Exponential stability of traveling fronts in monostable reaction-advection-diffusionequations with non-local delay

Shi-Liang Wu; Wan-Tong Li; San-Yang Liu

doi:10.3934/dcdsb.2012.17.347

Article Contents

2012, Volume 17, Issue 1: 347-366. Doi: 10.3934/dcdsb.2012.17.347

This issue Previous Article Generalized Jacobi rational spectral methods with essential imposition of Neumann boundary conditions in unbounded domains Next Article Stability and Hopf bifurcations for a delayed diffusion system in population dynamics

Exponential stability of traveling fronts in monostable reaction-advection-diffusion equations with non-local delay

1.
Department of Mathematics, Xidian University, Xi’an, Shaanxi 710071
2.
School of Mathematic and Statistics, Lanzhou University, Lanzhou, Gansu 730000
3.
Department of Applied Mathematics, Xidian University, Xi'an 710071

Received: June 2009

Revised: May 2011

Published: January 2012

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Abstract

This paper is concerned with the exponential stability of traveling fronts in monostable reaction-advection-diffusion equations with nonlocal delay. The existence and comparison theorem of solutions of the corresponding Cauchy problem in a weighted Sobolev space are first established for the systems on $\mathbb{R}$ by appealing to the theory of semigroup and abstract functional differential equations. The exponential stability of traveling fronts is then proved by the comparison principle and the (technical) weighted energy method. Comparing with the previous results, our results recovers and/or improves a number of existing ones. Finally, we apply our results to some biological and epidemic models and obtain some new results.

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Mathematics Subject Classification: Primary: 35K57, 35B40; Secondary: 35R20, 92D25.

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