On the global solubility of the Monge-Ampfere hyperbolic equations

摘要

This paper is devoted to the solubility of the Cauchy problem for the Monge-Ampere hyperbolic equations, in particular, for quasi-linear equations with two independent variables. It is proved that this problem has a unique maximal solution in the class of multi-valued solutions. The notion of a multi-valued solution goes back to Monge, Lie, et al. It is an historical predecessor of the notion of a generalized solution in the Sobolev-Schwartz sense. The relationship between multi-valued and generalized solutions of linear differential equations is established.