Spatial Problem of Darboux Type for One Model Equation of Third Order

For a hyperbolic type model equation of third order a Darboux type problem is investigated in a dihedral angle. It is shown that there exists a real number ρ0 such that for α > ρ0 the problem under consideration is uniquely solvable in the Frechet space. In the case where the coefficients are constants, Bochner’s method is developed in multidimensional domains, and used to prove the uniquely solvability of the problem both in Frechet and in Banach spaces. § 1. Statement of the Problem Let us consider a partial differential equation of hyperbolic type uxyz = F (1.1) in R3, where F is a given function and u is an unknown real function. For equation (1.1) the family of planes x = const, y = const, z = const is characteristic, while the directions determined by the unit vectors e1, e2, e3 of the coordinate axes are bicharacteristic. In the space R3 let S0 i : pi(x, y, z) ≡ α0 i x + β0 i y + γ0 i z = 0, i = 1, 2, be arbitrarily given planes passing through the origin. Assume that ν0 1 ∦ ν0 2 , |ν0 i | 6= 0, where ν0 i ≡ (α0 i , β0 i , γ0 i ), i = 1, 2. The space R3 is partitioned by the planes S0 i , i = 1, 2, into four dihedral angles. We consider equation (1.1) in one of these angles D0 which, without loss of generality, is assumed to be given in the form D0 ≡ {(x, y, z) ∈ R3 : α0 i x + β0 i y + γ0 i z > 0, i = 1, 2}. For the domain D0 we make the following assumptions: 1991 Mathematics Subject Classification. 35L35.