The theory of partial differential equations is a wide and rapidly
developing branch of contemporary mathematics. Problems related to
partial differential equations of order higher than one are so diverse
that a general theory can hardly be built up. There are several
essentially different kinds of differential equations called elliptic,
hyperbolic, and parabolic. Regarding the construction of solutions of
Cauchy, mixed and boundary value problems, each kind of equation
exhibits entirely different properties. Cauchy problems for hyperbolic
equations and systems with variable coefficients have been studied in
classical works of Petrovskii, Leret, Courant, Gording. Mixed problems
for hyperbolic equations were considered by Vishik, Ladyzhenskaya, and
that for general two- dimensional equations were investigated by
Bitsadze, Vishik, Gol'dberg, Ladyzhenskaya, Myshkis, and others. In last
decade the theory of solvability on the whole of boundary value problems
for nonlinear differential equations has received intensive development.
Significant results for nonlinear elliptic and parabolic equations of
second order were obtained in works of Gvazava, Ladyzhenskaya,
Nakhushev, Oleinik, Skripnik, and others. Concerning the solvability in
general of nonlinear hyperbolic equations, which are connected to the
theory of local and nonlocal boundary value problems for hyperbolic
equations, there are only partial results obtained by Bronshtein,
Pokhozhev, Nakhushev.