Abstract: We establish a Stable Manifold Theorem, with consequent exponential decay to equilibrium, for a class of degenerate evolution equations $Au'+u=D(u,u)$ with A bounded, self-adjoint, and one-to-one, but not invertible, and $D$ a bounded, symmetric bilinear map. This is related to a number of other scenarios investigated recently for which the associated linearized ODE $Au'+u=0$ is ill-posed with respect to the Cauchy problem. The particular case studied here pertains to the steady Boltzmann equation, yielding exponential decay of large-amplitude shock and boundary layers.