Existence and sharp regularity results for linear parabolic non-autonomous integro-differential equations

Abstract

We study existence, uniqueness and maximal regularity of the strict solutionu∈C 1([0,T],E) of the integro-differential equation\(u'(t) - A(t)u(t) - \int {_0^1 } B(t,s)u(s)ds = f(t),t \in [0,T],\) with the initial datumu(0)=x, in a Banach spaceE, {itA(itt)}f∈|0,1| is a family of generators of analytic semigroups whose domainsD A(t) are not constant int as well as (possibly) not dense inE, whereas {itB(itt)}0≦11≦T is a family of closed linear operators withD B(t,s) ⊇D A(s) ∨t∈[s, T]. We prove necessary and sufficient conditions for existence of the strict solution and for Holder continuity of its derivative; well-posedness of the problem with respect to the Holder norms is also shown.