## Analysis & PDE

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

### Wave and Klein–Gordon equations on hyperbolic spaces

#### Abstract

We consider the Klein–Gordon equation associated with the Laplace–Beltrami operator $Δ$ on real hyperbolic spaces of dimension $n≥2$; 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
Accepted: 1 March 2014
First available in Project Euclid: 20 December 2017

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

#### 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

#### References

• J.-P. Anker and V. Pierfelice, “Nonlinear Schrödinger equation on real hyperbolic spaces”, Ann. Inst. H. Poincaré Anal. Non Linéaire 26:5 (2009), 1853–1869.
• J.-P. Anker and V. Pierfelice, “Wave and Klein–Gordon equations on Damek–Ricci spaces”. Work in progress.
• J.-P. Anker, V. Pierfelice, and M. Vallarino, “Schrödinger equations on Damek–Ricci spaces”, Comm. Partial Differential Equations 36:6 (2011), 976–997.
• J.-P. Anker, V. Pierfelice, and M. Vallarino, “The wave equation on hyperbolic spaces”, J. Differential Equations 252:10 (2012), 5613–5661.
• J.-P. Anker, V. Pierfelice, and M. Vallarino, “The wave equation on Damek–Ricci spaces”, Ann. Mat. Pura. Appl. (2014).
• H. Bahouri and P. Gérard, “High frequency approximation of solutions to critical nonlinear wave equations”, Amer. J. Math. 121:1 (1999), 131–175.
• M. Christ and A. Kiselev, “Maximal functions associated to filtrations”, J. Funct. Anal. 179:2 (2001), 409–425.
• P. D'Ancona, V. Georgiev, and H. Kubo, “Weighted decay estimates for the wave equation”, J. Differential Equations 177:1 (2001), 146–208.
• J. Fontaine, “Une équation semi-linéaire des ondes sur ${\mathbb H}\sp 3$”, C. R. Acad. Sci. Paris Sér. I Math. 319:9 (1994), 945–948.
• J. Fontaine, “A semilinear wave equation on hyperbolic spaces”, Comm. Partial Differential Equations 22:3-4 (1997), 633–659.
• V. Georgiev, Semilinear hyperbolic equations, MSJ Memoirs 7, Mathematical Society of Japan, Tokyo, 2000.
• V. Georgiev, H. Lindblad, and C. D. Sogge, “Weighted Strichartz estimates and global existence for semilinear wave equations”, Amer. J. Math. 119:6 (1997), 1291–1319.
• J. Ginibre and G. Velo, “The global Cauchy problem for the nonlinear Klein–Gordon equation”, Math. Z. 189:4 (1985), 487–505.
• J. Ginibre and G. Velo, “Generalized Strichartz inequalities for the wave equation”, J. Funct. Anal. 133:1 (1995), 50–68.
• A. Hassani, Équation des ondes sur les espaces symétriques Riemanniens de type non compact, thesis, Université de Nanterre - Paris X, 2011, http://tel.archives-ouvertes.fr/tel-00669082.
• A. Hassani, “Wave equation on Riemannian symmetric spaces”, J. Math. Phys. 52:4 (2011), Article ID #043514.
• S. Helgason, Geometric analysis on symmetric spaces, Mathematical Surveys and Monographs 39, American Mathematical Society, Providence, RI, 1994.
• S. Helgason, Groups and geometric analysis: integral geometry, invariant differential operators, and spherical functions, Mathematical Surveys and Monographs 83, American Mathematical Society, Providence, RI, 2000. Corrected reprint of the 1984 original.
• S. Helgason, Differential geometry, Lie groups, and symmetric spaces, Graduate Studies in Mathematics 34, American Mathematical Society, Providence, RI, 2001. Corrected reprint of the 1978 original.
• L. H örmander, The analysis of linear partial differential operators, III: Pseudo-differential operators, Springer, Berlin, 2007. Reprint of the 1994 edition.
• A. D. Ionescu, “Fourier integral operators on noncompact symmetric spaces of real rank one”, J. Funct. Anal. 174:2 (2000), 274–300.
• A. D. Ionescu and G. Staffilani, “Semilinear Schrödinger flows on hyperbolic spaces: scattering in $\mathbb H\sp 1$”, Math. Ann. 345:1 (2009), 133–158.
• F. John, “Blow-up of solutions of nonlinear wave equations in three space dimensions”, Manuscripta Math. 28:1-3 (1979), 235–268.
• L. Kapitanski, “Weak and yet weaker solutions of semilinear wave equations”, Comm. Partial Differential Equations 19:9-10 (1994), 1629–1676.
• M. Keel and T. Tao, “Endpoint Strichartz estimates”, Amer. J. Math. 120 (1998), 955–980.
• S. Klainerman and G. Ponce, “Global, small amplitude solutions to nonlinear evolution equations”, Comm. Pure Appl. Math. 36:1 (1983), 133–141.
• T. H. Koornwinder, “Jacobi functions and analysis on noncompact semisimple Lie groups”, pp. 1–85 in Special functions: group theoretical aspects and applications, edited by R. A. Askey et al., Mathematics and its Applications 18, Reidel, Dordrecht, 1984.
• 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.
• S. Machihara, M. Nakamura, and T. Ozawa, “Small global solutions for nonlinear Dirac equations”, Differential Integral Equations 17:5-6 (2004), 623–636.
• J. Metcalfe and M. Taylor, “Nonlinear waves on 3D hyperbolic space”, Trans. Amer. Math. Soc. 363:7 (2011), 3489–3529.
• J. Metcalfe and M. Taylor, “Dispersive wave estimates on 3D hyperbolic space”, Proc. Amer. Math. Soc. 140:11 (2012), 3861–3866.
• K. Nakanishi, “Scattering theory for the nonlinear Klein–Gordon equation with Sobolev critical power”, Internat. Math. Res. Notices 1999 (1999), 31–60.
• V. Pierfelice, “Weighted Strichartz estimates for the Schrödinger and wave equations on Damek–Ricci spaces”, Math. Z. 260:2 (2008), 377–392.
• T. C. Sideris, “Nonexistence of global solutions to semilinear wave equations in high dimensions”, J. Differential Equations 52:3 (1984), 378–406.
• W. A. Strauss, Nonlinear wave equations, CBMS Regional Conference Series in Mathematics 73, American Mathematical Society, Providence, RI, 1989.
• D. Tataru, “Strichartz estimates in the hyperbolic space and global existence for the semilinear wave equation”, Trans. Amer. Math. Soc. 353:2 (2001), 795–807.
• H. Triebel, Theory of function spaces, II, Monographs in Mathematics 84, Birkhäuser, Basel, 1992.