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Existence and Optimization of the Critical Speed for Travelling Front Solutions with Convection in Unbounded Cylinders / VIANNEY DOMENECH

Swansea University Author: VIANNEY DOMENECH

DOI (Published version): 10.23889/SUThesis.66077

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For n > 1, we consider a reaction-diffusion equationut = Δu + α(y)∇ · G(u) + f(u), (0.2)in an unbounded cylinder Ω := R×D, where D ⊂ Rn−1 is a smooth bounded domain, with a presence of a convection term, under both Neumann and Dirichlet boundary conditions on ∂Ω. For both types of boundary condit...

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Published: Swansea University, Wales, UK 2023
Institution: Swansea University
Degree level: Doctoral
Degree name: Ph.D
Supervisor: Elaine, C. and Emmanuel, R.
URI: https://cronfa.swan.ac.uk/Record/cronfa66077
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spelling v2 66077 2024-04-18 Existence and Optimization of the Critical Speed for Travelling Front Solutions with Convection in Unbounded Cylinders ae89368a3fbb20ee9f3502809a33b342 VIANNEY DOMENECH VIANNEY DOMENECH true false 2024-04-18 For n > 1, we consider a reaction-diffusion equationut = Δu + α(y)∇ · G(u) + f(u), (0.2)in an unbounded cylinder Ω := R×D, where D ⊂ Rn−1 is a smooth bounded domain, with a presence of a convection term, under both Neumann and Dirichlet boundary conditions on ∂Ω. For both types of boundary condition, we consider two different forms of convection term, namely : α(y)∇·G(u) and∇ · (α(y)G(u)). The reaction term f is “monostable”. In both Neumann and Dirichlet cases, we prove that there exists a critical speed c⋆ ∈ R such that there exists a travelling front solution of the form u(x, t) = w(x1 −ct, y) with speed c if and only if c ≥ c⋆, where x1 is the coordinate corresponding to theaxis of the cylinder. The critical speed c⋆ often plays an important role for monostable problems by characterizing the long-time behaviour of the initial value problem. The existence of travelling waves for all c ≥ c⋆ is typical of monostable problems such as the prototype Fisher-KPP equation.We give a min-max formula for the speed c⋆. For both types of boundary conditions, we prove that c⋆ is bounded below by a quantity c′ which is related to a certain eigenvalue problem, associated with the linearized problem around 0. Note that under Dirichlet boundary conditions, an extra assumption is needed to ensure that c′ exists, namely, f′(0) has to be greater than the principal eigenvalue of the linearized operator. We discuss two special cases where the equality c⋆ = c′ holds. Under both Neumann and Dirichlet boundary conditions, the first special case is when G = (G1, 0, · · ·, 0), assuming the so-called KPP condition for f and that α(y)G′ 1(u) ≥ α(y)G′ 1(0), for all y ∈ D and all u ∈ (0, 1). The second case is treated only under Neumann boundary conditions : when G′ 1(0) = 0, assuming the KPP condition for f, and that α(y)G′ 1(u) ≥ 0, for all y ∈ D and u ∈ (0, 1). Note that in that case, we give an explicit formula : c⋆ = c′ = 2 p f′(0). Under Dirichlet boundary conditions, we highlight the influence of the domain D, the reaction term f and the convection term α(y)∇ · G(u) on the critical speed c⋆. In the special case where G = (G1, 0, ···, 0), using that c⋆ = c′, we use the eigenvalue problem related to c′ to establish some optimization results for c⋆. E-Thesis Swansea University, Wales, UK Travelling front, Convection, Cylinders, Optimization, Neumann boundary conditions,Dirichlet boundary conditions 21 12 2023 2023-12-21 10.23889/SUThesis.66077 COLLEGE NANME COLLEGE CODE Swansea University Elaine, C. and Emmanuel, R. Doctoral Ph.D Université Grenoble-Alpes / Swansea University (Research Partnership Scholarship) Université Grenoble-Alpes / Swansea University (Research Partnership Scholarship) 2024-06-20T16:38:26.2167152 2024-04-18T12:21:21.1176046 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics VIANNEY DOMENECH 1 66077__30715__5090bbf717784945a5fbded3c16a3793.pdf 2023_Domenech _V_final.66077.pdf 2024-06-20T16:37:23.2786746 Output 884157 application/pdf E-Thesis – open access true Copyright: The Author, Vianney Domenech 2023 true eng
