American Institute of Mathematical Sciences

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January  2006, 14(1): 75-92. doi: 10.3934/dcds.2006.14.75

Asymptotic properties and classification of bistable fronts with Lipschitz level sets

François Hamel 1, , Régis Monneau 2, and  Jean-Michel Roquejoffre 3,

1. 

LATP (UMR CNRS 6632), Faculté des Sciences et Techniques, Université Aix-Marseille III, F-13397 Marseille Cedex 20
2. 

CERMICS-ENPC, 6-8 avenue B. Pascal, Cité Descartes, F-77455 Marne-La-Vallée Cedex 2
3. 

Laboratoire M.I.P. (UMR CNRS 5640) and Institut Universitaire de France, Université Paul Sabatier, 118 route de Narbonne, F-31062 Toulouse Cedex 4, France

Received  October 2004 Revised  March 2005 Published  October 2005

In this paper we study solutions to reaction-diffusion equations in the bistable case, defined on the whole space in dimension $N$. The existence of solutions with cylindric symmetry is already known. Here we prove the uniqueness of these cylindric solutions whose level sets are curved Lipschitz graphs. Using a centre manifold-like argument, we also give the precise asymptotics of these level sets at infinity. In dimension 2, we classify all solutions under weak conditions at infinity. Finally, we also provide an alternative proof of the existence of these solutions in dimension 2, based on a continuation argument.
Keywords: Bistable fronts, centre manifold., elliptic equations, sliding method, classification.
Mathematics Subject Classification: Primary: 35B05, 35B40, 35B50, 35J6.
Citation: François Hamel, Régis Monneau, Jean-Michel Roquejoffre. Asymptotic properties and classification of bistable fronts with Lipschitz level sets. Discrete & Continuous Dynamical Systems - A, 2006, 14 (1) : 75-92. doi: 10.3934/dcds.2006.14.75
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