Asymptotic Behavior for Scalar Viscous Conservation Laws with Boundary Effect

We consider the asymptotic stability of viscous shock waveφfor scalar viscous conservation lawsut+f(u)x=uxxon the half-space (−∞, 0) with boundary valuesu x=−∞=u−,u x=0=u+. Our problem is divided into three cases depending on the sign of shock speedsof the shock (u−, u+). Whens 0, the asymptotic state ofubecomesφ(•+d(t)), whered(t) depends implicitly on the initial datau(x, 0) and is related to the boundary layer of the solution at the boundaryx=0. The stability of this state fors<0 will be shown by applying the weighted energy method. Fors=0 a conjecture ond(t) will be presented. The cases>0 is also treated.