Asymptotic behavior for scalar viscous conservation laws with boundary effect

We consider the asymptotic stability of viscous shock wave φ for scalar viscous conservation laws u_t+f(u)_χ=u_χχ on the half-space (-∞, 0) with boundary values u|_χ=-∞=u_-,u|_χ=0=u₊. Our problem is divided into three cases depending on the sign of shock speed s of the shock (u_-, u₊). When s≤0, the asymptotic state of u becomes φ(·+d(t)), where d(t) depends implicitly on the initial data u(χ, 0) and is related to the boundary layer of the solution at the boundary χ=0. The stability of this state for s<0 will be shown by applying the weighted energy method. For s=0 a conjecture on d(t) will be presented. The case s>0 is also treated.