The effective boundary conditions and their lifespan of the logistic diffusion equation on a coated body

Li, Huicong; Wang, Xuefeng; Wu, Yanxia

doi:10.1016/j.jde.2014.07.004

Cited by 6 publications

(3 citation statements)

References 16 publications

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“…[32]). As an interesting consequence, the dynamical boundary condition for the heat equation can be interpreted as the fast reaction limit of a volume-surface reaction-diffusion system; see [11,20,23,24] for alternative derivations. Note that, up to the best of our knowledge, [18] is the only existing result concerning fast reaction limits for VSRD systems.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Fast reaction limit of a volume–surface reaction–diffusion system towards a heat equation with dynamical boundary conditions

Henneke

Tang

2016

ASY

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The fast reaction limit of a volume-surface reaction-diffusion system is rigorously investigated. The system is motivated by proteins localisation in stem cell division. By using Ball's energy equation method, we show that as the reaction rate constant goes to infinity, the solution of the original system converges to the solution of a heat equation with dynamical boundary condition. As a consequence, the dynamical boundary condition can be interpreted as a fast reaction limit of a volume-surface reaction-diffusion system.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Fast reaction limit of a volume–surface reaction–diffusion system towards a heat equation with dynamical boundary conditions

Henneke

Tang

2016

ASY

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“…To derive the effective boundary conditions on the x-axis, unlike in the case of 'boundary reinforcement problem' in a bounded domain, considered in [22], we also need the following higher order estimates. Lemma 2.5.…”

Section: Now We Are Ready To Presentmentioning

confidence: 99%

“…As a result, one naturally expects a Wentzell type effective boundary condition. In the scenario of σδ → b ∈ [0, ∞], Robin type effective boundary conditions appear, as studied by the authors in the case of a bounded domain in [22,24,26,27,35].…”

Section: Introductionmentioning

confidence: 99%

Using effective boundary conditions to model fast diffusion on a road in a large field

Wang

2017

Nonlinearity

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We consider a logistic diffusion equation on the plane consisting of two components, a 'road' that is parallel to the x-axis, and a 'field', in each of which the diffusion rate differs significantly. Compared to the size of the field, the width δ of the road is assumed to be small. Thus in this diffusion equation multiple scales appear in two places: the spatial variable and the diffusion parameter. Such an equation is not easy to solve numerically, and it is not easy to see the effects of the road. Recently, Berestycki, Roquejoffre and Rossi provide a model which is meant to resolve these issues. In this paper we first use the idea of effective boundary conditions (EBCs) to propose, rigorously, a different model: we study the limit of the solution of the original logistic equation as δ → 0, obtaining a limiting model, in which the road now is the x-axis with EBCs imposed on it. This effective problem has no multiple scales and hence should be easier to solve numerically. Moreover, to see the effects of the road, we further investigate the asymptotic propagation speed of the effective model, showing that the road indeed enhances the spreading speed along its direction, provided that the diffusion rate on the road is of order O δ −1 .

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Bulk-surface coupling: Derivation of two models

Wang

et al. 2021

Journal of Differential Equations

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The effective boundary conditions and their lifespan of the logistic diffusion equation on a coated body

Cited by 6 publications

References 16 publications

Fast reaction limit of a volume–surface reaction–diffusion system towards a heat equation with dynamical boundary conditions

Fast reaction limit of a volume–surface reaction–diffusion system towards a heat equation with dynamical boundary conditions

Using effective boundary conditions to model fast diffusion on a road in a large field

Bulk-surface coupling: Derivation of two models

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