Global dynamics beyond the ground energy for the nonlinear Klein-Gordon equation

We study global behavior of the nonlinear Klein-Gordon equation with a focusing cubic power in three dimensions, in the energy space under the restriction of radial symmetry and an energy upper bound slightly above that of the ground state. We give a complete classification of the solutions into 9 non-empty sets according to whether they blow-up, scatter to 0, or scatter to the ground states, in the forward and backward time directions, and the splitting is given in terms of the stable and the unstable manifoldsof the ground states. This is joint work with Wilhelm Schlag.