First-order hyperbolic pseudodifferential equations with generalized symbols ✩

We consider the Cauchy problem for a hyperbolic pseudodifferential operator whose symbol is generalized, resembling a representative of a Colombeau generalized function. Such equations arise, for example, after a reduction-decoupling of second-order model systems of differential equations in seismology. We prove existence of a unique generalized solution under log-type growth conditions on the symbol, thereby extending known results for the case of differential operators [J. Math. Anal. Appl. 160 (1991) 93–106, J. Math. Anal. Appl. 142 (1989) 452–467].  2004 Elsevier Inc. All rights reserved.