Quasilinear evolutionary equations and continuous interpolation spaces

In this paper we analyze the abstract parabolic evolutionary equations Dt ðu xÞ þ AðuÞu 1⁄4 f ðuÞ þ hðtÞ; uð0Þ 1⁄4 x; in continuous interpolation spaces allowing a singularity as tk0: Here Dt denotes the timederivative of order aAð0; 2Þ: We first give a treatment of fractional derivatives in the spaces Lðð0;TÞ;XÞ and then consider these derivatives in spaces of continuous functions having (at most) a prescribed singularity as tk0: The corresponding trace spaces are characterized and the dependence on a is demonstrated. Via maximal regularity results on the linear equation Dt ðu xÞ þ Au 1⁄4 f ; uð0Þ 1⁄4 x; we arrive at results on existence, uniqueness and continuation on the quasilinear equation. Finally, an example is presented. r 2003 Elsevier Inc. All rights reserved.