On Well-posedness of the Linear Cauchy Problem with the Distributional Right-hand Side and Discontinuous Coefficients

The classical result on well-posedness of Cauchy problem for the linear ordinary differential system with the distributional right-hand side and smooth matrix of coefficients plays fundamental role in many applications of distribution theory to ordinary and partial differential equations. In the present paper we generalize this result to the case of system (0.1) x − A(t)x = f, where f is a distribution, A ∈ G, G is the space of functions possessing at most first-kind discontinuities together with all their derivatives defined almost everywhere. The left-hand side of system (0.1) contains the product of a distribution and, in general, a discontinuous function, which is undefined in the classical space D of distributions with the smooth test functions. As a result, the Cauchy problem for (0.1) in general has no solution in D. In what follows, we consider system (0.1) in the space R of distributions with G-test functions, whose elements admit continuous multiplication by functions in G, and show that there exists the unique solution of the Cauchy problem for (0.1) which depends continuously on f . The proof of this result requires investigation of structure of the kernel of operator of restriction of distributions from R to D, of properties of operation of multiplication and of properties of multi-valued (yet, in a sense, continuous) operation of differentiation in R.