On the Solvability of the Multidimensional Version of the First Darboux Problem for a Model Second-order Degenerating Hyperbolic Equation

A multidimensional version of the first Darboux problem is considered for a model second order degenerating hyperbolic equation. Using the technique of functional spaces with a negative norm, the correct formulation of this problem in the Sobolev weighted space is proved. In a space of variables x1, x2, t let us consider a second-order degenerating hyperbolic equation of the type Lu ≡ utt − |x2|ux1x1 − ux2x2 + a1ux1 + a2ux2 + a3ut + a4u = F, (1) where ai, i = 1, . . . , 4, F are given real functions and u is an unknown real function, m = const > 0. Denote by D : x2 < t < 1− x2, 0 < x2 < 2 , an unbounded domain lying in a space x2 > 0 and bounded by characteristic surfaces S1 : t − x2 = 0, 0 < x2 < 2 , S2 : t + x2 − 1 = 0, 0 < x2 < 1 2 , of equation (1) and by a plane surface S0 : x2 = 0, 0 < t < 1, of time type with an equation degenerating on it. The coefficients ai, i = 1, . . . , 4, of equation (1) in the domain D are assumed to be bounded functions of the class C1(D). For equation (1) let us consider a multidimensional version of the first Darboux problem formulated as follows: find in the domain D a solution u(x1, x2, t) of equation (1) satisfying the boundary condition u ∣ ∣ S0∪S1 = 0. (2) 1991 Mathematics Subject Classification. 35L80.