Vol. 2, No. 3, 2009

Download this article
Download this article For screen
For printing
Recent Issues

Volume 10
Issue 6, 1285–1538
Issue 5, 1017–1284
Issue 4, 757–1015
Issue 3, 513–756
Issue 2, 253–512
Issue 1, 1–252

Volume 9, 8 issues

Volume 8, 8 issues

Volume 7, 8 issues

Volume 6, 8 issues

Volume 5, 5 issues

Volume 4, 5 issues

Volume 3, 4 issues

Volume 2, 3 issues

Volume 1, 3 issues

The Journal
Cover
About the Cover
Editorial Board
Editors’ Interests
About the Journal
Scientific Advantages
Submission Guidelines
Submission Form
Subscriptions
Editorial Login
Contacts
Author Index
To Appear
 
ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
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