Nontrivial large-time behaviour in bistable reaction-diffusion equations

Bistable reaction-diffusion equations are known to admit one-dimensional travelling waves which are globally stable to one-dimensional perturbations-Fife and McLeod [7]. These planar waves are also stable to two-dimensional perturbations-Xin [30], Levermore-Xin [19], Kapitula [16]-provided that these perturbations decay, in the direction transverse to the wave, in an integrable fashion. In this paper, we first prove that this result breaks down when the integrability condition is removed, and we exhibit a large-time dynamics similar to that of the heat equation. We then apply this result to the study of the large-time behaviour of conical-shaped fronts in the plane, and exhibit cases where the dynamics is given by that of two advection-diffusion equations.