Boundary Effect on a Stationary Viscous Shock Wave for Scalar Viscous Conservation Laws

The initial-boundary value problem on the negative half-line R ut fŽu.x uxx , Žx, t. R Ž0, . uŽ0, t. u , Ž . uŽx, 0. u0Žx. ½ u , x u , x 0 is considered, subsequently to T.-P. Liu and K. Nishihara Ž1997, J. Differential Equations 133, 296 320.. Here, the flux f is a smooth function satisfying fŽu . 0 and the Oleinik shock condition fŽ . 0 for u u if u u or fŽ . 0 for u u if u u . In this situation the corresponding Cauchy problem on the whole line R Ž , .toŽ .has a stationary viscous shock wave Žx x0.for any fixed x0. Our aim in this paper is to show that the solution uŽx, t. to Ž . behaves as Žx dŽt.. with dŽt. OŽln t. as t under the suitable smallness conditions. When f u2 2, the fact was shown by T.-P. Liu and S.-H. YuŽ1997, Arch. Rational Mech. Anal. 139, 57 82., based on the Hopf Cole transformation. Our proof is based on the weighted energy method.