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.
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 195051)
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 Spacetime (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)
Sorry for adverstising myself. But Sylvie BenzoniGavage and myself published in 2007 a rather complete monograph on the subject. In particular, we treat initialboundary value problems, first linear and then nonlinear. We apply the theory to the stability of multidimensional shock waves, in particular in gas dynamics.
S. BenzoniGavage & D. S. Multidimensional hyperbolic partial differential equations. First order systems and applications. Oxford Mathematical Monographs, Oxford University Press (2007, xxiv + 512 pages, ISBN10: 019921123X, ISBN 13: 9780199211234).

$\begingroup$ Thanks! I'm now reading your two volumes of system of conservation laws, I found it clear and interesting. Thank you for your reply. I will read your recommended book. $\endgroup$ – sam Nov 24 '14 at 1:37
One that comes to mind is A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Springer 1984.

$\begingroup$ Yes. Majda's book is helpful. But it does not give a whole introduction to the standard methods to the mathematical analysis of hyperbolic equations. I hope to get a whole view of the modern theory. $\endgroup$ – sam Nov 24 '14 at 1:39
It is surprisingly difficult to find expositions on first order hyperbolic systems of PDE's. There are two standard types: symmetric hyperbolic and strictly hyperbolic. Both are studied using $L^2$ energy estimates.
Symmetric hyperbolic systems were first studied by K. O. Friederichs:
http://onlinelibrary.wiley.com/doi/10.1002/cpa.3160070206/abstract
I recall that this paper is quite readable.
Strictly hyperbolic systems were first studied by Lax, but the modern approach is to use pseudodifferential operators to obtain the energy estimates. By now I can't remember where I learned this from. Books I recall looking at were Michael Taylor's book, Pseudodifferential Operators and JeanFrancois Treves' Introduction to Pseudodifferential and Fourier Integral Operators: Pseudodifferential Operators

$\begingroup$ Thank you, Deane. I'm not familiar with Pesudodifferential operators. I think I will have a look at it. $\endgroup$ – sam Nov 24 '14 at 1:42
I suggest: S. Alinhac, Hyperbolic partial differential equations, Springer Universitext, 2009. The classic PDE book by F. John also gives a solid introduction to hyperbolic equations and systems, however his style of writing differs somewhat from todays.

$\begingroup$ Thank you. I have read F. John's book. It's a standard text for undergraduates. I want to study the deeper theory of hyperbolic equations. I think Alinhac's book may be helpful. $\endgroup$ – sam Nov 24 '14 at 1:44