On the focusing critical semi-linear wave equation

The wave equation ∂ttψ − Δψ − ψ5 = 0 in ℝ3 is known to exhibit finite time blowup for data of negative energy. Furthermore, it admits the special static solutions φ(x, a) = (3a) ¼ (1 + a|x|2)−½ for all a > 0 which are linearly unstable. We view these functions as a curve in the energy space ˙H1 ×L2. We prove the existence of a family of perturbations of this curve that lead to global solutions possessing a well-defined long time asymptotic behavior as the sum of a bulk term plus a scattering term. Moreover, this family forms a co-dimension one manifold M of small diameter in a suitable topology. Loosely speaking, M acts as a center-stable manifold with the curve Φ(·, a) as an attractor in M.