Schauder estimates for equationswith fractional derivatives

Abstract:

The equation \begin{equation*} D^\alpha _t (u-h_1) + D^\beta _x(u-h_2) =f,\quad 0< \alpha ,\beta < 1, \quad t,x \geq 0,\tag {$*$} \end{equation*} where $D^\alpha _t$ and $D^\beta _x$ are fractional derivatives of order $\alpha$ and $\beta$ is studied. It is shown that if $f=f(\underline {t}, \underline {x})$, $h_1=h_1(\underline {x})$, and $h_2=h_2(\underline {t})$ are Hölder-continuous and $f(0,0) =0$, then there is a solution such that $D^\alpha _t u$ and $D^\beta _x u$ are Hölder-continuous as well. This is proved by first considering an abstract fractional evolution equation and then applying the results obtained to ($*$). Finally the solution of ($*$) with $f=1$ is studied.