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 α u t t + u t = u x x − V ′ ( u ) on R . This equation has travelling front solutions of the form u ( x , t ) = h ( x − s t ) . 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.