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Propagating speeds of bistable transition fronts in spatially periodic media

Abstract

This paper is concerned with the propagating speeds of transition fronts in \(\mathbb {R}^N\) for spatially periodic bistable reaction–diffusion equations. The notion of transition fronts generalizes the standard notions of traveling fronts. Under the a priori assumption that there exist pulsating fronts for every direction e with nonzero speeds, we show some continuity and differentiability properties of the front speeds and profiles with respect to the direction e. Finally, we prove that the propagating speed of any transition front is larger than the infimum of speeds of pulsating fronts and less than the supremum of speeds of pulsating fronts.

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Acknowledgements

The author was supported by the China Scholarship Council for 3 years of study at Aix Marseille Université. The author is grateful to Professor François Hamel for his patient discussions and helpful suggestions, and to the anonymous referee for interesting comments which led to an improvement of the article.

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Correspondence to Hongjun Guo.

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Communicated by P. Rabinowitz.

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Guo, H. Propagating speeds of bistable transition fronts in spatially periodic media. Calc. Var. 57, 47 (2018). https://doi.org/10.1007/s00526-018-1327-9

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  • DOI: https://doi.org/10.1007/s00526-018-1327-9

Mathematics Subject Classification

  • 35A18
  • 35B27
  • 35C07
  • 35K57