On the Dirichlet Problem in a Characteristic Rectangle for Fourth Order Linear Singular Hyperbolic Equations

In the rectangle D = (0, a) × (0, b) with the boundary Γ the Dirichlet problem ∂4u ∂x2∂y2 = p(x, y)u + q(x, y), u(x, y) = 0 for (x, y) ∈ Γ is considered, where p and q : D → R are locally summable functions and may have nonintegrable singularities on Γ. The effective conditions guaranteeing the unique solvability of this problem and the stability of its solution with respect to small perturbations of the coefficients of the equation under consideration are established. § 1. Formulation of the Problem and Main Results In the open rectangle D = (0, a) × (0, b) consider the linear hyperbolic equation ∂4u ∂x2∂y2 = p0(x, y)u + q(x, y), (1.1) where p and q are real functions, Lebesgue summable on [δ, a− δ]× [δ, b− δ] for any small δ > 0. We do not exclude the case, where p and q are not summable on D and have singularities on the boundary of D. In this sense equation (1.1) is singular. Let Γ be the boundary of D. In the present paper for equation (1.1) we study the homogeneous Dirichlet problem u(x, y) = 0 for (x, y) ∈ Γ. (1.2) 1991 Mathematics Subject Classification. 35L55.