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

Abstract

We prove global existence of smooth solutions of the 3D loglog energy-supercritical wave equation ${\partial }_{tt}u-△u=-u^5{\log }^c(log(10+u^2))$ 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.

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