Local stability of travelling fronts for a damped wave equation

The paper is concerned with the long-time behaviour of the travelling fronts of the damped wave equation αutt + ut = uxx − V′(u) on ℝ. The long-time asymptotics of the solutions of this equation are quite similar to those of the corresponding reaction-diffusion equation ut = uxx − V′(u). Whereas a lot is known about the local stability of travelling fronts in parabolic systems, for the hyperbolic equations it is only briefly discussed when the potential V is of bistable type. However, for the combustion or monostable type of V, the problem is much more complicated. In this paper, a local stability result for travelling fronts of this equation with combustion type of nonlinearity is established. And then, the result is extended to the damped wave equation with a case of monostable pushed front.