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Linear Stability of Traveling Waves in First-Order Hyperbolic PDEs

  • Jens Rottmann-Matthes
Article

Abstract

In the analysis of traveling waves it is 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.

Keywords

Hyperbolic partial differential equations Traveling waves Partial differential algebraic equations Linear stability Asymptotic behavior Resolvent estimates 

Mathematics Subject Classification (2010)

35B40 35P15 35L45 65P40 

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© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Bielefeld UniversityBielefeldGermany

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