Maximal space regularity for abstract linear non-autonomous parabolic equations

The linear non-autonomous evolution equation $\text{u′(t) ? A(t) u(t) = ?(t), t ∈ 0, T]}$, with the initial datum u(0) = x, in the space C(0, T], E), where E is a Banach space and {A(t)} is a family of infinitesimal generators of bounded analytic semigroups is considered; the domains D(A(t)) are supposed constant in t and possibly not dense in E. Maximal regularity of the strict and classical solutions, i.e., regularity of u′ and A(·)u(·) with values in the interpolation spaces D_A(0)(θ, ∞) and D_A(0)(θ) between D(A(0)) and E, is studied. A characterization of such spaces in a concrete case is also given.