TITLE: Books on the analysis of hyperbolic partial differential equations
QUESTION [12 upvotes]: Most of the present books on pde analysis deal with the elliptic partial differential equations. Is there some book related to rigorous analysis with hyperbolic pdes, and especially hyperbolic systems of pdes? I want to some great books on this subject for research. Thank you.

REPLY [10 votes]: It's funny that a similar question hasn't already appeared on MO. Other answers already give some good suggestions. Here's a bunch more. Note that the older ones may not be considered very pedagogical or rigorous by today's standards.

Hadamard, J. Lectures on Cauchy's Problem in Linear Partial Differential Equations (Yale University Press, 1923)
Courant, R. and Hilbert, D. Methods of mathematical physics. Vol. II: Partial differential equations (Interscience, 1962; German original 1937)
Petrovsky, I. G. (also as Petrowsky) Lectures on Partial Differential Equations (Interscience, 1954; Russian original 1950--51)
Leray, J. Hyperbolic differential equations (Institute for Advanced Study, Princeton, 1953)
John, F. Partial differential equations (Springer, 1971)
Lax, P. D. Hyperbolic Partial Differential Equations (AMS, 2006; original notes 1963)
Garabedian, P. R. Partial differential equations (Wiley, 1964)
Friedlander, F. G. The Wave Equation on a Curved Space-time (CUP, 1975)
Günther, P. Huygens' Principle and Hyperbolic Equations (Academic Press, 1988)
Hörmander, L. Lectures on Nonlinear Hyperbolic Differential Equations (Springer, 1997; original notes 1987)

A bunch more have appeared in more recent years, some of which have already been mentioned. Here are a few other noteworthy ones.

Christodoulou, D. The Action Principle and Partial Differential Equations (PUP, 2000)
Bär, C., Ginoux, N. and Pfäffle, F. Wave Equations on Lorentzian Manifolds and Quantization (EMS, 2007)
Dafermos, C. M. Hyperbolic Conservation Laws in Continuum Physics (Springer, 2010)
Klainerman, S. Lecture Notes in Analysis (lecture notes, Princeton, 2011)
Rauch, Hyperbolic PDEs and Geometric Optics (AMS, 2012)