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

We prove global existence of the solutions of the loglog energy-supercritical Klein-Gordon equation partial derivative(tt)u - Delta u + u = -vertical bar u vertical bar(4/n-2) u log(gamma) (log(10 + vertical bar u vertical bar(2))), with n is an element of {3, 4, 5}, 0 < gamma < gamma(n), and data (u(0), u(1)) is an element of H-k x Hk-1 for k > 1 (resp. 7/3 > k > 1) if n is an element of {3, 4} (resp. n = 5). The proof is by contradiction. Assuming that blow-up occurs at a maximal time of existence, we perform an analysis close to this time in order to find a finite bound of a Strichartz-type norm, which eventually leads to a contradiction with the blow-up assumption.