An important question in mathematical relativity theory is that of the nature of spacetime singularities. The equations of general relativity, the Einstein equations, are essentially hyperbolic in nature and the study of spacetime singularities is naturally related to blow-up phenomena for nonlinear hyperbolic systems. These connections are explained and recent progress in applying the theory of hyperbolic equations in this field is presented. A direction which has turned out to be fruitful is that of constructing large families of solutions of the Einstein equations with singularities of a simple type by solving singular hyperbolic systems. Heuristic considerations indicate, however, that the generic case will be much more complicated and require different techniques.

@article{JEDP_2000____A14_0, author = {Rendall, Alan D.}, title = {Blow-up for solutions of hyperbolic {PDE} and spacetime singularities}, journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles}, eid = {14}, publisher = {Universit\'e de Nantes}, year = {2000}, zbl = {01808704}, mrnumber = {2001j:58051}, language = {en}, url = {http://www.numdam.org/item/JEDP_2000____A14_0/} }

Rendall, Alan D. Blow-up for solutions of hyperbolic PDE and spacetime singularities. Journées équations aux dérivées partielles (2000), article no. 14, 12 p. http://www.numdam.org/item/JEDP_2000____A14_0/

[1] Blowup for nonlinear hyperbolic equations. Birkhäuser, Boston. | MR | Zbl

1995[2] Quiescent cosmological singularities. Preprint gr-qc/0001047. | Zbl

, 2000[3] A general solution of the Einstein equations with a time singularity. Adv. Phys. 31, 639-667.

, , , and , 1982[4] Numerical investigation of singularities. In M. Francaviglia et. al. (eds.) Proceedings of the 14th International Conference on General Relativity and Gravitation. World Scientific, Singapore. | Zbl

, 1997[5] On uniqueness in the large of solutions of the Einstein equations («Strong Cosmic Censorship») Proc. CMA 27, ANU, Canberra. | MR | Zbl

1991[6] The Cauchy problem for the Einstein equations. In B.G. Schmidt (ed) Einstein's Field Equations and Their Physical Implications. Lecture Notes in Physics 540. Springer, Berlin. | MR | Zbl

, 2000[7] Blow-up surfaces for nonlinear wave equations I. Comm. PDE 18, 431-452. | MR | Zbl

and 1993[8] Blow-up surfaces for nonlinear wave equations II. Comm. PDE 18, 1869-1899. | Zbl

and 1993[9] Fuchsian equations in Sobolev spaces and blow-up. J. Diff. Eq. 125, 299-327. | MR | Zbl

1996[10] The blow-up problem for exponential nonlinearities. Comm. PDE 21, 125-162. | MR | Zbl

1996[11] Nonlinear Wave Equations. Marcel Dekker, New York. | MR | Zbl

1996[12] Analytic description of singularities in Gowdy spacetimes. Class. Quantum Grav. 15, 1339-1355. | Zbl

, 1998[13] Global properties of Gowdy spacetimes with T3 x R topology. Ann. Phys. (NY) 132, 87-107. | MR

1980[14] On the behaviour of the universe near the cosmological singularity. Class. Quantum Grav. 12, 2505-2517. | MR | Zbl

1995[15] Fuchsian analysis of singularities in Gowdy spacetimes beyond analyticity. Preprint gr-qc/0004044. | MR | Zbl

2000[16] Curvature blow-up in Bianchi VIII and IX vacuum spacetimes. Class. Quantum Grav. 17, 713-731. | MR | Zbl

2000[17] Mixmaster behavior in inhomogeneous cosmological spacetimes. Phys. Rev. Lett. 80, 2984-2987.

, and , 1998