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Journal of Evolution Equations

, Volume 18, Issue 2, pp 923–946 | Cite as

Asymptotic stability of traveling wave solutions for nonlocal viscous conservation laws with explicit decay rates

  • Franz Achleitner
  • Yoshihiro Ueda
Open Access
Article
  • 126 Downloads

Abstract

We consider scalar conservation laws with nonlocal diffusion of Riesz–Feller type such as the fractal Burgers equation. The existence of traveling wave solutions with monotone decreasing profile has been established recently (in special cases). We show the local asymptotic stability of these traveling wave solutions in a Sobolev space setting by constructing a Lyapunov functional. Most importantly, we derive the algebraic-in-time decay of the norm of such perturbations with explicit algebraic-in-time decay rates.

Keywords

Nonlocal evolution equations Riesz–Feller operator Fractional Laplacian Traveling wave solutions Asymptotic stability Decay rates 

Mathematics Subject Classification

47J35 26A33 35C07 

Notes

Acknowledgements

Open access funding provided by University of Vienna. The first author was partially supported by Austrian Science Fund (FWF) under Grant P28661 and the FWF-funded SFB #F65.

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© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Faculty of MathematicsUniversity of ViennaViennaAustria
  2. 2.Faculty of Maritime SciencesKobe UniversityKobeJapan

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