and investigation of behaviour near the boundary are also extended to a ball with time-dependent radius, and a time-dependent box.We then consider time-periodic bounded domains Ω(t). The long-time be-haviour is determined by a principal periodic eigenvalue µ, for which we derive several bounds and also consider the large and small frequency limits. For the nonlinear problem, we prove that the solution converges to either zero or a unique positive periodic solution u∗.The nonlinear problem is also studied on a bounded domain in RN moving at constant velocity c, and we derive several properties of the positive stationary limit Uc.Results describing long-time behaviour for the nonlinear equation are then extended to certain other types of time-dependent domain that have non-constant velocity and non-constant length, and to time-dependent cylinders.
published_date 2022-10-11T04:20:29Z
_version_ 1763754355594362880
score 11.036706

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