Blow-up for hyperbolic systems in diagonal form (Q5959075)

This paper deals with the blow up of the classical solutions to the Cauchy problem of a quasilinear strictly hyperbolic system of the type \(N\times N\) in one space dimension. The system is written in a diagonal form, and the initial data are periodic and nonconstant. Under several conditions imposed on the eigenvalue \(\lambda_1(u)\) it is proved that the first derivatives of the solutions blow up in finite time, and the life span of that solutions is estimated from above. These conditions can be interpreted as a sort of weak genuine nonlinearity with respect to a characteristic vector field.