Гладкі розв’язки гіперболічних за Шиловим систем

Abstract

We consider a wide class of linear partial differential equations hyperbolic by Shilov, which covers the class of hyperbolic by Petrovsky systems with constant coefficients, and also the class of Gording equations. For such systems, the problem of finding smooth classical solutions, which are vector functions with compact support or rapidly decreasing at infinity, is investigated. Studies are carried out by the Fourier transform method in combination with the theory of spaces of the type S and S’ Gelfand I. M. and Shilov G.E. basic and generalized functions. The components of the fundamental solution of the Cauchy problem for such systems belong to the Dirac space of generalized functions, and also are convolvers in some spaces of the main Gelfand and Shilov functions. This made it possible to establish in the classical sense the correct solvability of the Cauchy problem in each such space of basic functions. That is, to prove the existence, uniqueness and continuous dependence on the initial data of the classical solution of a hyperbolic system in space of basic functions, provided that its boundary value on the initial hyperplane is an element of this space. At the same time, the solution tends to the initial vector of the function as the time variable approaches zero in the sense of the topology of this space. This result, in particular, makes it possible to draw the important conclusion that within the framework of a space of the S type, evolutionary processes with no external influence, which are described by Shilov hyperbolic system, may, over time, retain those qualitative characteristics that they owned at the initial stage of evolution.