Global dynamics above the ground state energy for the focusing nonlinear Klein–Gordon equation

The analysis of global dynamics of nonlinear dispersive equations has a long history starting from small solutions. In this paper we study the focusing, cubic, nonlinear Klein–Gordon equation in with large radial data in the energy space. This equation admits a unique positive stationary solution Q, called the ground state. In 1975 Payne and Sattinger showed that solutions u(t) with energy strictly below that of the ground state are divided into two classes, depending on a suitable functional K(u): If K(u)<0, then one has finite time blow-up, if K(u) 0 global existence; moreover, these sets are invariant under the flow. Recently, Ibrahim, Masmoudi and the first author [22] improved this result by establishing scattering to zero for K[u] 0 by means of a variant of the Kenig–Merle method (Kenig and Merle, 2006, 2008 [25] and [26]). In this paper we go slightly beyond the ground state energy and we give a complete description of the evolution in that case. For example, in a small neighborhood of Q one encounters the following trichotomy: On one side of a center-stable manifold one has finite time blow-up for t 0, on the other side scattering to zero, and on the manifold itself one has scattering to Q, both as t→+∞. In total, the class of data with energy at most slightly above that of Q is divided into nine disjoint non-empty sets each displaying different asymptotic behavior as t→±∞, which includes solutions blowing up in one time direction and scattering to zero on the other. The analogue of the solutions found by Duyckaerts and Merle (2009, 2008) [13] and [14] for the energy critical wave and Schrödinger equations appear here as the unique one-dimensional stable/unstable manifolds approaching ±Q exponentially as t→∞ or t→−∞, respectively. The main technical ingredient in our proof is a “one-pass” theorem which excludes the existence of (almost) homoclinic orbits between Q (as well as −Q) and (almost) heteroclinic orbits connecting Q with −Q. In a companion paper (Nakanishi and Schlag, 2010 [31]) we establish analogous properties for the NLS equation.