Skip to main content
SpringerLink
Log in

Nonexistence of Global Solutions for Fractional Differential Problems with Power Type Source Term

  • Published:
Mediterranean Journal of Mathematics Aims and scope Submit manuscript

Abstract

In this paper, we study the nonexistence of nontrivial global solutions for a nonlinear fractional differential. The equation involves two Hadamard fractional derivatives of different orders and a nonlinear source term. Our results are obtained using the test-function method, several properties of fractional derivatives and some integral inequalities. An example is provided to illustrate our findings.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Agarwal, R.P., Benchohra, M., Hamani, S.: A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions. Acta Appl. Math. 109(3), 973–1033 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  2. Agarwal, R.P., Belmekki, M., Benchohra, M.: A survey on semilinear differential equations and inclusions involving Riemann-Liouville fractional derivative. Adv. Differ. Equations 2009(1), 1–47 (2009)

    MathSciNet  MATH  Google Scholar 

  3. Butzer, P.L., Kilbas, A.A., Trujillo, J.J.: Mellin transform analysis and integration by parts for Hadamard-type fractional integral. J. Math. Anal. Appl. 270(1), 1–15 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  4. Furati, K.M., Tatar, N.-E.: An existence result for a nonlocal fractional differential problem. J. Fract. Calc. 26, 43–51 (2004)

    MathSciNet  MATH  Google Scholar 

  5. Furati, K.M., Tatar, N.-E.: Behavior of solutions for a weighted Cauchy-type fractional differential problem. J. Fract. Calc. 28, 23–42 (2005)

    MathSciNet  MATH  Google Scholar 

  6. Furati, K.M., Kassim, M.D., Tatar, N.-E.: Existence and uniqueness for a problem involving Hilfer fractional derivative. Comput. Math. Appl. 64, 1616–1626 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. Furati, K.M., Kassim, M.D., Tatar, N.-E.: Non-existence of global solutions for a differential equation involving Hilfer fractional derivative. Electron. J. Differ. Equations 2013(235), 1–10 (2013)

    MathSciNet  MATH  Google Scholar 

  8. Kassim, M.D., Ali, S.M., Abdo, M.S., Jarad, F.: Nonexistence results of Caputo-type fractional problem. Adv. Differ. Equations 2021, 1–12 (2021)

    Article  MathSciNet  Google Scholar 

  9. Kassim, M.D., Furati, K.M., Tatar, N.-E.: On a differential equation involving Hilfer–Hadamard fractional derivative. Abstr. Appl. Anal. 2012, 1–17 (2012)

    MathSciNet  MATH  Google Scholar 

  10. Kassim, M.D., Tatar, N.-E.: Well-posedness and stability for a differential problem with Hilfer–Hadamard fractional derivative. Abstr. Appl. Anal. 2013, 1–12 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kassim, M.D., Furati, K.M., Tatar, N.-E.: Asymptotic behavior of solutions to nonlinear fractional differential equations. Math. Model. Anal. 21(5), 610–629 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kassim, M.D., Furati, K.M., Tatar, N.-E.: Asymptotic behavior of solutions to nonlinear initial-value fractional differential problems. Electron. J. Differ. Equations 2016(291), 1–14 (2016)

    MathSciNet  MATH  Google Scholar 

  13. Kassim, M.D., Furati, K.M., Tatar, N.-E.: Nonexistence of global solutions for a fractional differential problem. J. Comput. Appl. Math. 314, 61–68 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kassim, M.D., Furati, K.M., Tatar, N.-E.: Non-existence for fractionally damped fractional differential problems. Acta Math. Sci. 37(1), 119–130 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kassim, M.D., Tatar, N.-E.: Stability of logarithmic type for a Hadamard fractional differential problem. J. Pseudo Differ. Oper. Appl. 11(1), 447–466 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kassim, M.D., Tatar, N.-E.: Convergence of solutions of fractional differential equations to power-type functions. Electron. J. Differ. Equations 2020(111), 1–14 (2020)

