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The spatial critical points not moving along the heat flow

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Abstract

We consider solutions of the heat equation, in domains inR N, and their spatial critical points. In particular, we show that a solutionu has a spatial critical point not moving along the heat flow if and only ifu satisfies some balance law. Furthermore, in the case of Dirichlet, Neumann, and Robin homogeneous initial-boundary value problems on bounded domains, we prove that if the origin is a spatial critical point never moving for sufficiently many compactly supported initial data satisfying the balance law with respect to the origin, then the domain must be a ball centered at the origin.

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Authors and Affiliations

  1. Dipartimento di Matematica “U. Dini”, Università di Firenze, Viale Morgagni 67/A, 50134, Firenze, Italy

    Rolando Magnanini

  2. Department of Mathematics Faculty of Science, Ehime University, 2-5 Bunkyo-cho Matsuyama-shi, 790-77, Ehime, Japan

    Shigeru Sakaguchi

Authors
  1. Rolando Magnanini
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  2. Shigeru Sakaguchi
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Correspondence to Rolando Magnanini.

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Magnanini, R., Sakaguchi, S. The spatial critical points not moving along the heat flow. J. Anal. Math. 71, 237–261 (1997). https://doi.org/10.1007/BF02788032

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  • DOI: https://doi.org/10.1007/BF02788032

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