Let say I have a hyperbolic system of conservation law. How do I show there is a blow up in finite time? For a single conservation law, I think, I could just show that there is collision of characteristics given a certain velocity for example $a(u)=u$ in Burgers' equation.


I would also like to know good books or any other references on this topic (blow up in finite time for hyperbolic conservation law). I have with me books by Lax, Evans, and Leveque (Finite Volume ...). Thanks.



blow up in finite time of hyperbolic system of conservation law
For general hyperbolic systems in one space and on time dimension, the result is treated in 


John, F.  

Formation of singularities in one-dimensional nonlinear wave propagation  

Comm. Pure Appl. Math., 1974, 27, 377-405


The higher dimensional case is not completely understood at present. There's a lot of work by Alinhac on what he terms "geometric blowup", which is the direct analogue of intersections of characteristics in the one spatial dimension case. 


Alinhac, S.  

Blowup for nonlinear hyperbolic equations  

Birkhäuser Boston Inc., 1995, xiv+113


Alinhac, S.  

A minicourse on global existence and blowup of classical solutions to multidimensional quasilinear wave equations  

Journées ``Équations aux Dérivées Partielles'' (Forges-les-Eaux, 2002), Univ. Nantes, 2002, Exp. No. I, 33


One of the reasons that higher dimensional hyperbolic systems are complicated is that there the equation is dispersive. Dispersion gives a decay mechanism that can compete against the self-resonance (take a derivative of the Burger's equation you get Riccati) driving blow-up. And sometimes you win, and sometimes you lose. 


