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 nonstrictly 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, and of Kreiss, Kreiss, and Petersson. As an example we present an application to the Hodgkin-Huxley model.

Keywords

traveling waves, Parabolic-hyperbolic partial differential equations, Degenerate partial differential equations, Partial differential algebraic equations, Nonlinear stability, Asymptotic behavior, Resolvent estimates

2010 Mathematics Subject Classification

35B35, 35B40, 35M31, 35P05

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Published 7 March 2012