STABILITY OF TRAVELING WAVES FOR A DAMPED HYPERBOLIC EQUATION

We consider a nonlinear damped hyperbolic equation in R-n, 1 less than or equal to n less than or equal to 4, depending on a positive parame ter epsilon. If epsilon = 0, this equation reduces to the well-known p arabolic KPP equation. We remark that, after a change of variables, th e hyperbolic equation has the same family of one-dimensional travellin g waves (or fronts) as the KPP equation. Using various energy function als, we show that these fronts are locally stable under perturbations in appropriate weighted Sobolev spaces. Moreover, the decay rate in ti me of the perturbed solutions towards the front of minimal speed c = 2 is shown to be polynomial. In the one-dimensional case, if epsilon < 1/4, we can apply a Maximum Principle for hyperbolic equations and pro ve a global stability result. We also prove that the decay rate of the perturbed solutions towards the fronts is polynomial, for all c > 2.