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Stability of travelling waves for a damped hyperbolic equation

  • Th. Gallay;
  • G.. Raugel;

Abstract.

We consider a nonlinear damped hyperbolic equation in \( {\bf R}^n, 1 \le n \le 4 \), depending on a positive parameter \( \epsilon \). If \( \epsilon = 0 \), this equation reduces to the well-known parabolic KPP equation. We remark that, after a change of variables, the hyperbolic equation has the same family of one-dimensional travelling waves (or fronts) as the KPP equation. Using various energy functionals, we show that 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 perturbed solutions towards the fronts is polynomial, for all c > 2.

Key words. Travelling waves, nonlinear stability, hyperbolic equations. 

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Copyright information

© Birkhäuser Verlag, Basel, 1997

Authors and Affiliations

  • Th. Gallay;
    • 1
  • G.. Raugel;
    • 2
  1. 1.CNRS et Université de Paris-Sud, URA 760, Analyse Numérique et EDP, Bâtiment 425, F-91405 Orsay, France, e-mail: Thierry.Gallay@math.u-psud.fr FR
  2. 2.CNRS et Université de Paris-Sud, URA 760, Analyse Numérique et EDP, Bâtiment 425, F-91405 Orsay, France, e-mail: Genevieve.Raugel@math.u-psud.fr FR

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