L\sp p-maximal regularity for non-autonomous evolution equations

Let D, X be two Banach spaces such that D is continuously embedded in X and let {A(t)} be a family of linear continuous operators from D to X. We assume that each operator A(t) has maximal regularity. Then we show under some additional hypothesis that the non-autonomous problem (P) u_t +A(t)u = f, u(0) = x, is well-posed in Lp. If the operators A(t) are accretive, we show that conversely, well-posedness of (P ) implies that A(t) has maximal regularity for all t. We also consider the non-autonomous second order problem u_{tt}+ B(t)u_t +A(t)u = f, u(0) = x, u_t(0) = y, for which we prove similar regularity and perturbation results.