Multidimensional versions of the Cauchy characteristic problem, the Darboux problems, and the Sobolev problem for a class of second order semilinear hyperbolic systems are investigated. Depending on the type of nonlinearity, spatial dimension and structure of the hyperbolic system, the cases for which these problems are globally solvable, are singled out. Moreover, the cases of the absence of solutions of these problems are also considered. The questions of the solvability of some nonlocal in time problems for multidimensional second order semilinear hyperbolic equations are studied. The particular cases of the above-mentioned problems are the periodic and antiperiodic problems.
Mathematics Subject Classification: 35L05, 35L20, 35L51, 35L71
Key words and phrases: Characteristic Cauchy problem, Darboux problems, Sobolev problem, nonlocal problems, multidimensional hyperbolic equations and systems, global and local solvability, uniqueness, existence and nonexistence of solutions