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Transversality of stable and Nehari manifolds for a semilinear heat equation |
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| Authors: | Flavio Dickstein Noriko Mizoguchi Philippe Souplet Fred Weissler |
| Institution: | 1. Departamento de Matematica Aplicada, Universidade Federal do Rio de Janeiro, Caixa Postal 68530, Rio de Janeiro, CEP 21945-970, Brazil 2. Department of Mathematics, Tokyo Gakugei University, Koganei, Tokyo, 184-8501, Japan 3. Precursory Research for Embryonic Science and Technology (PRESTO), Japan Science and Technology Agency (JST), Tokyo, Japan 4. Universit?? Paris 13, CNRS UMR 7539, Institut Galil??e, LAGA, 93430, Villetaneuse, France |
| 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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