On a free boundary problem for a nonlocal reaction-diffusion model

Jia-Feng Cao; Wan-Tong Li; Meng Zhao

doi:10.3934/dcdsb.2018128

Article Contents

2018, Volume 23, Issue 10: 4117-4139. Doi: 10.3934/dcdsb.2018128

This issue Previous Article Time asymptotics of structured populations with diffusion and dynamic boundary conditions Next Article Global phase portraits of a degenerate Bogdanov-Takens system with symmetry (Ⅱ)

On a free boundary problem for a nonlocal reaction-diffusion model

School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China

Received: June 30, 2017

Revised: October 31, 2017

Early access: April 2018

Published: September 2018

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Abstract

This paper is concerned with the spreading or vanishing dichotomy of a species which is characterized by a reaction-diffusion Volterra model with nonlocal spatial convolution and double free boundaries. Compared with classical reaction-diffusion equations, the main difficulty here is the lack of a comparison principle in nonlocal reaction-diffusion equations. By establishing some suitable comparison principles over some different parabolic regions, we get the sufficient conditions that ensure the species spreading or vanishing, as well as the estimates of the spreading speed if species spreading happens. Particularly, we establish the global attractivity of the unique positive equilibrium by a method of successive improvement of lower and upper solutions.

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Mathematics Subject Classification: 35K57, 35R20, 92D25.

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