Study on monostable and bistable reaction-diffusion equations by iteration of travelling wave maps

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:263

Article

Yi, Taishan¹  Chen, Yuming² 

[1] Sun Yat Sen Univ, Sch Math Zhuhai, Zhuhai 519082, Guangdong, Peoples R China

[2] Wilfrid Laurier Univ, Dept Math, Waterloo, ON N2L 3C5, Canada

关键词: Dirichlet problem;    Heterogeneous steady state;    Monostable/bistable reaction-diffusion equation;    Spreading speed;    Travelling wave map;   

DOI  :  10.1016/j.jde.2017.08.017

来源: Elsevier

【 摘 要 】

In this paper, based on the iterative properties of travelling wave maps, we develop a new method to obtain spreading speeds and asymptotic propagation for monostable and bistable reaction-diffusion equations. Precisely, for Dirichlet problems of monostable reaction-diffusion equations on the half line, by making links between travelling wave maps and integral operators associated with the Dirichlet diffusion kernel (the latter is NOT invariant under translation), we obtain some iteration properties of the Dirichlet diffusion and some a priori estimates on nontrivial solutions of Dirichlet problems under travelling wave transformation. We then provide the asymptotic behavior of nontrivial solutions in the space-time region for Dirichlet problems. These enable us to develop a unified method to obtain results on heterogeneous steady states, travelling waves, spreading speeds, and asymptotic spreading behavior for Dirichlet problem of monostable reaction-diffusion equations on R+ as well as of monostable/bistable reaction-diffusion equations on R. (C) 2017 Elsevier Inc. All rights reserved.

【 授权许可】

Free