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2009 Global existence of smooth solutions of a 3D log-log energy-supercritical wave equation
Tristan Roy
Anal. PDE 2(3): 261-280 (2009). DOI: 10.2140/apde.2009.2.261

Abstract

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

t t u u = u 5 log c ( log ( 1 0 + u 2 ) )

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.

Citation

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Tristan Roy. "Global existence of smooth solutions of a 3D log-log energy-supercritical wave equation." Anal. PDE 2 (3) 261 - 280, 2009. https://doi.org/10.2140/apde.2009.2.261

Information

Received: 4 November 2008; Revised: 7 June 2009; Accepted: 21 July 2009; Published: 2009
First available in Project Euclid: 20 December 2017

zbMATH: 1195.35222
MathSciNet: MR2603799
Digital Object Identifier: 10.2140/apde.2009.2.261

Subjects:
Primary: 35Q55

Keywords: global regularity , log-log energy supercritical wave equation

Rights: Copyright © 2009 Mathematical Sciences Publishers

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