Global stability of travelling fronts for a damped wave equation with bistable nonlinearity

Abstract

We consider the damped wave equation αutt+ut = uxx−V'(u) on the whole real line, where V is a bistable potential. This equation has travelling front solutions of the form u(x, t) = h(x − st) which describe a moving interface between two different steady states of the system, one of which being the global minimum of V . We show that, if the initial data are sufficiently close to the profile of a front for large |x|, the solution of the damped wave equation converges uniformly on R to a travelling front as t→+∞. The proof of this global stability result is inspired by a recent work of E. Risler [38] and relies on the fact that our system has a Lyapunov function in any Galilean frame.