Global dynamics away from the ground state for the energy-critical nonlinear wave equation

We study global behavior of radial solutions for the nonlinear wave equation with the focusing energy critical nonlinearity in three and five space dimensions. Assuming that the solution has energy at most slightly more than the ground states and gets away from them in the energy space, we can classify its behavior into four cases, according to whether it blows up in finite time or scatters to zero, in forward or backward time direction. We prove that initial data for each case constitute a non-empty open set in the energy space. This is an extension of the recent results by the latter two authors on the subcritical nonlinear Klein- Gordon and Schr¨odinger equations, except for the part of the center manifolds. The key step is to prove the “one-pass” theorem, which states that the transition from the scattering region to the blow-up region can take place at most once along each trajectory. The main new ingredients are the control of the scaling parameter and the blow-up characterization by Duyckaerts, Kenig, and Merle.