Some existence and regularity results for abstract non-autonomous parabolic equations

We study existence, uniqueness and regularity of the strict, classical and strong solutions $\text{u? C(¦0, T ¦,E)}$ of the non-autonomous evolution equation $\text{u′(t) ? A(t)u(t)=?(t)}$, with the initial datum u(0) = x, in a Banach space E, under the classical Kato-Tanabe assumptions. The domains of the operators A(t) are not needed to be dense in E. We prove necessary and sufficient conditions for existence and Hölder regularity of the solution and its derivative.