Lp-maximal regularity for non-autonomous evolution equations

AbstractLet A:[0,τ]→L(D,X) 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(P)u˙+A(t)u=fa.e. on (0,τ),u(0)=x, is well-posed in Lp; i.e. for all 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) solution of (P), where 1<p<∞. 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∈[0,τ]. We also consider the non-autonomous second order problemu¨+B(t)u˙+A(t)u=fa.e. on (0,τ),u(0)=x,u˙(0)=y, for which we prove similar regularity and perturbation results