Using effective boundary conditions to model fast diffusion on a road in a large field. (15th September 2017)

• Record Type: Journal Article
• Authors: Li, Huicong; Wang, Xuefeng
• Is Part Of: Nonlinearity. Volume 30:Number 10(2017:Oct.)
• Page Start: 3853
• Page End: 3894
• Publication Date: 2017-09-15
• Subjects: logistic diffusion equation -- asymptotic behavior -- effective boundary condition -- propagation speed -- comparison principle -- regularity; 35K57 -- 35B40 -- 35Q92; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/1361-6544/aa82ee
• Languages: English
• ISSNs: 0951-7715

Abstract

Abstract: 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 ) .