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 Lu = −D6t u + ∆u − λu (Dt =∂/∂t , ∆ = Σni=1∂2/∂x2i – 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 Lu = 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.

Keywords

• a sixth order quasi-hyperbolic equation
• eigenvalues
• eigenfunctions
• nontrivial solutions