Well-posedness of linear partial differential equations with unbounded delay operators (Q1827089)

The authors study the well-posedness of the following delay equation \[ \begin{gathered} \dot u(t)= Au(t)+ Bu(t- r),\quad t> 0,\\ u(0)= x,\quad u(\theta)= f(\theta),\quad \theta\in [-r,0],\end{gathered}\tag{1} \] where \((A,D(A))\) generates a strongly continuous semigroup \(T(t)\) on a Banach space \(X\) and the delay operator \((B,D(B))\) is an unbounded and closed linear operator on \(X\). To show the well-posedness of (1), the authors use the well-known method of transforming this equation into an abstract differential equation in a phase space. Moreover, as an example, they apply their abstract results to the linear elastic system with delays in the damping term.