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

Pengyu Chen; Xuping Zhang; Yongxiang Li

doi:10.3934/cpaa.2018094

Article Contents

2018, Volume 17, Issue 5: 1975-1992. Doi: 10.3934/cpaa.2018094

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

Department of Mathematics, Northwest Normal University, Lanzhou 730070, China

* Corresponding author
* Corresponding author

Received: July 31, 2017

Revised: October 31, 2017

Published: April 2018

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Abstract

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.

Keywords:

Fractional non-autonomous evolution equation,
analytic semigroup,
measure of noncompactness,
volterra integro-differential,
mild solution.

Mathematics Subject Classification: Primary: 35R11; Secondary: 47H08, 47J35.

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