Global Existence of Solutions of a Loglog Energy-Supercritical Klein–Gordon Equation

We prove global existence of the solutions of the loglog energysupercritical Klein-Gordon equations ∂ttu−△u+u = −|u| 4 n−2 u log log(10+ |u|2), with n ∈ {3, 4, 5}, 0 < γ < γn, and data (u0, u1) ∈ H × Hk−1, for k > 1 (resp. 7 3 > k > 1) if n ∈ {3, 4} (resp. n = 5). The proof is by contradiction. Assuming that the blow-up occurs at a time of maximal existence, we perform an analysis close to this time in order to find a finite bound of a Strichartz-type norm by using arguments in [9, 16, 17], which eventually leads to a contradiction with the blow-up assumption.