Global dynamics above the ground state energy for the energy-critical Klein-Gordon equation

Consider the focusing energy-critical Klein-Gordon equation in dimension d ∈ 3, 4, 5 d \ 3, 4, 5 \ ∂ t t u − Δ u + u a m p ; = | u | 4 d − 2 u, u (0, x) a m p ; ≔ f 0 (x), ∂ t u (0, x) a m p ; ≔ f 1 (x) equation* {cases ₓₓ u - u + u Acta Math. 201 (2008), pp. 147–212], and Krieger, Nakanishi, and Schlag Discrete Contin. Dyn. Syst. 33 (2013), pp. 2423–2450 to prove scattering as t → ± ∞ t. There are three main differences between this paper and that of Krieger, Nakanishi, and Schlag Discrete Contin. Dyn. Syst. 33 (2013), pp. 2423–2450. The first one is the lack of scaling symmetry. The second one appears in the proof of the ejection lemma: one has to control the mass in the ejection process. The third one appears in the proof of the one-pass lemma: in the worst scenario, one cannot use the equipartition of energy and therefore one has to prove a decay estimate which allows to use an argument of Bourgain J. Amer. Math. Soc. 12 (1999), pp. 145–171.