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

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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.

Article information

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10]

global regularity log-log energy supercritical wave equation


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.

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