2019
Том 71
№ 9

# Shklyar A. Ya.

Articles: 2
Article (Russian)

### Well-posedness of the cauchy problem for complete second-order operator-differential equations

Ukr. Mat. Zh. - 1996. - 48, № 7. - pp. 999-1006

For the equation y″(t)+Ay′(t)+By(t)=0, where A and B are arbitrary commuting normal operators in a Hilbert space H, we obtain a necessary and sufficient condition for well-posedness of the Cauchy problem in the space of initial data D(B)×(D(A)∩D(|B|1/2)) and for weak well-posedness of the Cauchy problem in H×H_(|A|+|B|1/2+1). This condition is expressed in terms of location of the joint spectrum of the operators A and B in C 2. In terms of location of the spectrum of the operator pencil z 2+Az+B in C 1, such a condition cannot be written.

Article (Russian)

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

Ukr. Mat. Zh. - 1993. - 45, № 5. - pp. 704–714

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.