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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 |
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 [
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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.
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E. G. Bajlekova, Fractional Evolution Equations in Banach Spaces, Ph. D thesis, Department of Mathematics, Eindhoven University of Technology, 2001. |
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J. |
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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.
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[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.
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[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.
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[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.
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[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.
|
show all references
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. |
[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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