Correctness of the Cauchy problem for trinomial higher-order operator differential equations

Abstract

Criteria are established for the correctness of the Cauchy problem for the equations $y^{(2n)} + Ay^{(n)} + By = 0, \quad t \in [0, \infty)$ where $n > 1, А, В$ are arbitrary commuting self-adjoint operators in a Hilbert space. For $n = 2$, the criterion is illustrated by the example of the equation describing the dynamics of an exponentially stratified rotating compressible fluid.