S. Kharibegashvili
abstract:
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.
Mathematics Subject Classification: 35L05, 35L20, 35L50, 35L70, 35L80, 35Q60
Key words and phrases: Hyperbolic equations and systems, hyperbolic systems with symmetric principal part, multidimensional versions of the Darboux and Goursat problems, degenerating hyperbolic equations of the second order, nonlocal problems, existence or nonexistence of global solutions for nonlinear wave equations