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Transition fronts for periodic bistable reaction-diffusion equations

  • Weiwei Ding
  • François HamelEmail author
  • Xiao-Qiang Zhao
Article

Abstract

This paper is concerned with the existence and qualitative properties of transition fronts for spatially periodic reaction-diffusion equations with bistable nonlinearities. The notion of transition fronts connecting two stable steady states generalizes the standard notion of pulsating fronts. In this paper, we prove that the time-global solutions in the class of transition fronts share some common features. In particular, we establish a uniform estimate for the mean speed of transition fronts, independently of the spatial scale. Under the a priori existence of a pulsating front with nonzero speed or under a more general condition guaranteeing the existence of such a pulsating front, we show that transition fronts are reduced to pulsating fronts, and thus are unique up to shift in time. On the other hand, when the spatial period is large, we also obtain the existence of a new type of transition fronts which are not pulsating fronts. This example, which is the first one in periodic media, shows that even in periodic media, the notion of generalized transition fronts is needed to describe the set of solutions connecting two stable steady states.

Mathematics Subject Classification

35B27 35B30 35C07 35K57 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Weiwei Ding
    • 1
    • 2
  • François Hamel
    • 1
    Email author
  • Xiao-Qiang Zhao
    • 3
  1. 1.Aix Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373MarseilleFrance
  2. 2.School of Mathematical SciencesUniversity of Science and Technology of ChinaHefeiPeople’s Republic of China
  3. 3.Department of Mathematics and StatisticsMemorial University of NewfoundlandSt. John’sCanada

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