Advances in Differential Equations

Asymptotic stability of traveling waves for viscous conservation laws with dispersion

Jun Pan and Gerald Warnecke

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This work is concerned with the asymptotic stability of traveling waves for scalar viscous conservation laws with a convex flux function and a dispersion term. First we prove the existence of solutions locally in time of the initial-value problem for initial data near a constant solution by Fourier analysis. Using the semigroup method the local existence for initial data that are an $L^2$ perturbation of a traveling-wave profile is proved. We also obtain a regularity property of these solutions. The solution operator generated by the linearized equation plays a crucial role. Using the energy method we establish a priori estimates. These estimates, when combined with the local existence, lead to the desired global-in-time existence as well as the time-asymptotic decay of solutions with initial data close to a monotone traveling wave.

Article information

Adv. Differential Equations, Volume 9, Number 9-10 (2004), 1167-1184.

First available in Project Euclid: 18 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35Q53: KdV-like equations (Korteweg-de Vries) [See also 37K10]
Secondary: 35B35: Stability 47N20: Applications to differential and integral equations 76E99: None of the above, but in this section


Pan, Jun; Warnecke, Gerald. Asymptotic stability of traveling waves for viscous conservation laws with dispersion. Adv. Differential Equations 9 (2004), no. 9-10, 1167--1184.

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