This paper is concerned with the asymptotic stability of degenerate stationary waves for viscous gases in the half space. We discuss the following two cases: (1) viscous conservation laws and (2) damped wave equations with nonlinear convection. In each case, we prove that the solution converges to the corresponding degenerate stationary wave at the rate t-α/4 as t → ∞, provided that the initial perturbation is in the weighted space L2α=L2(ℝ+;(1+x)αdx). This convergence rate t-α/4 is weaker than the one for the non-degenerate case and requires the restriction α < α*(q), where α*(q) is the critical value depending only on the degeneracy exponent q. Such a restriction is reasonable because the corresponding linearized operator for viscous conservation laws cannot be dissipative in L2α for α > α *(q) with another critical value α*(q). Our stability analysis is based on the space-time weighted energy method in which the spatial weight is chosen as a function of the degenerate stationary wave.
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