Asymptotic stability of traveling waves for viscous scalar conservation laws

Abstract

There are many research about Viscous scalar conservation law $u_{t}+f(u)_{x}=\mu u_{xx}$. In this paper, we study stability and asymptotic stability of travelling solutions for viscous scalar conservation law. The major result shows that if Rankine Hugoniot condition, the generalized shock condition and some assumptions hold, there exist a solution approaches to the travelling solution, satisfy stability and asymptotic stability problems at corresponding rate. The important feature of this paper is to employ an appropriate weight function to show the stability and asymptotic behavior of the viscous shock waves.