Preprint of the project:
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.
