## Analysis & PDE

• Anal. PDE
• Volume 2, Number 3 (2009), 261-280.

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

Tristan Roy

#### 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 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 $Lt∞H̃2(ℝ3)$ norm, with $H̃2(ℝ3):=Ḣ2(ℝ3)∩Ḣ1(ℝ3)$. 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 $Lt∞H̃2(ℝ3)$ norm of the solution by combining the long time estimate with an induction on time of the Strichartz estimates.

#### Article information

Source
Anal. PDE, Volume 2, Number 3 (2009), 261-280.

Dates
Revised: 7 June 2009
Accepted: 21 July 2009
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.apde/1513798037

Digital Object Identifier
doi:10.2140/apde.2009.2.261

Mathematical Reviews number (MathSciNet)
MR2603799

Zentralblatt MATH identifier
1195.35222

#### Citation

Roy, Tristan. Global existence of smooth solutions of a 3D log-log energy-supercritical wave equation. Anal. PDE 2 (2009), no. 3, 261--280. doi:10.2140/apde.2009.2.261. https://projecteuclid.org/euclid.apde/1513798037

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