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Transition fronts in unbounded domains with multiple branches

Calculus of Variations and Partial Differential Equations volume 59, Article number: 160 (2020) Cite this article

Abstract

This paper is concerned with the existence and uniqueness of transition fronts of a general reaction-diffusion-advection equation in domains with multiple branches. In this paper, every branch in the domain is not necessary to be straight and we use the notions of almost-planar fronts to generalize the standard planar fronts. Under some assumptions of existence and uniqueness of almost-planar fronts with positive propagating speeds in extended branches, we prove the existence of entire solutions emanating from some almost-planar fronts in some branches. Then, we get that these entire solutions converge to almost-planar fronts in some of the rest branches as time increases if no blocking occurs in these branches. Finally, provided by the complete propagation of every front-like solution emanating from one almost-planar front in every branch, we prove that there is only one type of transition fronts, that is, the entire solutions emanating from some almost-planar fronts in some branches and converging to almost-planar fronts in the rest branches.

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Fig. 1

Notes

  1. A bounded set K is called star-shaped if there is x in the interior \(\mathrm {Int}(K)\) of K such that \(x+t(y-x)\in \mathrm {Int}(K)\) for all \(y\in \partial K\) and \(t\in [0,1)\). Then, we say that K is star-shaped with respect to the point x.

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Acknowledgements

Research partially supported by National Science Foundation (grant no. NSF1826801).

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Wyoming, Laramie, WY, USA

    Hongjun Guo

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  1. Hongjun Guo
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Correspondence to Hongjun Guo.

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Communicated by P. Rabinowitz.

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Guo, H. Transition fronts in unbounded domains with multiple branches. Calc. Var. 59, 160 (2020). https://doi.org/10.1007/s00526-020-01825-2

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  • DOI: https://doi.org/10.1007/s00526-020-01825-2

Mathematics Subject Classification

  • 35B51
  • 35J61
  • 35K57