Communications in Mathematical Sciences

Volume 15 (2017)

Number 4

Stability of traveling wave solutions of nonlinear conservation laws for image processing

Pages: 1073 – 1106

DOI: http://dx.doi.org/10.4310/CMS.2017.v15.n4.a8

Authors

Tong Li (Department of Mathematics, University of Iowa, Iowa City, Ia., U.S.A.)

Jeungeun Park (Department of Mathematics, University of Iowa, Iowa City, Ia., U.S.A.)

Abstract

This paper studies the stability of smooth traveling wave solutions to a nonlinear PDE problem in reducing image noise. Specifically, we prove that the solution to the Cauchy problem approaches to the traveling wave solution if the initial data is a small perturbation of the traveling wave. We use a weighted energy method to show that if the initial perturbation decays algebraically or exponentially as $\lvert x \rvert \to \infty$, then the Cauchy problem solution approaches to the traveling wave at corresponding rates as $t \to \infty$.

Keywords

nonlinear stability, nonlinear conservation laws, rate of decay, weighted energy estimates, image processing

2010 Mathematics Subject Classification

35B35, 35B40, 35C07, 35K15, 35L50, 35L65, 92C35

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Published 16 May 2017