Invariant manifolds for a singular ordinary differential equation

We study the singular ordinary differential equation where , the functions and are of class and ζ is a real valued function. The equation is singular because ζ(U) can attain the value 0. We focus on the solutions of (0.1) that belong to a small neighborhood of a point such that and . We investigate the existence of manifolds that are locally invariant for (0.1) and that contain orbits with a prescribed asymptotic behavior. Under suitable hypotheses on the set {U:ζ(U)=0}, we extend to the case of the singular ODE (0.1) the definitions of center manifold, center-stable manifold and of uniformly stable manifold. We prove that the solutions of (0.1) lying on each of these manifolds are regular: this is not trivial since we provide examples showing that, in general, a solution of (0.1) is not continuously differentiable. Finally, we show a decomposition result for a center-stable manifold and for the uniformly stable manifold. An application of our analysis concerns the study of the viscous profiles with small total variation for a class of mixed hyperbolic–parabolic systems in one space variable. Such a class includes the compressible Navier–Stokes equation