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# 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

#### 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