Formation of singularities for nonlinear hyperbolic \(2\times 2\) systems with periodic data (Q1382133)

The paper deals with the formation of singularities in the classical solutions of \(2\times 2\) hyperbolic systems with periodic initial data. The main result is a generalization of \textit{S. Klainerman} and \textit{A. Majda's} one [Commun. Pure Appl. Math. 33, 241-263 (1980; Zbl 0443.35040)]. Under an assumption on the characteristic fields allowing the genuine nonlinearity to fail at some isolated points, it shows that the \({\mathcal L}^1\) solution with small enough \({\mathcal L}^1\) periodic initial data blows up in finite time. It applies in particular to the nonlinear wave equation considered by Klainerman and Majda: \[ u_{tt} + k(u_x) u_{xx} = 0 , \] where \(k\) is assumed to be monotone and \(k'\) may have isolated zeros.