    MathSciNet  MATH  Google Scholar 

  17. Kassim, M.D., Tatar, N.-E.: Asymptotic behavior of solutions of fractional differential equations with Hadamard fractional derivatives. Fract. Calc. Appl. Anal. 24(2), 483–508 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations, vol. 204. Elsevier, Oxford (2006)

  19. Kilbas, A.A., Marichev, O.I., Samko, S.G.: Fractional Integral and Derivatives (Theory and Applications). Gordon and Breach, Switzerland (1993)

    MATH  Google Scholar 

  20. Kirane, M., Medved, M., Tatar, N.-E.: On the nonexistence of blowing-up solutions to a fractional functional differential equations. Georgian Math. J. 19, 127–144 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kirane, M., Tatar, N.-E.: Nonexistence of solutions to a hyperbolic equation with a time fractional damping. Z. für Anal. und ihre Anwendung. 25, 131–142 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kirane, M., Tatar, N.-E.: Absence of local and global solutions to an elliptic system with time-fractional dynamical boundary conditions. Sib. J. Math. 48(3), 477–488 (2007)

    Article  MATH  Google Scholar 

  23. Kirane, M., Laskri, Y., Tatar, N.-E.: Critical exponents of Fujita type for certain evolution equations and systems with spatio-temporal fractional derivatives. J. Math. Anal. Appl. 312(2), 488–501 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  24. Laskri, Y., Tatar, N.-E.: The critical exponent for an ordinary fractional differential problem. Comput. Math. Appl. 59, 1266–1270 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  25. Messaoudi, S.A., Said-Houari, B., Tatar, N.-E.: Global existence and asymptotic behavior for a fractional differential equation. Appl. Math. Comput. 188(2), 1955–1962 (2007)

    MathSciNet  MATH  Google Scholar 

  26. Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)

    MATH  Google Scholar 

  27. Mitidieri, E., Pohozaev, S.I.: A priori estimates and blow-up of solutions to non-linear partial differential equations and inequalities. Proc. Steklov Inst. Math. 234, 3–383 (2001)

    MATH  Google Scholar 

  28. Oldham, K.B., Spanier, J.: The Fractional Calculus Theory and Applications of Differentiation and Integration to Arbitrary Order. Elsevier, Amsterdam (1974)

    MATH  Google Scholar 

  29. Podlubny, I.: Fractional Differential Equations: an Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, vol. 198. Elsevier, Amsterdam (1998)

  30. Tatar, N.-E.: Nonexistence results for a fractional problem arising in thermal diffusion in fractal media. Chaos Solitons Fract. 36, 1205–1214 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  31. Tatar, N.-E.: Existence results for an evolution problem with fractional nonlocal conditions. Comput. Math. Appl. 60, 2971–2982 (2010)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors are very grateful to the anonymous referees who helped in improving the first version of the paper by their valuable comments and suggestions. The first author is grateful to Imam Abdulrahman Bin Faisal University for the facilities and support provided. The second author would like to thank King Fahd University of Petroleum and Minerals for the financial support provided through project no. SB191023.

Author information

Authors and Affiliations

  1. Department of Basic Engineering Sciences, College of Engineering, Imam Abdulrahman Bin Faisal University, P.O. Box 1982, Dammam, 34151, Saudi Arabia

    Mohammed D. Kassim

  2. Department of Mathematics, Interdisciplinary Research Center for Intelligent Manufacturing and Robotics, King Fahd University of Petroleum and Minerals, Dhahran, 31261, Saudi Arabia

    Nasser-Eddine Tatar

Authors
  1. Mohammed D. Kassim
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nasser-Eddine Tatar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mohammed D. Kassim.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kassim, M.D., Tatar, NE. Nonexistence of Global Solutions for Fractional Differential Problems with Power Type Source Term. Mediterr. J. Math. 18, 238 (2021). https://doi.org/10.1007/s00009-021-01903-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00009-021-01903-5

Keywords

Mathematics Subject Classification

Search

Navigation