Journal of Mathematical Sciences, Vol. 171, No. 1, 2010 BOUNDARY-VALUE PROBLEMS FOR FOURTH-ORDER EQUATIONS OF HYPERBOLIC AND COMPOSITE TYPES V. I. Korzyuk, O. A. Konopel’ko, and E. S. Cheb UDC 517.951 517.956 Abstract. Boundary-value problems for fourth-order linear partial differential equations of hyperbolic and composite types are studied. The method of energy inequalities and averaging operators with variable step is used to prove existence and uniqueness theorems for strong solutions. The Riesz theorem on the representation of linear continuous functionals in Hilbert spaces is used to prove the existence and uniqueness theorems for generalized solutions. CONTENTS Introduction ............................................ 89 1. Setting of the Problem for Equation (1) ............................ 90 2. Energy Inequalities ........................................ 93 3. Strong Solutions ......................................... 96 4. Strong Solutions in Other Function Spaces ........................... 103 5. Boundary-Value Conditions for Fourth-Order Equations of the Composite Type ...... 108 References ............................................. 111 Introduction For functions u : R n+1 ∋ x =(x 0 ,x 1 ,...,x n ) → u(x) ∈ R, where R n+1 is the (n + 1)-dimensional Euclidean space of independent variables x, equations of the following kind are considered: L (1) u ≡  ∂ 2 ∂x 2 0 - a 2 A  ∂ 2 ∂x 2 0 - b 2 A  u + A (3) u = f (x) (1) and L (2) u ≡  ∂ 2 ∂x 2 0 - a 2 A  ∂ 2 ∂x 2 0 + b 2 A  u + A (3) u = f (x), (2) where A = n ∑ i,j =1 ∂ ∂x i  a (ij ) ∂ ∂x j  ,A (3) = ∑ |α|≤3 a (3) α (x)D α , α = (α 0 ,...,α n ) is a multi-index, D α = ∂ |α| ∂x α 0 0 ...∂x αn n , |α| = α 0 + ... + α n , the coefficients a (ij ) of the operator A form a matrix of a positive quadratic form (see Condition 1.1), a 2 ,b 2 ∈ R n , a 2 > 0, and b 2 > 0. Under the above conditions, Eq. (1) is hyperbolic with respect to the direction ζ = (1, 0,..., 0) along the axis x 0 , while Eq. (2) is of composite type. Boundary-value problems in cylindrical domains are mainly considered for those equations. A comprehensive literature is devoted to problems for hyperbolic equations. Mixed problems for higher-order equations were also considered. The Cauchy problem for partial differential equations was studied by Kovalevskaya, Petrovskii, Leray, Friedrichs, Ladyzhenskaya, G˚ arding, Volevich, Gindikin, and others. In [68], the investigation of the Cauchy problem for general hyperbolic equations and Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 36, Proceedings of the Fifth International Conference on Differential and Func- tional Differential Equations. Part 2, 2010. 1072–3374/10/1711–0089 c  2010 Springer Science+Business Media, Inc. 89