Vol. 2, No. 3, 2009

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

Tristan Roy

Vol. 2 (2009), No. 3, 261–280
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 and smooth initial data $\left(u\left(0\right)={u}_{0},\phantom{\rule{0.3em}{0ex}}{\partial }_{t}u\left(0\right)={u}_{1}\right)$. 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 ${L}_{t}^{\infty }{\stackrel{̃}{H}}^{2}\left({ℝ}^{3}\right)$ norm, with ${\stackrel{̃}{H}}^{2}\left({ℝ}^{3}\right):={Ḣ}^{2}\left({ℝ}^{3}\right)\cap {Ḣ}^{1}\left({ℝ}^{3}\right)$. 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 ${L}_{t}^{\infty }{\stackrel{̃}{H}}^{2}\left({ℝ}^{3}\right)$ norm of the solution by combining the long time estimate with an induction on time of the Strichartz estimates.

Keywords
global regularity, log-log energy supercritical wave equation
Primary: 35Q55