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Bistable traveling wave passing an obstacle: perturbation results

We study the existence of generalized transition fronts for a bistable reaction diffusion equation in an exterior domain, i.e the whole space minus some compact set K. In the paper, Bistable traveling waves around an obstacle from 2009, H. Berestycki, F. Hamel and H. Matano proved the existence of a generalized transition front when K is smooth enough and satisfies some geometric properties. We are interested in a extension of this result for some perturbation of K. We prove that as soon as the perturbation is close to K in the C^{2,alpha}-topology the result still applies and give a counter-example when the perturbation is not smooth enough.