Bistable traveling wave passing an obstacle: perturbation results

We study the existence of generalized transition fronts for a bistable reaction diffusion equation, ut − ∆u = f(u), in a heterogeneous medium, Ω = RN\K where K is a compact set of RN , with Neumann boundary condition and t ∈ R. In the paper, Bistable traveling waves around an obstacle (2009), H. Berestycki, F. Hamel and H. Matano prove the existence of a generalized transition front when K is smooth enough and satisfies some geometric properties. We are interested in an extension of this result when Ωε = R\Kε and Kε is a small perturbation of K. We prove that as soon as Kε is close to K in the C2,α topology generalized transition front still exist while it does not if the perturbation is not smooth enough. 2010 Mathematics Subject Classification: 35B40, 35K57, 35C07, 35B51, 35B53 .