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January  2012, 17(1): 347-366. doi: 10.3934/dcdsb.2012.17.347

## 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, China 2 School of Mathematic and Statistics, Lanzhou University, Lanzhou, Gansu 730000 3 Department of Applied Mathematics, Xidian University, Xi'an 710071, China

Received  June 2009 Revised  May 2011 Published  October 2011

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.
Citation: Shi-Liang Wu, Wan-Tong Li, San-Yang Liu. Exponential stability of traveling fronts in monostable reaction-advection-diffusion equations with non-local delay. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 347-366. doi: 10.3934/dcdsb.2012.17.347
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##### References:
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