Quasilinear evolutionary equations and continuous interpolation spaces

In this paper we analyze the abstract parabolic evolutionary equations D-t(alpha)(u - x) + A(u)u =f (u) + h(t), u(0) = x, in continuous interpolation spaces allowing a singularity as tdown arrow0. Here D-t(alpha) denotes the time-derivative of order alpha is an element of (0, 2). We first give a treatment of fractional derivatives in the spaces L-p((0, T); X) and then consider these derivatives in spaces of continuous functions having (at most) a prescribed singularity as tdown arrow0. The corresponding trace spaces are characterized and the dependence on alpha is demonstrated. Via maximal regularity results on the linear equation D-t(alpha)(u - x) + Au =f, u(0) = x, we arrive at results on existence, uniqueness and continuation on the quasilinear equation. Finally, an example is presented. (C) 2003 Elsevier Inc. All rights reserved.