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

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 a priori upper bound of its $L_{t}^{\infty} {\tilde{H}}^{2} (ℝ^{3})$ norm, with ${\tilde{H}}^{2} (ℝ^{3}) : = Ḣ^{2} (ℝ^{3}) \cap Ḣ^{1} (ℝ^{3})$ . 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 $L_{t}^{\infty} {\tilde{H}}^{2} (ℝ^{3})$ 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

Mathematical Subject Classification 2000

Primary: 35Q55

Milestones

Received: 4 November 2008

Revised: 7 June 2009

Accepted: 21 July 2009

Published: 9 February 2010

Authors

Tristan Roy
Department of Mathematics University of California Los Angeles, CA 90095 United States

Vol. 2, No. 3, 2009

Tristan Roy

Abstract

Keywords

Mathematical Subject Classification 2000

Milestones

Authors