A blowup alternative result for fractional nonautonomous evolution equation of Volterra type

In this article, we consider a class of fractional non-autonomous integro-differential evolution equation of Volterra type in a Banach space \begin{document}$E$\end{document} , where the operators in linear part (possibly unbounded) depend on time \begin{document}$t$\end{document} . Combining the theory of fractional calculus, operator semigroups and measure of noncompactness with Sadovskii's fixed point theorem, we firstly proved the local existence of mild solutions for corresponding fractional non-autonomous integro-differential evolution equation. Based on the local existence result and a piecewise extended method, we obtained a blowup alternative result for fractional non-autonomous integro-differential evolution equation of Volterra type. Finally, as a sample of application, these results are applied to a time fractional non-autonomous partial integro-differential equation of Volterra type with homogeneous Dirichlet boundary condition. This paper is a continuation of Heard and Rakin [ 13 , J. Differential Equations, 1988] and the results obtained essentially improve and extend some related conclusions in this area.