Th. Gallay, G. Raugel
Stability of Travelling Waves for a Damped Hyperbolic Equation
(151K, uuencoded compressed Postscript)

ABSTRACT.  We consider a nonlinear damped hyperbolic equation in $\real^n$, 
$1 \le n \le 4$, depending on a positive parameter $\epsilon$. 
If we set $\epsilon=0$, this equation reduces to the well-known 
Kolmogorov-Petrovski-Piskunov equation. We remark 
that, after a change of variables, this hyperbolic equation has 
the same family of one-dimensional travelling waves as the KPP equation. 
Using various energy functionals, we show that, if $\epsilon >0$, 
these fronts are locally stable under perturbations in appropriate 
weighted Sobolev spaces. Moreover, the decay rate in time 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 prove a global 
stability result. We also prove that the decay rate of the perturbated 
solutions towards the fronts is polynomial, for all $c > 2$.