Transversality of stable and Nehari manifolds for a semilinear heat equation

Dickstein, Flavio; Mizoguchi, Noriko; Souplet, Philippe; Weissler, Fred
November 2011
Calculus of Variations & Partial Differential Equations;Nov2011, Vol. 42 Issue 3/4, p547
Academic Journal
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 (). 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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