Nonexistence of global solutions for a fractional system of strongly coupled integro-differential equations

In this paper, we study the nonexistence of nontrivial global solutions for a system of two strongly coupledfractional differential equations. Each equation involves two fractional derivatives and a nonlinear source term. Thefractional derivatives are of Caputo type of subfirst orders. The nonlinear sources are nonlocal in time. They have theform of a convolution of a polynomial of the state with a (possibly singular) kernel. The system under considerationis a generalization of many interesting special systems of equations whose solutions do not exist globally in time. Weestablish some criteria under which no nontrivial global solutions exist. Several integral inequalities and estimations arederived and the test function method is adopted. Special cases and examples are given to illustrate the results.

PDF

___

[1] Aghajani A, Jalilian Y, Trujillo JJ. On the existence of solutions of fractional integro-differential equations. Fractional Calculus and Applied Analysis 2012; 15 (1): 44-69. doi: 10.2478/s13540-012-0005-4
[2] Ahmad AM, Furati KM, Tatar NE. On the nonexistence of global solutions for a class of fractional integro-differential problems. Advances in Difference Equations 2017; 2017(1): 59. doi: 10.1186/s13662-017-1105-6
[3] Alsaedi A, Ahmad B, Kirane M, Al Musalhi F, Alzahrani F. Blowing-up solutions for a nonlinear time-fractional system. Bulletin of Mathematical Sciences 2017; 7 (2), 201-10. doi: 10.1007/s13373-016-0087-0
[4] Bai Z, Chen Y, Lian H, Sun S. On the existence of blow up solutions for a class of fractional differential equations. Fractional Calculus and Applied Analysis 2014; 17 (4), 1175-1187. doi: 10.2478/s13540-014-0220-2
[5] Furati KM, Tatar NE. An existence result for a nonlocal fractional differential problem. Journal of Fractional Calculus 2004; 26: 43-51.
[6] Furati KM, Kirane M. Necessary conditions for the existence of global solutions to systems of fractional differential equations. Fractional Calculus and Applied Analysis 2008; 11 (3): 281-298.
[7] Henderson J, Luca R. Nonexistence of positive solutions for a system of coupled fractional boundary value problems. Boundary Value Problems 2015; 2015 (1): 138. doi: 10.1186/s13661-015-0403-8
[8] Kadem A, Kirane M, Kirk MC, Olmstead WE. Blowing-up solutions to systems of fractional differential and integral equations with exponential non-linearities. IMA Journal of Applied Mathematics 2014; 79 (6): 1077-1088.
[9] Kassim MD, Furati KM, Tatar NE. Non-existence for fractionally damped fractional differential problems. Acta Mathematica Scientia 2017; 37 (1):119-30. doi: 10.1016/S0252-9602(16)30120-5
[10] Kilbas AA, Srivastava HM, Trujillo JJ. Theory and Applications of Fractional Differential Equations. Amsterdam, the Netherlands: Elsevier, 2006.
[11] Kirane M, Ahmad B, Alsaedi A, Al-Yami M. Non-existence of global solutions to a system of fractional diffusion equations. Acta Applicandae Mathematicae 2014; 133 (1): 235-248. doi: 10.1007/s10440-014-9865-4
[12] Kirane M, Medved’ M, Tatar NE. On the nonexistence of blowing-up solutions to a fractional functional differential equations. Georgian Mathematical Journal 2012; 19 (1): 127-144. doi: 10.1515/gmj-2012-0006
[13] Kirane M, Malik SA. The profile of blowing-up solutions to a nonlinear system of fractional differential equations. Nonlinear Analysis: Theory, Methods & Applications 2010; 73 (12): 3723-3736. doi: 10.1016/j.na.2010.06.088
[14] Ma J. Blow-up solutions of nonlinear Volterra integro-differential equations. Mathematical and Computer Modelling 2011; 54 (11-12): 2551-2559. doi: 10.1016/j.mcm.2011.06.020
[15] Mennouni A, Youkana A. Finite time blow-up of solutions for a nonlinear system of fractional differential equations. Electronic Journal of Differential Equations 2017; 2017 (152), 1-15.
[16] Mitidieric E, Pohozaev SI. A priori estimates and blow-up of solutions to nonlinear partial differential equations and inequalities. Trudy Matematicheskogo Instituta Imeni VA Steklova 2001; 234: 1-383.
[17] Samko SG, Kilbas AA, Marichev OI. Fractional Integrals and Derivatives: Theory and Applications. Yverdon-les- Bains, Switzerland: Gordon and Breach Science Publishers, Yverdon; 1993.
[18] Tatar NE. Existence results for an evolution problem with fractional nonlocal conditions. Computers & Mathematics with Applications 2010; 60 (11): 2971-2982. doi: 10.1016/j.camwa.2010.09.057

ISSN: 1300-0098
Yayın Aralığı: Yılda 6 Sayı
Yayıncı: TÜBİTAK

Arşiv