Global existence of smooth solutions of a 3D log-log energy-supercritical wave equation

Abstract

We prove global existence of smooth solutions of the 3D log-log energy-supercritical wave equation

$${\partial }_{tt}u-△u=-u^5{log}^c\left(log\left(10+u^2\right)\right)$$

with $0<c<8∕225$ and smooth initial data $\left(u\left(0\right)=u_0,{\partial }_{t}u\left(0\right)=u_1\right)$. 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 a priori upper bound of its $L_t^{\infty }{\tilde{H}}^2\left({ℝ}^3\right)$ norm, with ${\tilde{H}}^2\left({ℝ}^3\right):={Ḣ}^2\left({ℝ}^3\right)\cap {Ḣ}^1\left({ℝ}^3\right)$. The proof of this long time estimate relies upon the use of some potential decay estimates and a modification of an argument by Tao. Then we find an a posteriori upper bound of the $L_t^{\infty }{\tilde{H}}^2\left({ℝ}^3\right)$ norm of the solution by combining the long time estimate with an induction on time of the Strichartz estimates.