Integrated semigroups and linear partial differential equations with delay (Q1827123)

The author studies an equation of the type \[ u'(t)= Au(t)+ \Phi u_t, \quad t\geq 0, \qquad u(0)=x, \quad u_0=f, \tag{1} \] where \(A\) is an (unbounded) operator on a Banach space \(X\), \(\Phi\) is the delay operator, \(u_t\) is the history function and \(x\in X\), \(f\in L^p ([-1,1],Z)\) are the initial values at time 0. Here, \(Z\) is a Banach space such that \(D(A) \hookrightarrow Z\hookrightarrow X\) with continuous embeddings. Using a new perturbation theorem for \(\alpha\)-times integrated semigroup, the author establishes existence results for solutions of the delay equation (1).