Abstract
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.
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References
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Shklyar, A.Y. Well-posedness of the cauchy problem for complete second-order operator-differential equations. Ukr Math J 48, 1131–1139 (1996). https://doi.org/10.1007/BF02390969
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DOI: https://doi.org/10.1007/BF02390969