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Th. Gallay, G. Raugel
Stability of Travelling Waves for a Damped Hyperbolic Equation
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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 $c > 2$.