Abstract :
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.
