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

ttu u = u5 logc(log(10 + u2))

with 0 < c < 8225 and smooth initial data (u(0) = u0,tu(0) = u1). First we control the Lt4Lx12 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 LtH̃2(3) norm, with H̃2(3) := 2(3) 1(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 LtH̃2(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