Layers in the Presence of Conservation Laws

Alin Pogan, Arnd Scheel

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9 Scopus citations

Abstract

We study standing layers in systems where a reaction-diffusion equation couples to a scalar conservation law. Our results give spectral stability and instability results depending only on relative monotonicity of the two components of the system. We also prove the robustness of layers and their stability properties. Our results classify stability properties of layers in most such systems. Our method is based on tracking the point spectrum during a homotopy to a simple, decoupled system. Main difficulty is the possibility of eigenvalues disappearing in a branch point of the essential spectrum. This phenomenon is investigated using a Lyapunov-Schmidt reduction method on exponentially weighted spaces combined with a matching procedure for the far-field.

Original languageEnglish (US)
Pages (from-to)249-287
Number of pages39
JournalJournal of Dynamics and Differential Equations
Volume24
Issue number2
DOIs
StatePublished - Jun 2012

Bibliographical note

Funding Information:
Acknowledgments We would like to thank the anonymous referee for a careful reading of the manuscript and many helpful comments. We also gratefully acknowledge support by the National Science Foundation under grant DMS-0806614.

Keywords

  • Conservation law
  • Heteroclinic solutions
  • Lyapunov-Schmidt reduction
  • Stability

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