Fast reaction limit of reaction-diffusion systems

Hideki Murakawa

doi:10.3934/dcdss.2020405

Article Contents

2021, Volume 14, Issue 3: 1047-1062. Doi: 10.3934/dcdss.2020405

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Fast reaction limit of reaction-diffusion systems

Hideki Murakawa

Faculty of Advanced Science and Technology, Ryukoku University, 1-5 Yokotani, Seta Oe-cho, Otsu, Shiga 520-2194, Japan

Received: December 31, 2018

Revised: April 30, 2020

Early access: July 2020

Published: March 2021

This work was supported by JSPS KAKENHI Grant nos. 26287025, 15H03635 and 17K05368. Most of the work was performed during a visit of the author to Imperial College London thanks to JST CREST Grant No. JPMJCR14D3. The support of JST and the hospitality of Imperial College London are warmly acknowledged

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Abstract

Singular limit problems of reaction-diffusion systems have been studied in cases where the effects of the reaction terms are very large compared with those of the other terms. Such problems appear in literature in various fields such as chemistry, ecology, biology, geology and approximation theory. In this paper, we deal with the singular limit of a general reaction-diffusion system including many problems in the literature. We formulate the problem, derive the limit equation and establish a rigorous mathematical theory.

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Mathematics Subject Classification: Primary: 35K57, 35K65; Secondary: 35K51, 35R35, 80A22, 80A30.

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Figure 1. Level sets $ F(u,v) = 0 $, which are the graphs $ \{ (u,v)\ |\ v\in \alpha (u)\} $ (thick lines), and vector fields of (9) for some $ F $. (a) $ F(u,v) = F_2(u,-v) $ with $ F_2 $ of (8) (the limit of (1) is represented by the one-phase Stefan problem), (b) $ F $ of (11) with $ \beta $ of (10) (the limit of (1) is represented by the two-phase Stefan problem), (c) $ F $ of (11) with $ \beta $ of (12) (the limit of (1) is represented by the porous medium equation)

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