Asymptotic behavior for scalar viscous conservation laws with boundary effect (Q678059)

The authors consider the asymptotic stability for scalar viscous conservation laws \(u_t + f(u)_x = u_{xx}\) on the half-space \((-\infty, 0)\). The problem includes also boundary effects, the prescribed boundary values are \(u(-\infty) = u_-\) and \(u(0) = u_+\). The rigorous analysis is divided into three cases depending on the sign of the shock speed \(s\) of the shock \((u_-, u_+)\). For the case \(s\leq 0\) the form of asymptotic state of the solution is determined precisely and is related to the boundary layer of the solution at \(x=0\). For the case \(s < 0\) the stability of the solution is shown using weighted energy method. On the other hand for the case \(s=0\) only a conjecture on the form of asymptotic state can be presented. The case \(s>0\) is more simpler than the other cases and is treated also in the paper.