Fractional Evolution Equations Governed by Coercive Differential Operators

Abstract

This paper is concerned with evolution equations of fractional order ${\mathbf{\text{D}}}^{\alpha }u\left(t\right)=Au\left(t\right)$; $u\left(0\right)=u_0,u^{\prime }\left(0\right)=0$, where $A$ is a differential operator corresponding to a coercive polynomial taking values in a sector of angle less than $\pi$ and $1<\alpha <2$. We show that such equations are well posed in the sense that there always exists an $\alpha$-times resolvent family for the operator $A$.