The Cauchy Problem for General Hyperbolic Systems in the Complete Scale of Sobolev Type Spaces

Abstract

Ever since the well-known work of S. L. Sobolev [Sob] generalized functions have frequently been used to study the Cauchy problem for hyperbolic equations (we mention here [Ler], [Går], [Vla], [Hör], the survey [VoG], and the bibliography given there). In this note the Cauchy problem for a system strictly hyperbolic in the Leray—Volevich sense is studied in the complete scale of spaces of Sobolev type depending on real parameters s and τ; s characterizes the order smoothness of a solution in all variables, while τ describes the additional smoothness in the tangential variables. The solutions is the ‘more generalized’ the smaller s and τ; for sufficiently large s and τ the solution is an ordinary classical solution of the problem under consideration. In [R7] and [R8] such problems were studied for a single equation.