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

Full-text: Open access

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.

Article information

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

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

Permanent link to this document
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

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

Keywords
global regularity log-log energy supercritical wave equation

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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References

  • H. Bahouri and P. Gérard, “High frequency approximation of solutions to critical nonlinear wave equations”, Amer. J. Math. 121:1 (1999), 131–175.
  • J. Bourgain, “Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case”, J. Amer. Math. Soc. 12:1 (1999), 145–171.
  • J. Ginibre and G. Velo, “Generalized Strichartz inequalities for the wave equation”, J. Funct. Anal. 133:1 (1995), 50–68.
  • M. G. Grillakis, “Regularity and asymptotic behaviour of the wave equation with a critical nonlinearity”, Ann. of Math. $(2)$ 132:3 (1990), 485–509.
  • M. G. Grillakis, “Regularity for the wave equation with a critical nonlinearity”, Comm. Pure Appl. Math. 45:6 (1992), 749–774.
  • L. Kapitanski, “Global and unique weak solutions of nonlinear wave equations”, Math. Res. Lett. 1:2 (1994), 211–223.
  • M. Keel and T. Tao, “Endpoint Strichartz estimates”, Amer. J. Math. 120:5 (1998), 955–980. http:www.ams.org/mathscinet-getitem?mr=2000d:35018MR 2000d:35018
  • H. Lindblad and C. D. Sogge, “On existence and scattering with minimal regularity for semilinear wave equations”, J. Funct. Anal. 130:2 (1995), 357–426.
  • J. Rauch, “I: The $u\sp{5}$ Klein–Gordon equation; II: Anomalous singularities for semilinear wave equations”, pp. 335–364 in Nonlinear partial differential equations and their applications (Paris, 1978/1979), vol. 1, edited by H. Brezis and J. L. Lions, Res. Notes in Math. 53, Pitman, Boston, MA, 1981.
  • J. Shatah and M. Struwe, “Regularity results for nonlinear wave equations”, Ann. of Math. $(2)$ 138:3 (1993), 503–518.
  • J. Shatah and M. Struwe, “Well-posedness in the energy space for semilinear wave equations with critical growth”, Internat. Math. Res. Notices 7 (1994), 303–309.
  • J. Shatah and M. Struwe, Geometric wave equations, Courant Lecture Notes in Mathematics 2, Courant Institute of Mathematical Sciences, New York, 1998.
  • C. D. Sogge, Lectures on nonlinear wave equations, Monographs in Analysis 2, International Press, Boston, MA, 1995.
  • M. Struwe, “Globally regular solutions to the $u\sp 5$ Klein-Gordon equation”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. $(4)$ 15:3 (1988), 495–513.
  • T. Tao, “Spacetime bounds for the energy-critical nonlinear wave equation in three spatial dimensions”, Dyn. Partial Differ. Equ. 3:2 (2006), 93–110.
  • T. Tao, “Global regularity for a logarithmically supercritical defocusing nonlinear wave equation for spherically symmetric data”, J. Hyperbolic Differ. Equ. 4:2 (2007), 259–265.