Mathematica Balkanica ————————— New Series Vol. 26, 2012, Fasc. 1-2 Strict L p Solutions for Nonautonomous Fractional Evolution Equations Emilia Bazhlekova Presented at 6 th International Conference “TMSF’ 2011” Nonautonomous problems are important especially as a transient case between the linear and the nonlinear theory. We study the nonautonomous linear problem for the fractional evolution equation D α t u(t)+ A(t)u(t)= f (t), a.a. t ∈ (0,T ), where D α t is the Riemann-Liouville fractional derivative of order α ∈ (0, 1), {A(t)} t∈[0,T ] is a family of linear closed operators densely defined on a Banach space X and the forcing function f (t) ∈ L p (0,T ; X). Strict L p solvability of this problem is proved for a suitable class of operators A(t). The proof is based on L p regularity estimates for the corresponding autonomous problem. MSC 2010: 26A33, 34A08, 34K37 Key Words: fractional evolution equation, maximal L p regularity, real interpolation space, randomized bounded family of operators 1. Introduction The notion of maximal L p regularity plays an important role in the func- tional analytic approach to parabolic partial differential equations. Many initial and boundary value problems can be reduced to an abstract Cauchy problem of the form u ′ (t)+ Au(t)= f (t),t ∈ I, u(0) = 0, (1) where I = (0,T ), T> 0, −A generates a bounded analytic semigroup on a Banach space X and f and u are X -valued functions defined on I . It is well known that (1) has a strong solution for all locally Bochner integrable f , but in many applications we need that u ′ has the same “smoothness” as f . This property is called maximal regularity. In particular, one says that problem (1) has maximal L p regularity on I if for every f ∈ L p (I ; X ) there exists one and only one u ∈ L p (I ; D(A)) ∩ W 1,p (I ; X ) satisfying (1). Here