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

Abstract

Consider the focusing energy-critical Klein-Gordon equation in dimension d ∈ { 3 , 4 , 5 } d \in \{ 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 ) \begin{equation*} \begin {cases} \partial _{tt} u - \Delta u + u & = |u|^{\frac {4}{d-2}} u, \\ u(0,x) & ≔f_{0}(x), \\ \partial _{t} u(0,x) & ≔f_{1}(x) \end{cases} \end{equation*} with data ( f 0 , f 1 ) ∈ H ≔ H 1 × L 2 (f_{0},f_{1}) \in \mathcal {H} ≔H^{1} \times L^{2} . We describe the global dynamics of real-valued solutions of which the energy is slightly larger than that of the ground states’. We classify the flows of the solutions that are ejected from a small neighborhood of the ground states or that are away from them. The classification relies upon a modification of the arguments of Payne and Sattinger [Israel J. Math. 22 (1975), pp. 273–303] to prove blow-up in finite time, and a modification of the arguments of Ibrahim, Masmoudi, and Nakanishi [Anal. PDE 4 (2011), pp. 405–460], Kenig and Merle [Invent. Math. 166 (2006), pp. 645–675; 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 \rightarrow \pm \infty . 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].