LpLp-maximal regularity for non-autonomous evolution equations

Let A:[0,τ]→L(D,X)A:[0,τ]→L(D,X) be strongly measurable and bounded, where D, X   are Banach spaces such that D↪XD↪X. We assume that the operator A(t)A(t) has maximal regularity for all t∈[0,τ]t∈[0,τ]. Then we show under some additional hypothesis (viz. relative continuity) that the non-autonomous problem(P)u˙+A(t)u=fa.e. on (0,τ),u(0)=x, is well-posed in LpLp; i.e. for all f∈Lp(0,τ;X)f∈Lp(0,τ;X) and all x∈(X,D)1p∗,p there exists a unique u∈W1,p(0,τ;X)∩Lp(0,τ;D)u∈W1,p(0,τ;X)∩Lp(0,τ;D) solution of (P)(P), where 1