Numerische Analyse äquivarianter
Evolutionsgleichungen
10-049 Jens Rottmann-Matthes.
Linear stability of traveling waves in nonstrictly hyperbolic PDEs
In the analysis of traveling waves it is very common that coupled
parabolic-hyperbolic problems occur, where the hyperbolic part is not strictly hyperbolic. For example,
this happens whenever a reaction diffusion equation with more than one non diffusing component is
considered in a co-moving frame. In this paper we analyze the stability of traveling waves in
nonstrictly hyperbolic PDEs by reformulating the problem as a partial differential algebraic equation
(PDAE). We prove uniform resolvent estimates for the original PDE problem and for the PDAE by using
exponential dichotomies. It is shown that the zero eigenvalue of the linearization is removed from the
spectrum in the PDAE formulation and, therefore, the PDAE problem is better suited for the stability
analysis. This is rigorously done via the vector valued Laplace transform which also leads to optimal
rates. The linear stability result presented here is a major step in the proof of nonlinear stability.
