Differential properties of a generalized solution to a hyperbolic system of first-order differential equations

Abstract

We study some questions of the qualitative theory of differential equations. A Cauchy problem is considered for a hyperbolic system of two first-order differential equations whose right-hand sides contain some discontinuous functions. A generalized solution is defined as a continuous solution to the corresponding system of integral equations. We prove the existence and uniqueness of a generalized solution and study the differential properties of the obtained solution. In particular, its first-order partial derivatives are unbounded near certain parts of the characteristic lines. We observe that this property contradicts the common approach which uses the reduction of a system of two first-order equations to a single second-order equation.