On the abstract nonautonomous parabolic cauchy problem in the case of constant domains

Summary We study existence, uniqueness and regularity of the strict, classical and strong solution u C([0,T],E) of the non-autonomous evolution equation u(t)–A(t)u(t)= f(t), with the initial datum, u(0)=x, in a Banach space E, where {A(t)} is a family of infinitesimal generators of analytic semi-groups whose domains are constant in t and possibly not dense in E. We prove necessary and sufficient conditions for existence and Hölder regularity of the solutions and their first derivative.