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Cylindrically symmetric travelling fronts in a periodic reaction—diffusion equation with bistable nonlinearity

Part of: Partial differential equations Parabolic equations and systems Representations of solutions

Published online by Cambridge University Press:  28 August 2015

Zhi-Cheng Wang
Zhi-Cheng Wang*
Affiliation:
School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, People’s Republic of China (wangzhch@lzu.edu.cn)
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Abstract

This paper is concerned with the existence, non-existence and qualitative properties of cylindrically symmetric travelling fronts for time-periodic reaction–diffusion equations with bistable nonlinearity in ℝm with m ≥ 2. It should be mentioned that the existence and stability of two-dimensional time-periodic V-shaped travelling fronts and three-dimensional time-periodic pyramidal travelling fronts have been studied previously. In this paper we consider two cases: the first is that the wave speed of a one-dimensional travelling front is positive and the second is that the one-dimensional wave speed is zero. For both cases we establish the existence, non-existence and qualitative properties of cylindrically symmetric travelling fronts. In particular, for the first case we furthermore show the asymptotic behaviours of level sets of the cylindrically symmetric travelling fronts.

Keywords

reaction–diffusion equationtime-periodic nonlinearitybistable nonlinearitycylindrically symmetric travelling front

MSC classification

Primary: 35C07: Traveling wave solutions
Secondary: 35K57: Reaction-diffusion equations 35B10: Periodic solutions 35B08: Entire solutions 92D25: Population dynamics (general)
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 145 , Issue 5 , October 2015 , pp. 1053 - 1090
Copyright
Copyright © Royal Society of Edinburgh 2015 

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