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Traveling wave solutions to some reaction diffusion equations with fractional Laplacians

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Abstract

We show the nonexistence of traveling wave solutions in the combustion model with fractional Laplacian \(\displaystyle (-\Delta )^s\) when \(\displaystyle s\in (0,1/2]\). Our method can be used to give a direct and simple proof of the nonexistence of traveling fronts for the usual Fisher-KPP nonlinearity. Also we prove the existence and nonexistence of traveling wave solutions for different ranges of the fractional power \(s\) for the generalized Fisher–KPP type model.

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Acknowledgments

This work was partially supported by a grant from the Simons Foundation (Award # 199305) and a NSF IPA award. The authors would also like to thank the anonymous referee for helpful suggesions for the revision of the manuscript.

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Authors and Affiliations

  1. Department of Mathematics, University of Connecticut, 196 Auditorium Road, Unit 3009, Storrs, CT, 06269-3009, USA

    Changfeng Gui & Tingting Huan

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  1. Changfeng Gui
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  2. Tingting Huan
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Correspondence to Changfeng Gui.

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Communicated by A. Malchiodi.

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Gui, C., Huan, T. Traveling wave solutions to some reaction diffusion equations with fractional Laplacians. Calc. Var. 54, 251–273 (2015). https://doi.org/10.1007/s00526-014-0785-y

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  • DOI: https://doi.org/10.1007/s00526-014-0785-y

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