**Sergo Kharibegashvili**

##
Some Local and Nonlocal Multidimensional Problems for a Class of Semilinear
Hyperbolic Equations and Systems

**abstract:**

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