The correctness of degenerate Cauchy problems with generators of exponentially bounded semigroups (Q2387082)

Let \(X\) be a Banach space and let \(A : D(A) \subset X \to X\) be a linear closed multi-valued operator on \(X\). The author studies the correctness of the Cauchy problem \[ u^\prime(t) \in Au(t), \quad t\geq 0, \qquad u(0)=x. \] The main result of the paper establishes the connections between correctness of the above Cauchy problem in the distributional space and the existence of a degenerate exponentially bounded \(n\)-times integrated semigroup generated by \(A\). The author applies his result in order to prove a correctness theorem on certain subsets of \(D(A)\).