Global Existence Of Smooth Solutions Of A 3D Loglog Energy-Supercritical Wave Equation

We prove global existence of smooth solutions of the 3D loglog energy-supercritical wave equation $\partial_{tt} u - \triangle u = -u^{5} \log^{c} (log(10+u^{2})) $ with $0 < c < {8/225}$ and smooth initial data $(u(0)=u_{0}, \partial_{t} u(0)=u_{1})$. First we control the $L_{t}^{4} L_{x}^{12}$ norm of the solution on an arbitrary size time interval by an expression depending on the energy and an \textit{a priori} upper bound of its $L_{t}^{\infty} \tilde{H}^{2}(\mathbb{R}^{3})$ norm, with $\tilde{H}^{2}(\mathbb{R}^{3}):=\dot{H}^{2}(\mathbb{R}^{3}) \cap \dot{H}^{1}(\mathbb{R}^{3})$. The proof of this long time estimate relies upon the use of some potential decay estimates \cite{bahger, shatstruwe} and a modification of an argument in \cite{taolog}. Then we find an \textit{a posteriori} upper bound of the $L_{t}^{\infty} \tilde{H}^{2}(\mathbb{R}^{3})$ norm of the solution by combining the long time estimate with an induction on time of the Strichartz estimates.