title Existence and Optimization of the Critical Speed for Travelling Front Solutions with Convection in Unbounded Cylinders
spellingShingle Existence and Optimization of the Critical Speed for Travelling Front Solutions with Convection in Unbounded Cylinders
VIANNEY DOMENECH
title_short Existence and Optimization of the Critical Speed for Travelling Front Solutions with Convection in Unbounded Cylinders
title_full Existence and Optimization of the Critical Speed for Travelling Front Solutions with Convection in Unbounded Cylinders
title_fullStr Existence and Optimization of the Critical Speed for Travelling Front Solutions with Convection in Unbounded Cylinders
title_full_unstemmed Existence and Optimization of the Critical Speed for Travelling Front Solutions with Convection in Unbounded Cylinders
title_sort Existence and Optimization of the Critical Speed for Travelling Front Solutions with Convection in Unbounded Cylinders
author_id_str_mv ae89368a3fbb20ee9f3502809a33b342
author_id_fullname_str_mv ae89368a3fbb20ee9f3502809a33b342_***_VIANNEY DOMENECH
author VIANNEY DOMENECH
author2 VIANNEY DOMENECH
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publishDate 2023
institution Swansea University
doi_str_mv 10.23889/SUThesis.66077
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 For n > 1, we consider a reaction-diffusion equationut = Δu + α(y)∇ · G(u) + f(u), (0.2)in an unbounded cylinder Ω := R×D, where D ⊂ Rn−1 is a smooth bounded domain, with a presence of a convection term, under both Neumann and Dirichlet boundary conditions on ∂Ω. For both types of boundary condition, we consider two different forms of convection term, namely : α(y)∇·G(u) and∇ · (α(y)G(u)). The reaction term f is “monostable”. In both Neumann and Dirichlet cases, we prove that there exists a critical speed c⋆ ∈ R such that there exists a travelling front solution of the form u(x, t) = w(x1 −ct, y) with speed c if and only if c ≥ c⋆, where x1 is the coordinate corresponding to theaxis of the cylinder. The critical speed c⋆ often plays an important role for monostable problems by characterizing the long-time behaviour of the initial value problem. The existence of travelling waves for all c ≥ c⋆ is typical of monostable problems such as the prototype Fisher-KPP equation.We give a min-max formula for the speed c⋆. For both types of boundary conditions, we prove that c⋆ is bounded below by a quantity c′ which is related to a certain eigenvalue problem, associated with the linearized problem around 0. Note that under Dirichlet boundary conditions, an extra assumption is needed to ensure that c′ exists, namely, f′(0) has to be greater than the principal eigenvalue of the linearized operator. We discuss two special cases where the equality c⋆ = c′ holds. Under both Neumann and Dirichlet boundary conditions, the first special case is when G = (G1, 0, · · ·, 0), assuming the so-called KPP condition for f and that α(y)G′ 1(u) ≥ α(y)G′ 1(0), for all y ∈ D and all u ∈ (0, 1). The second case is treated only under Neumann boundary conditions : when G′ 1(0) = 0, assuming the KPP condition for f, and that α(y)G′ 1(u) ≥ 0, for all y ∈ D and u ∈ (0, 1). Note that in that case, we give an explicit formula : c⋆ = c′ = 2 p f′(0). Under Dirichlet boundary conditions, we highlight the influence of the domain D, the reaction term f and the convection term α(y)∇ · G(u) on the critical speed c⋆. In the special case where G = (G1, 0, ···, 0), using that c⋆ = c′, we use the eigenvalue problem related to c′ to establish some optimization results for c⋆.
published_date 2023-12-21T16:38:25Z
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score 11.013776

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