The method of integrated semigroups for Cauchy problems in Banach spaces
The author considers the second-order operator equations \[ u''(t)=Au'(t)+Bu(t), \qquad Qu''(t)=Au'(t)+Bu(t), \] where \(t\geq 0\), \(A\), \(B\), and \(Q\) are linear operators with \(\ker Q\neq\{0\}\) (in other words, the second equation is degenerate). Necessary and sufficient conditions for the well-posedness of the Cauchy problem for these equations are given in the original Banach space and in some subspaces. These conditions are stated in terms of the resolvent \(R(\lambda ^2)=(\lambda ^2-\lambda A -B)^{-1}\) of the operators \(A\) and \(B\).
second-order operator-differential equation
scientific article; zbMATH DE number 1285615