Existence of classical solutions to nonautonomous nonlocal parabolic problems

Abstract

We obtain a new existence theorem for classical solutions to nonautonomous equations with nonlocal initial conditions u ′ ( t ) = A ( t ) u ( t ) + f ( t , u ( t ) ) , t ∈ ( s , T ] , u ( s ) + g ( u ) = u 0 , in a Banach space X, where T > s ⩾ 0 , f , g are given X-valued functions, and A ( t ) is a sectorial operator (not necessarily densely defined) in X for each t ∈ [ 0 , T ] . Both Banach's contraction principle and Schauder's fixed point theorem, as well as the theory of evolution families and interpolation spaces, are employed in our approach. A concrete example is shown to illustrate the existence theorem.