Quasilinear evolutionary equations and continuous interpolation spaces

Authors' abstract: In this paper we analyze the abstract parabolic evolutionary equations $$D_t^\alpha(u-x)+A(u)u=f(u)+h(t),\quad u(0)=x,$$ in continuous interpolation spaces allowing a singularity as $t\downarrow 0$. Here $D_t^\alpha$ denotes the time-derivative of order $\alpha\in(0,2)$. 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 $t\downarrow 0$. The corresponding trace spaces are characterized and the dependence on $\alpha$ is demonstrated. Via maximal regularity results on the linear equation $$ D_t^\alpha(u-x)+Au=f,\quad u(0)=x, $$ we arrive at results on existence, uniqueness and continuation on the quasilinear equation. Finally, an example is presented. [ Irina V.Melnikova (Ekaterinburg) ]