Convergence to travelling fronts of solutions of the p-system with viscosity in the presence of a boundary

We study the asymptotic behavior as time goes to infinity of solutions to the initial-boundary-value problem on the half space R+ for a one-dimensional model system for the isentropic flow of a compressible viscous gas, the so-called p-system with viscosity. As boundary conditions, we prescribe the constant state at infinity and require that the velocity be zero at the boundary x = 0. When the velocity at infinity is negative and satisfies a condition on the magnitude, we prove that if the initial data are suitably close to those for the corresponding outgoing viscous shock profile, which is suitably far from the boundary, then a unique solution exists globally in time and tends toward the properly shifted viscous shock profile as the time goes to infinity. The proof is given by an elementary energy method.