A BLOWUP ALTERNATIVE RESULT FOR FRACTIONAL NON-AUTONOMOUS 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 E, where the operators in linear part (possibly unbounded) depend on time t. 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.