Classical solutions for partial functional differential equations with nonautonomous past (Q1865727)

In [Discrete Contin. Dyn. Syst. 8, 953--966 (2002; Zbl 1013.35080)], \textit{S. Brendle} and \textit{R. Nagel} studied the partial functional differential equation with nonautonomous past \[ \begin{cases} \frac{\partial}{\partial t} u(t,s)= \frac{\partial}{\partial s} u(t,s)+A(s)\,u(t,s),&\quad s\leq 0,\;\;t\geq 0,\\ \frac{\partial}{\partial t} u(t,0)=Bu(t,0)+\Phi u(t,\cdot),&\quad t\geq 0, \end{cases} \] where \(A(s)\) are unbounded operators on a Banach space \(X\), \(B\) is the generator of a strongly continuous semigroup \((S(t))_{t\geq 0}\) on \(X\) and \(\Phi\), the delay operator, is a linear operator from a space of \(X\)-valued functions on \({\mathbb R}_{-}\) into \(X\). They obtained the mild solutions of the above equation by constructing an appropriate semigroup. The author of the present paper proves that the same semigroup yields the classical solution of the above equation under appropriate assumptions.