Lp-maximal regularity for non-autonomous evolution equations

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