Boundary effect on a stationary viscous shock wave for scalar viscous conservation laws

The initial-boundary value problem on the negative half-line R_-[formula]is considered, subsequently to T.-P. Liu and K. Nishihara (1997, J. Differential Equations133, 296-320). Here, the flux f is a smooth function satisfying f(u_±)=0 and the Oleinik shock condition f(φ)<0 for u₊<φ<u_- if u₊<u_- or f(φ)0 for u₊φu_- if u₊<u_-. In this situation the corresponding Cauchy problem on the whole line R=(-∞,∞) to (*) has a stationary viscous shock wave φ(x+x₀) for any fixed x₀. Our aim in this paper is to show that the solution u(x,t) to (*) behaves as φ(x+d(t)) with d(t)=O(lnt) as t→∞ under the suitable smallness conditions. When f=u²/2, the fact was shown by T.-P. Liu and S.-H. Yu (1997, Arch. Rational Mech. Anal.139, 57-82), based on the Hopf-Cole transformation. Our proof is based on the weighted energy method.