Global stability of travelling fronts for a damped wave equation

The paper is concerned with the long-time behaviour of the solutions of the damped wave equation αutt+ut=uxx−V′(u) on R. This equation has travelling front solutions of the form u(x,t)=h(x−st). Gallay and Joly have proved in Gallay and Joly (2009) [7] that when the nonlinearity V(u) is of bistable type, 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→+∞. In this paper, we establish a global stability result under more general assumptions on the function V, which include in particular nonlinearities of combustion type. We impose, however, more restrictive conditions on the initial data in the region x≫1. We also apply our method to the case of a monostable pushed front.