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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

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

* Corresponding author

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.

Citation: Pengyu Chen, Xuping Zhang, Yongxiang Li. A blowup alternative result for fractional nonautonomous evolution equation of Volterra type. Communications on Pure and Applied Analysis, 2018, 17 (5) : 1975-1992. doi: 10.3934/cpaa.2018094
##### References:
 [1] R. P. Agarwal, M. 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. [3] J. and K. Goebel, Measures of Noncompactness in Banach Spaces, In Lecture Notes in Pure and Applied Mathematics, Volume 60, Marcel Dekker, New York, 1980. [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-Borai, K. 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. Li, C. Chen and F. B. Li, On fractional powers of generators of fractional resolvent families, J. Funct. Anal., 259 (2010), 2702-2726. [19] K. Li, J. 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. Mei, J. 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. Wang, D. H. Chen and T. J. Xiao, Abstract fractional Cauchy problems with almost sectorial operators, J. Differential Equations, 252 (2012), 202-235. [28] R. N. Wang, T. 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. Wang, Y. 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. Zhu, L. 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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##### References:
 [1] R. P. Agarwal, M. 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. [3] J. and K. Goebel, Measures of Noncompactness in Banach Spaces, In Lecture Notes in Pure and Applied Mathematics, Volume 60, Marcel Dekker, New York, 1980. [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-Borai, K. 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. Li, C. Chen and F. B. Li, On fractional powers of generators of fractional resolvent families, J. Funct. Anal., 259 (2010), 2702-2726. [19] K. Li, J. 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. Mei, J. 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. Wang, D. H. Chen and T. J. Xiao, Abstract fractional Cauchy problems with almost sectorial operators, J. Differential Equations, 252 (2012), 202-235. [28] R. N. Wang, T. 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. Wang, Y. 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. Zhu, L. 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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