Maximal regularity for non-autonomous Cauchy problems in weighted spaces

Abstract

We consider the regularity for the non-autonomous Cauchy problem u′(t) + A(t)u(t) = ƒ(t) (t ∈ [0, τ]), u(0) = u0. The time dependent operator A(t) is associated with (time dependent) sesquilinear forms on a Hilbert space H. We prove the maximal regularity result in temporally weighted L2-spaces and other regularity properties for the solution of the problem under minimal regularity assumptions on the forms and the initial value u0. Our results are motivated by boundary value problems.