Explosion for some semilinear wave equations

In this note we show that known results on the propagation of singularities for semilinear wave equations suffice for the construction of explosive solutions with arbitrarily small data. In particular, for the quadratically nonlinear wave equation in one space dimension utt - uxx = Au; + Bup, + Cu: 3 Q(u,, u,) (1) we show that the only nonlinearity without such explosions is Nirenberg’s equation B = 0, C = -A which is transformed to the linear wave equation by the substitution u = eA”. This nonlinearity is also singled out by special compactness properties [3] and its role in the equipartition of energy Cl]. Another much studied equation for which we supply explosive solutions is the Carleman model of transport theory, 8, s 8, f 8,, a_+u* = r(u: -242).