Blow up for a class of quasilinear wave equations in one space dimension

For suitable sigma and F, we prove that all classical solutions of the quas ilinear wave equation phi(tt) - (sigma(phi(x)))(x) = F(phi), with initial d ata of compact support, develop singularities in finite time. The assumptio ns on a and F include in particular the model case phi(tt) - phi(xx)(1 + ph i(x)(2))= epsilon phi(q+1), for q greater than or equal to 2, and epsilon = +/- 1. The starting point of the proof is to write the equation under the form of a first order system of two equations, in which F(phi) appears as a nonlocal term. Then, we present a new idea to control the effect of this p erturbation term, and we conclude the proof by using well-known methods dev eloped for 2 x 2 systems of conservation laws. Copyright (C) 2000 John Wile y & Sons, Ltd.