On the focusing critical semi-linear wave equation

The focusing critical wave equation in three dimensions exhibits a special class of static solutions which are linearly unstable. These solutions decay like an inverse first power. We construct small codimension one stable manifolds in the class of radial perturbations for these static solutions. We show that initial data on this manifold lead to global solutions. Moreover, the long-time behavior of these solutions can be described explicitly. In particular, the family of static solutions acts as an attractor for these solutions. On a technical level, we remark that the linearized operator exhibits a zero energy resonance, and we develop the tools needed for that case.