Asymptotic stability of the viscous shock wave for the one-dimensional compressible Navier–Stokes equations with density dependent viscous coefficient on the half space

In this paper, taking the boundary effect into consideration, we investigate the asymptotic stability of a viscous shock wave for one-dimensional isentropic compressible Navier–Stokes equations with density dependent viscous coefficient. Under the assumption that the viscous coefficient is given as a power function of the density, we prove that, for any positive power index of the viscous coefficient, if the amplitude of the corresponding outgoing viscous shock wave is suitably small and the initial data are close to the outgoing viscous shock wave which is far from the boundary, then a global solution exists uniquely in time and tends towards the properly shifted viscous shock wave as the time goes to infinity. The proof is given by an elementary energy method.