V-shaped fronts around an obstacle

Mathematische Annalen volume 379, pages661–689(2021)Cite this article

Abstract

In this paper, we investigate V-shaped fronts around an obstacle K. We first prove that there exist solutions emanating from any homogeneous transition front including V-shaped front for exterior domains \(\varOmega ={\mathbb {R}}^N{\setminus } K\). By providing the complete propagation of the V-shaped front, we prove that the V-shaped front can recover after passing the obstacle.

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Fig. 1

Notes

  1. 1.

    The obstacle K is called star-shaped if either \(K=\emptyset \) or 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 \(t\in [0,1)\).

  2. 2.

    The obstacle K is called directionally convex with respect to a hyperplane \(H=\{x\in {\mathbb {R}}^N: x\cdot e=a\}\), with \(e\in \mathbb {S}^{N-1}\) and \(a\in {\mathbb {R}}\), if for every line \(\varSigma \) parallel to e, the set \(K\cap \varSigma \) is either a single line segment or empty and if \(K\cap H\) is equal to the orthogonal projection of K onto H.

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Acknowledgements

Research partially supported by National Science Foundation (grant no. DMS-1514752). The authors are grateful to the anonymous referees for interesting comments which led to an improvement of the article.

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Affiliations

  1. Department of Mathematics, University of Miami, Coral Gables, FL, USA

    Hongjun Guo

  2. Research Institute for Interdisciplinary Science, Okayama University, 3-1-1 Tsushima-naka, Kita-ku, Okayama, 700-8530, Japan

    Harunori Monobe

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  1. Hongjun Guo
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  2. Harunori Monobe
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Correspondence to Hongjun Guo.

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Guo, H., Monobe, H. V-shaped fronts around an obstacle. Math. Ann. 379, 661–689 (2021). https://doi.org/10.1007/s00208-019-01944-y

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Mathematics Subject Classification

  • 35A18
  • 35B08
  • 35B30
  • 35C07
  • 35K57