On the Solvability of a Darboux Type Non-characteristic Spatial Problem for the Wave Equation

The question of the correct formulation of a Darboux type non-characteristic spatial problem for the wave equation is investigated. The correct solvability of the problem is proved in the Sobolev space for surfaces of the temporal type on which Darboux type boundary conditions are given. In the space of variables x1, x2, t let us consider the wave equation ƒu ≡ ∂ 2u ∂t2 − ∂ 2u ∂x1 − ∂ 2u ∂x2 2 = F, (1) where F and u are known and desired real functions, respectively. We denote by D : k1t < x2 < k2t, 0 < t < t0, −1 < ki = const < 1, i = 1, 2, k1 < k2, a domain lying in the half-space t > 0 and bounded by the plane surfaces Si : kit − x2 = 0, 0 ≤ t ≤ t0, i = 1, 2, of the temporal type and by the plane t = t0. We shall consider a Darboux type problem formulated as follows: In the domain D find a solution u(x1, x2, t) of equation (1) by the boundary conditions u ∣ ∣ Si = fi, i = 1, 2, (2) where fi, i = 1, 2, are the known real functions on Si and (f1−f2)|S1∩S2 = 0. It should be noted that in [1–5] Darboux type problems are studied for the cases where at least one of the surfaces S1 and S2 is the characteristic surface of equation (1) passing through the Ox1-axis. Other multi-dimensional analogues of the Darboux problem are treated in [6–8]. 1991 Mathematics Subject Classification. 35L20.