Spectral problem for the sixth order nonclassical differential equations

Abstract

In this article we investigate the correctness of boundary value problems for a sixth order quasi-hyperbolic equation in the Sobolev space L_u = −D⁶_t u + ∆u − λu (D_t =∂/∂t , ∆ = Σⁿ_i=1∂²/∂x²_i – Laplace operator, λ – real parameter). For the given operator L two spectral problems are introduced and uniqueness of these problems is established. The eigenvalues and eigenfunctions of the first spectral problem are calculated for the sixth order quasi-hyperbolic equation. In this work we show that the equation L_u = 0 for λ < 0 under uniform conditions has a countable set of nontrivial solutions. Usually, this does not happen when the operator L is an ordinary hyperbolic operator.