Preprint des Projektes: SFB 701: Spektrale Strukturen und Topologische Methoden in der Mathematik - Projekt B3

Numerische Analyse äquivarianter Evolutionsgleichungen

12-005 Jens Rottmann-Matthes.
Stability of parabolic-hyperbolic traveling waves

In this paper we investigate nonlinear stability of traveling waves in general parabolic-hyperbolic coupled systems where we allow for a non-strictly hyperbolic part.
We show that the problem is locally well-posed in a neighborhood of the traveling wave and prove that nonlinear stability follows from stability of the point spectrum and a simple algebraic condition on the coefficients of the linearization. We also obtain rates of convergence that are directly related to the spectral gap. The proof is based on a trick to reformulate the PDE as a partial differential algebraic equation for which the zero eigenvalue is removed from the spectrum. Then the Laplace-technique becomes applicable and resolvent estimates are used to prove stability.
Our results apply to pulses as well as fronts and generalize earlier results of Bates and Jones [2] and of Kreiss, Kreiss, and Petersson [13]. As an example we present an application to the Hodgkin-Huxley model.