Stability of Travelling Waves for a Damped Hyperbolic Equation

Abstract

We consider a nonlinear damped hyperbolic equation in ${\Re }^n$, $1\le n\le 4$, 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 $c>2$.

32 pages, plain TeX

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