Maslov-Heaviside operator method and \(C_0\)-operator Duhamel integral (Q393861)

Let \(U(t)\), \(t\geq 0\) be a \(C_{0}\)-semigroup of operators on a Banach space \(E\), with generator \(A\). The authors apply the Maslov operator method to \(A\) to obtain an exact solution \(u\) for a broad class of problems satisfying \(Au =\sum a_{m} A^{m}u = f\), \(a_{m} \in C\) for all \(f\in E\). Theses solutions are represented in the form of a Duhamel integral and it is shown that these problems are uniformly well posed for differential operators with coefficients having singularities. The authors also provide examples of \(C_{0}\)-semigroups with generators whose coefficients degenerate or increase to infinity, but with an exact unique solution that can be represented in the form of a Duhamel integral.