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September 2018, 17(5): 1975-1992. doi: 10.3934/cpaa.2018094
A blowup alternative result for fractional nonautonomous evolution equation of Volterra type
Pengyu Chen , , Xuping Zhang and Yongxiang Li
Department of Mathematics, Northwest Normal University, Lanzhou 730070, China
Received August 2017 Revised November 2017 Published April 2018
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.
Citation: Pengyu Chen, Xuping Zhang, Yongxiang Li. A blowup alternative result for fractional nonautonomous evolution equation of Volterra type. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1975-1992. doi: 10.3934/cpaa.2018094
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Pengyu Chen Xuping Zhang Yongxiang Li