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.
Categories: math.AP