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Transversality of stable and Nehari manifolds for a semilinear heat equation

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Abstract

It is well known that for the subcritical semilinear heat equation, negative initial energy is a sufficient condition for finite time blowup of the solution. We show that this is no longer true when the energy functional is replaced with the Nehari functional, thus answering negatively a question left open by Gazzola and Weth (2005). Our proof proceeds by showing that the local stable manifold of any non-zero steady state solution intersects the Nehari manifold transversally. As a consequence, there exist solutions converging to any given steady state, with initial Nehari energy either negative or positive.

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Authors and Affiliations

  1. Departamento de Matematica Aplicada, Universidade Federal do Rio de Janeiro, Caixa Postal 68530, Rio de Janeiro, CEP 21945-970, Brazil

    Flavio Dickstein

  2. Department of Mathematics, Tokyo Gakugei University, Koganei, Tokyo, 184-8501, Japan

    Noriko Mizoguchi

  3. Precursory Research for Embryonic Science and Technology (PRESTO), Japan Science and Technology Agency (JST), Tokyo, Japan

    Noriko Mizoguchi

  4. Université Paris 13, CNRS UMR 7539, Institut Galilée, LAGA, 93430, Villetaneuse, France

    Philippe Souplet & Fred Weissler

Authors
  1. Flavio Dickstein
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  2. Noriko Mizoguchi
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  3. Philippe Souplet
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  4. Fred Weissler
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Corresponding author

Correspondence to Fred Weissler.

Additional information

Communicated by P. Rabinowitz.

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Dickstein, F., Mizoguchi, N., Souplet, P. et al. Transversality of stable and Nehari manifolds for a semilinear heat equation. Calc. Var. 42, 547–562 (2011). https://doi.org/10.1007/s00526-011-0397-8

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  • DOI: https://doi.org/10.1007/s00526-011-0397-8

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