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

It is proved that the Dirichlet problem is correct in the characteristic rectangle Dab = [0, a] × [0, b] for the linear hyperbolic equation ∂4u ∂x2∂y2 = p0(x, y)u + p1(x, y) ∂u ∂x + p2(x, y) ∂u ∂y + +p3(x, y) ∂2u ∂x∂y + q(x, y) with the summable in Dab coefficients p0, p1, p2, p3 and q if and only if the corresponding homogeneous problem has only the trivial solution. The effective and optimal in some sense restrictions on p0, p1, p2 and p3 guaranteeing the correctness of the Dirichlet problem are established. § 1. Formulation of the Problem and Main Results The Dirichlet problem for second order hyperbolic equations and some higher order linear hyperbolic equations with constant coefficients has long been attracting the attention of mathematicians. The problem has been the subject of numerous studies (see [1–20] and the references therein) but still remains investigated very little for a wide class of hyperbolic equations. This class includes the fourth order hyperbolic equation ∂4u ∂x2∂y2 =p0(x, y)u+p1(x, y) ∂u ∂x +p2(x, y) ∂u ∂y +p3(x, y) ∂2u ∂x∂y +q(x, y) (1.1) for which the Dirichlet problem is considered here in the characteristic rectangle Dab = [0, a]× [0, b]. 1991 Mathematics Subject Classification. 35L55.