Scattering and Blow-up for the Focusing Nonlinear Klein–Gordon Equation with Complex-Valued Data

We consider the focusing nonlinear cubic Klein–Gordon equation in three spatial dimensions (NLKG). By the classical result of Payne and Sattinger (Israel J Math 22(3–4):273–303, 1975), one can distinguish global existence and blow-up for real-valued initial data with energy less than that of the ground state, by a variational characterization. Following the idea of Kenig and Merle (Acta Math 201(2):147–212, 2008), Ibrahim, Masmoudi, and Nakanishi (Anal PDE 4(3):405–460, 2011) proved scattering of real-valued solutions in the region of global existence. We will extend their classification to complex-valued solutions below the ground-state standing waves using the charge (see Theorem 1.1). We apply their method to obtain the scattering result. The difference is that they use only the energy, but we need to control both the energy and the charge. On the other hand, we cannot use their proof, which is based on Payne and Sattinger (Israel J Math 22(3–4):273–303, 1975), to obtain the blow-up result. Therefore, we combine the idea of Holmer and Roudenko (Commun Partial Differ Equ 35(5):878–905, 2010), who proved a similar classification for the nonlinear Schrödinger equation, with the idea of Ohta and Todorova (SIAM J Math Anal 38(6):1912–1931, 2007). Moreover, we extend the classification using not only the energy and the charge, but also the momentum (see Theorem 1.3). Due to these extensions, we can classify the solutions which have large energy.