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

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  • 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.

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


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  • [1] R. P. AgarwalM. Benchohra and S. Hamani, A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions, Acta Appl. Math., 109 (2010), 973-1033. 
    [2] E. G. Bajlekova, Fractional Evolution Equations in Banach Spaces, Ph. D thesis, Department of Mathematics, Eindhoven University of Technology, 2001.
    [4] P. M. Carvalho-Neto and G. Planas, Mild solutions to the time fractional Navier-Stokes equations in $\mathbb{R}^N$, J. Differential Equations, 259 (2015), 2948-2980. 
    [5] P. Chen and Y. Li, Monotone iterative technique for a class of semilinear evolution equations with nonlocal conditions, Results Math., 63 (2013), 731-744. 
    [6] P. Chen and Y. Li, Existence of mild solutions for fractional evolution equations with mixed monotone nonlocal conditions, Z. Angew. Math. Phys., 65 (2014), 711-728. 
    [7] K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, New York, 1985.
    [8] M. M. El-Borai, The fundamental solutions for fractional evolution equations of parabolic type, J. Appl. Math. Stoch. Anal., 3 (2004), 197-211. 
    [9] M. M. El-BoraiK. E. El-Nadi and E. G. El-Akabawy, On some fractional evolution equations, Comput. Math. Appl., 59 (2010), 1352-1355. 
    [10] A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, New York, NY, USA, 1969.
    [11] R. Gorenflo and F. Mainardi, Fractional calculus and stable probability distributions, Arch. Mech., 50 (1998), 377-388. 
    [12] H. Gou and B. Li, Local and global existence of mild solution to impulsive fractional semilinear integro-differential equation with noncompact semigroup, Commun. Nonlinear Sci. Numer. Simul., 42 (2017), 204-214. 
    [13] M. L. Heard and S. M. Rankin, A semi-linear parabolic integro-differential equation, J. Differential Equations, 71 (1988), 201-233. 
    [14] H. P. Heinz, On the behaviour of measure of noncompactness with respect to differentiation and integration of vector-valued functions, Nonlinear Anal., 7 (1983), 1351-1371. 
    [15] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., vol. 840, Springer-verlag, New York, 1981.
    [16] V. Lakshmikantham and S. Leela, Nonlinear Differential Equations in Abstract Spaces, Pergamon Press, New York, 1981.
    [17] Y. Li, Existence of solutions of initial value problems for abstract semilinear evolution equations, Acta Math. Sin., 48 (2005), 1089-1094 (in Chinese). 
    [18] M. LiC. Chen and F. B. Li, On fractional powers of generators of fractional resolvent families, J. Funct. Anal., 259 (2010), 2702-2726. 
    [19] K. LiJ. Peng and J. Jia, Cauchy problems for fractional differential equations with Riemann-Liouville fractional derivatives, J. Funct. Anal., 263 (2012), 476-510. 
    [20] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, in: North-Holland Mathematics Studies, vol. 204, Elsevier Science B. V., Amsterdam, 2006.
    [21] Z. MeiJ. Peng and Y. Zhang, An operator theoretical approach to Riemann-Liouville fractional Cauchy problem, Math. Nachr., 288 (2015), 784-797. 
    [22] Z. Ouyang, Existence and uniqueness of the solutions for a class of nonlinear fractional order partial differential equations with delay, Comput. Math. Appl., 61 (2011), 860-870. 
    [23] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-verlag, Berlin, 1983.
    [24] M. H. M. Rashid and A. Al-Omari, Local and global existence of mild solutions for impulsive fractional semi-linear integro-differential equation, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 3493-3503. 
    [25] H. Tanabe, Functional Analytic Methods for Partial Differential Equations, Marcel Dekker, New York, USA, 1997.
    [26] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, second ed., Springer-verlag, New York, 1997.
    [27] R. N. WangD. H. Chen and T. J. Xiao, Abstract fractional Cauchy problems with almost sectorial operators, J. Differential Equations, 252 (2012), 202-235. 
    [28] R. N. WangT. J. Xiao and J. Liang, A note on the fractional Cauchy problems with nonlocal conditions, Appl. Math. Lette., 24 (2011), 1435-1442. 
    [29] J. Wang and Y. Zhou, A class of fractional evolution equations and optimal controls, Nonlinear Anal. Real World Appl., 12 (2011), 262-272. 
    [30] J. WangY. Zhou and M. Fečkan, Abstract Cauchy problem for fractional differential equations, Nonlinear Dyn., 74 (2013), 685-700. 
    [31] Y. Zhou and F. Jiao, Existence of mild solutions for fractional neutral evolution equations, Comput. Math. Appl., 59 (2010), 1063-1077. 
    [32] B. ZhuL. Liu and Y. Wu, Local and global existence of mild solutions for a class of nonlinear fractional reaction-diffusion equations with delay, Appl. Math. Lett., 61 (2016), 73-79. 
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