A concrete realization of the slow-fast alternative for a semi linear heat equation with homogeneous Neumann boundary conditions

Marina Ghisi (1) , Massimo Gobbino (2) , Alain Haraux (3)

1 dm.unipi - Dipartimento di Matematica
2 dm.unipi - Dipartimento di Matematica
3 LJLL - Laboratoire Jacques-Louis Lions

We investigate the asymptotic behavior of solutions to a semilinear heat equation with homogeneous Neumann boundary conditions. It was recently shown that the nontrivial kernel of the linear part leads to the coexistence of fast solutions decaying to 0 exponentially (as time goes to infnity), and slow solutions decaying to 0 as negative powers of t. Here we provide a characterization of slow/fast solutions in terms of their sign, and we show that the set of initial data giving rise to fast solutions is a graph of codimension one in the phase space.

Domaines: Equations aux dérivées partielles [math.AP]; Analyse fonctionnelle [math.FA]; Systèmes dynamiques [math.DS]

Marina Ghisi, Massimo Gobbino, Alain Haraux. A concrete realization of the slow-fast alternative for a semi linear heat equation with homogeneous Neumann boundary conditions. 2016. ⟨hal-01349361⟩