Hölder regularity for non-autonomous fractional evolution equations

Abstract

This paper concerns a Hölder regularity result for non-autonomous fractional evolution equations (NFEEs) of the form $$^C\!D^\alpha _t u(t)+A(t)u(t)=f(t)$$ in the sense of Caputo’s fractional derivative. Under the assumption of the Acquistapace-Terremi conditions, we get a representation of solution that is closer to standard integral equation. A pair of families of solution operators will be constructed in terms of the Mittag-Leffler function, the Mainardi Wright-type function, the analytic semigroups generated by closed dense operator $$A(\cdot )$$ and two variation of parameters formulas. By a suitable definition of classical solutions, the solvability of the Cauchy problem is established. Moreover, we also prove the Hölder regularity of classical solutions.