On a Spatial Problem of Darboux Type for a Second-order Hyperbolic Equation

The theorem of unique solvability of a spatial problem of Darboux type in Sobolev space is proved for a second-order hyperbolic equation. In the space of variables x1, x2, t let us consider the second order hyperbolic equation Lu ≡ ƒu + aux1 + bux2 + cut + du = F, (1) where ƒ ≡ ∂ 2 ∂t2 − ∂ ∂x1 − ∂ 2 ∂x2 is a wave operator; the coefficients a, b, c, d and the right-hand side F of equation (1) are given real functions, and u is an unknown real function. Denote by D : kt < x2 < t, 0 < t < t0, −1 < k = const < 1, the domain lying in a half-space t > 0, which is bounded by a time-type plane surface S1 : kt − x2 = 0, 0 ≤ t ≤ t0, a characteristic surface S2 : t − x2 = 0, 0 ≤ t ≤ t0 of equation (1), and a plane t = t0. Let us consider the Darboux type problem formulated as follows: find in the domain D the solution u(x1, x2, t) of equation (1) under the boundary conditions u ∣ ∣ Si = fi, i = 1, 2, (2) where fi, i = 1, 2, are given real functions on Si; moreover (f1−f2)|S1∩S2 = 0. Note that in the class of analytic functions the problem (1),(2) is considered in [1]. In the case where S1 is a characteristic surface t + x2 = 0, 0 ≤ t ≤ t0, the problem (1),(2) is studied in [1–3]. Some multidimensional analogues of the Darboux problems are treated in [4–6]. In the present paper the problem (1),(2) is investigated in the Sobolev space W 1 2 (D). 1991 Mathematics Subject Classification. 35L20.