On the evolution operator for abstract parabolic equations (Q1109319)

The initial value problem for the differential equation in Banach space X is considered: \[ (1)\quad u'(t)=A(t)u(t)+f(t),\quad t_ 0<t\leq t_ 1;\quad u(t_ 0)=x. \] Here A(t):\({\mathcal D}\to X\) \((t_ 0<t\leq t_ 1)\) is a family of linear operators with constant domain \({\mathcal D}\). It is essential that \({\mathcal D}\) is not necessarily dense in X. It is supposed that for any \(t\in [t_ 0,t_ 1]\) there are \(\omega\in R\), \(\theta\in (\pi /2,\pi)\), \(M>0\) such that the resolvent set of A(t) contains th sector \(S=\{\lambda \in {\mathbb{C}}\), \(\lambda\neq \omega\), \(| \arg (\lambda -\omega)| <\theta\) and \(\| (\lambda -\omega)(\lambda -A)^{-1}\|_{L(X)}\leq M\}\) and the function belongs to the Hölder space \(C^{\alpha}([t_ 0,t_ 1]\), L(\({\mathcal D},X))\). The initial value x belongs to \({\mathcal D}\) or \(\bar {\mathcal D}.\) The evolution operator G(t,S) associted to a family A(t) is constructed. The existence and uniqueness of the strict, classical and strong solutions of the problem (1) for various classes of the functions f(t) and the regularity properties of these solutions are estalished. For the presentation of the solution the standard formula \[ u(t)=G(t,t_ 0)x_ 0+\int^{t}_{t_ 0}G(t,s)f(s)ds,\quad t\geq t_ 0 \] is applied.