Analysis & PDE

  • Anal. PDE
  • Volume 7, Number 4 (2014), 953-995.

Wave and Klein–Gordon equations on hyperbolic spaces

Jean-Philippe Anker and Vittoria Pierfelice

Full-text: Open access

Abstract

We consider the Klein–Gordon equation associated with the Laplace–Beltrami operator Δ on real hyperbolic spaces of dimension n2; as Δ has a spectral gap, the wave equation is a particular case of our study. After a careful kernel analysis, we obtain dispersive and Strichartz estimates for a large family of admissible couples. As an application, we prove global well-posedness results for the corresponding semilinear equation with low regularity data.

Article information

Source
Anal. PDE, Volume 7, Number 4 (2014), 953-995.

Dates
Received: 3 August 2013
Accepted: 1 March 2014
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.apde/1513731544

Digital Object Identifier
doi:10.2140/apde.2014.7.953

Mathematical Reviews number (MathSciNet)
MR3254350

Zentralblatt MATH identifier
1297.35138

Subjects
Primary: 35L05: Wave equation 43A85: Analysis on homogeneous spaces 43A90: Spherical functions [See also 22E45, 22E46, 33C55] 47J35: Nonlinear evolution equations [See also 34G20, 35K90, 35L90, 35Qxx, 35R20, 37Kxx, 37Lxx, 47H20, 58D25]
Secondary: 22E30: Analysis on real and complex Lie groups [See also 33C80, 43-XX] 35L71: Semilinear second-order hyperbolic equations 58D25: Equations in function spaces; evolution equations [See also 34Gxx, 35K90, 35L90, 35R15, 37Lxx, 47Jxx] 58J45: Hyperbolic equations [See also 35Lxx] 81Q05: Closed and approximate solutions to the Schrödinger, Dirac, Klein- Gordon and other equations of quantum mechanics

Keywords
hyperbolic space wave kernel semilinear wave equation semilinear Klein–Gordon equation dispersive estimate Strichartz estimate global well-posedness

Citation

Anker, Jean-Philippe; Pierfelice, Vittoria. Wave and Klein–Gordon equations on hyperbolic spaces. Anal. PDE 7 (2014), no. 4, 953--995. doi:10.2140/apde.2014.7.953. https://projecteuclid.org/euclid.apde/1513731544


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