SOME MULTIDIMENSIONAL PROBLEMS FOR HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS AND SYSTEMS

For a class of first and second order hyperbolic systems with symmetric principal part, to which belong systems of Maxwell and Dirac equations, crystal optics equations, equations of the mathematical theory of elasticity and so on which are well known from the mathematical physics, we have developed a method allowing one to give correct formulations of boundary value problems in dihedral angles and conical domains in Sobolev spaces. For second order hyperbolic equations of various types of degeneration, we study the multidimensional versions of the Goursat and Darboux problems in dihedral angles and conical domains in the corresponding Sobolev spaces with weight. For the wave equation with one or two spatial variables, the correctness of some nonlocal problems is shown. The existence or nonexistence of global solutions of the characteristic Cauchy problem in a conic domain is studied for multidimensional wave equations with power nonlinearity.