Non-Autonomous Maximal Regularity Under Besov Regularity in Time

Abstract

We consider the maximal regularity problem for non-autonomous Cauchy problems u′(t) + A(t)u(t) = f (t) t-a.e., u(0) = u0. Each operator A(t) arises from a time depending sesquilinear form a(t) on a Hilbert space H with constant domain V. We prove maximal Lp-regularity result for p ≤ 2 under min-imal regularity assumptions on the forms. Our main assumption is that (A(t))t∈[0,τ ] arepiecewise in the Besov space Bp1/2, 2 p with respect to the variable t. This improves previouslyknown results. We give three examples which illustrate our results. Mathematics Subject Classification (2010): 35K90, 35K45, 47D06.