Niveau: Supérieur, Doctorat, Bac+8
A variational proof of global stability for bistable travelling waves Thierry Gallay Institut Fourier Universite de Grenoble I 38402 Saint-Martin-d'Heres France Emmanuel Risler Institut Camille Jordan INSA de Lyon 69621 Villeurbanne France June 19, 2007 Abstract We give a variational proof of global stability for bistable travelling waves of scalar reaction-diffusion equations on the real line. In particular, we recover some of the classical results by P. Fife and J.B. McLeod (1977) without any use of the maximum principle. The method that is illustrated here in the simplest possi- ble setting has been successfully applied to more general parabolic or hyperbolic gradient-like systems. 1 Introduction The purpose of this work is to revisit the stability theory for travelling waves of reaction- diffusion systems on the real line. We are mainly interested in global stability results which assert that, for a wide class of initial data with a specified behavior at infinity, the solutions approach for large times a travelling wave with nonzero velocity. In the case of scalar reaction-diffusion equations, such properties have been established by Kolmogorov, Petrovski & Piskunov [11], by Kanel [9, 10], and by Fife & McLeod [4, 5] under various assumptions on the nonlinearity. The proofs of all these results use a priori estimates and comparison theorems based on the parabolic maximum principle.
- scalar equations
- travelling waves
- reaction- diffusion systems
- systems nor
- corresponding energy
- gradient
- global stability