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Existence, uniqueness and stability of transition fronts of non-local equations in time heterogeneous bistable media

Part of: Parabolic equations and systems Representations of solutions

Published online by Cambridge University Press:  28 August 2019

WENXIAN SHEN  and
ZHONGWEI SHEN Open the ORCID record for ZHONGWEI SHEN [Opens in a new window]
WENXIAN SHEN
Affiliation:
Department of Mathematics and Statistics, Auburn University, Auburn, AL36849, USA e-mail: wenxish@auburn.edu
ZHONGWEI SHEN
Affiliation:
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB T6G 2G1, Canada e-mail: zhongwei@ualberta.ca
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Abstract

The present paper is devoted to the study of the existence, the uniqueness and the stability of transition fronts of non-local dispersal equations in time heterogeneous media of bistable type under the unbalanced condition. We first study space non-increasing transition fronts and prove various important qualitative properties, including uniform steepness, stability, uniform stability and exponential decaying estimates. Then, we show that any transition front, after certain space shift, coincides with a space non-increasing transition front (if it exists), which implies the uniqueness, up-to-space shifts and monotonicity of transition fronts provided that a space non-increasing transition front exists. Moreover, we show that a transition front must be a periodic travelling front in periodic media and asymptotic speeds of transition fronts exist in uniquely ergodic media. Finally, we prove the existence of space non-increasing transition fronts, whose proof does not need the unbalanced condition.

Keywords

transition frontnon-local dispersal equationbistabletime heterogeneous media

MSC classification

Primary: 35C07: Traveling wave solutions 35K55: Nonlinear parabolic equations 35K57: Reaction-diffusion equations
Secondary: 92D25: Population dynamics (general)
Type
Papers
Information
European Journal of Applied Mathematics , Volume 31 , Issue 4 , August 2020 , pp. 601 - 645
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Copyright
© Cambridge University Press 2019

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Footnotes

†

Z. Shen is supported by a start-up grant from the University of Alberta and an NSERC discovery grant.

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