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Stability, uniqueness and recurrence of generalized traveling waves in time heterogeneous media of ignition type


Authors: Wenxian Shen and Zhongwei Shen
Journal: Trans. Amer. Math. Soc. 369 (2017), 2573-2613
MSC (2010): Primary 35C07, 35K55, 35K57, 92D25
DOI: https://doi.org/10.1090/tran/6726
Published electronically: June 29, 2016
MathSciNet review: 3592521
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Abstract: The present paper is devoted to the study of stability, uniqueness and recurrence of generalized traveling waves of reaction-diffusion equations in time heterogeneous media of ignition type, whose existence has been proven by the authors of the present paper in a previous work. It is first shown that generalized traveling waves exponentially attract wave-like initial data. Next, properties of generalized traveling waves, such as space monotonicity and exponential decay ahead of interface, are obtained. Uniqueness up to space translations of generalized traveling waves is then proven. Finally, it is shown that the wave profile and the front propagation velocity of the unique generalized traveling wave are of the same recurrence as the media. In particular, if the media is time almost periodic, then so are the wave profile and the front propagation velocity of the unique generalized traveling wave.


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Additional Information

Wenxian Shen
Affiliation: Department of Mathematics and Statistics, Auburn University, Auburn, Alabama 36849
Email: wenxish@auburn.edu

Zhongwei Shen
Affiliation: Department of Mathematics and Statistics, Auburn University, Auburn, Alabama 36849
Address at time of publication: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1, Canada
Email: zzs0004@auburn.edu, zhongwei@ualberta.ca

DOI: https://doi.org/10.1090/tran/6726
Keywords: Generalized traveling wave, stability, monotonicity, uniqueness, recurrence, almost periodicity
Received by editor(s): August 19, 2014
Received by editor(s) in revised form: April 15, 2015
Published electronically: June 29, 2016
Article copyright: © Copyright 2016 American Mathematical Society