On the focusing critical semi-linear wave equation
The wave equation partial derivative tt psi - Delta psi - psi(5) = 0 in R-3 is known to exhibit finite time blowup for data of negative energy. Furthermore, it admits the special static solutions phi(x, a) = (3a)(1/4) (1 + a\x\(2))- 1/2 for all a > 0 which are linearly unstable. We view these functions as a curve in the energy space (H) over dot(1) x L-2. 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 phi(., a) as an attractor in M.