Well-posedness of the cauchy problem for complete second-order operator-differential equations
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.
English version (Springer): Ukrainian Mathematical Journal 48 (1996), no. 7, pp 1131-1139.
Citation Example: Shklyar A. Ya. Well-posedness of the cauchy problem for complete second-order operator-differential equations // Ukr. Mat. Zh. - 1996. - 48, № 7. - pp. 999-1006.