Quasilinear evolutionary equations and continuous interpolation spaces

In this paper we analyze the abstract parabolic evolutionary equations D^(alpha)(t) (u-x)+A(u)u=f(u)+h(t), u(0)=x,in continuous interpolation spaces allowing a singularity as t(down arrow)0. Here D^(alpha)(t) denotes the time-derivative of order (alpha) (element of)(0,2). We first give a treatment of fractional derivatives in the spaces Lp ((0,T);X) and then consider these derivatives in spaces of continuous functions having (at most) a prescribed singularity as t(down arrow)0. The corresponding trace spaces are characterized and the dependence on (alpha) is demonstrated. Via maximal regularity results on the linear equation D^(alpha)(t) (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.