Communications in Mathematical Sciences
Volume 15 (2017)
Stability of traveling wave solutions of nonlinear conservation laws for image processing
Pages: 1073 – 1106
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 $t \to \infty$.
nonlinear stability, nonlinear conservation laws, rate of decay, weighted energy estimates, image processing
2010 Mathematics Subject Classification
35B35, 35B40, 35C07, 35K15, 35L50, 35L65, 92C35