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

Abstract

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 R3 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 E[u,u˙]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 [25,26]). 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) [13,14] 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.