T. Gallay et G. Raugel, STABILITY OF TRAVELING WAVES FOR A DAMPED HYPERBOLIC EQUATION, Zeitschrift fur angewandte Mathematik und Physik, 48(3), 1997, pp. 451-479
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.