Skip to main content
SpringerLink
Log in

Well-posedness and ill-posedness results for backward problem for fractional pseudo-parabolic equation

  • Original Research
  • Published:
Journal of Applied Mathematics and Computing Aims and scope Submit manuscript

Abstract

In this paper, we study a pseudo-parabolic equation with the Caputo fractional derivative. By applying the properties of Mittag–Leffler functions and the method of eigenvalue expansion, under a suitable definition of mild solution of our problem, we obtain the existence result and \(L^p\) regularity of the mild solution by using some Sobolev embeddings. Finally, we also give some examples to illustrate the proposed method.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  1. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Application of Fractional Differential Equations, North-Holland Mathematics Studies, vol. 204. Elsevier Science B.V, Amsterdam (2006)

    Google Scholar 

  2. Kiryakova, V.: Generalized Fractional Calculus and Applications, Pitman Research Notes in Mathematics 301. Longman, Harlow (1994)

    Google Scholar 

  3. Podlubny, I.: Fractional Differential Equations, Mathematics in Science and Engineering, vol. 198. Academic Press Inc, San Diego, CA (1990)

    Google Scholar 

  4. Li, L., Liu, G.J.: A generalized definition of Caputo derivatives and its application to fractional ODEs. SIAM J. Math. Anal. 50(3), 2867–2900 (2018)

    Article  MathSciNet  Google Scholar 

  5. Barenblat, G.I., Kochiva, I.: Basic concepts in the theory of seepage of homogeneous liquids in fissured rock. J. Appl. Math. Mech. 24(5), 1286–1303 (1960)

    Article  Google Scholar 

  6. Benjamin, T.B., Bona, J.L., Mahony, J.J.: Model equations for long waves in nonlinear dispersive systems. Philos. Trans. R. Soc. Lond. Ser. A 272, 47–78 (1972)

    Article  MathSciNet  Google Scholar 

  7. Ting, W.T.: Certain non-steady flows of second-order fluids Arch. Ration. Mech. Anal. 14, 1–26 (1963)

    Article  Google Scholar 

  8. Padron, V.: Effect of aggregation on population recovery modeled by a forward-backward pseudoparabolic equation Trans. Am. Math. Soc. 356, 2739–2756 (2004)

    Article  MathSciNet  Google Scholar 

  9. Cao, Y., Yin, J., Wang, C.: Cauchy problems of semilinear pseudo-parabolic equations. J. Differ. Equ. 246, 4568–4590 (2009)

    Article  MathSciNet  Google Scholar 

  10. Chen, H., Xu, H.: Global existence and blow-up of solutions for infinitely degenerate semilinear pseudo-parabolic equations with logarithmic nonlinearity. Discrete Contin. Dyn. Syst. 39(2), 1185–1203 (2019)

    Article  MathSciNet  Google Scholar 

  11. Chen, H., Tian, S.: Initial boundary value problem for a class of semilinear pseudo-parabolic equations with logarithmic nonlinearity. J. Differ. Equ. 258(12), 4424–4442 (2015)

    Article  MathSciNet  Google Scholar 

  12. Ding, H., Zhou, J.: Global existence and blow-up for a mixed pseudo-parabolic p-Laplacian type equation with logarithmic nonlinearity. J. Math. Anal. Appl. 478, 393–420 (2019)

    Article  MathSciNet  Google Scholar 

  13. Jin, L., Li, L., Fang, S.: The global existence and time-decay for the solutions of the fractional pseudo-parabolic equation. Comput. Math. Appl. 73(10), 2221–2232 (2017)

    Article  MathSciNet  Google Scholar 

  14. Zhu, X., Li, F., Li, Y.: Global solutions and blow up solutions to a class of pseudo-parabolic equations with nonlocal term. Appl. Math. Comput. 329, 38–51 (2018)

    MathSciNet  MATH  Google Scholar 

  15. Abbaszadeh, M., Dehghan, M.: Analysis of mixed finite element method (MFEM) for solving the generalized fractional reaction-diffusion equation on nonrectangular domains. Comput. Math. Appl. 78(5), 1531–1547 (2019)

    Article  MathSciNet  Google Scholar 

  16. Abbaszadeh, M., Dehghan, M.: Numerical and analytical investigations for neutral delay fractional damped diffusion-wave equation based on the stabilized interpolating element free Galerkin (IEFG) method. Appl. Numer. Math. 145, 488–506 (2019)

    Article  MathSciNet  Google Scholar 

  17. Abbaszadeh, M., Dehghan, M.: A reduced order finite difference method for solving space-fractional reaction-diffusion systems: the Gray–Scott model. Eur. Phys. J. Plus 134, 620 (2019)

    Article  Google Scholar 

  18. Aissania, K., Benchohrab, M., Nieto, J.J.: Controllability for impulsive fractional evolution inclusions with state-dependent delay. Adv. Theory Nonlinear Anal. Appl. 3(1), 18–34 (2019)

    Google Scholar 

  19. Kaddoura, T.H., Hakema, A.: Sufficient conditions of non global solution for fractional damped wave equations with non-linear memory. Adv. Theory Nonlinear Anal. Appl. 2(4), 224–237 (2018)

    Google Scholar 

  20. Sakamoto, K., Yamamoto, M.: Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems. J. Math. Anal. Appl. 382, 426–447 (2011)

    Article  MathSciNet  Google Scholar 

  21. Tuan, N.H., Huynh, L.N., Ngoc, T.B., Zhou, Y.: On a backward problem for nonlinear fractional diffusion equations. Appl. Math. Lett. 92, 76–84 (2019)

    Article  MathSciNet  Google Scholar 

  22. Zhou, Y., He, J.W., Ahmad, B., Tuan, N.H.: Existence and regularity results of a backward problem for fractional diffusion equations. Math. Meth. Appl. Sci. 42(18), 5945–7545 (2019)

    Article  MathSciNet  Google Scholar 

  23. Tuan, N.H., Debbouche, A., Ngoc, T.B.: Existence and regularity of final value problems for time fractional wave equations. Comput. Math. Appl. 78(5), 1396–1414 (2019)

    Article  MathSciNet  Google Scholar 

  24. Beshtokov, M.K.: To boundary-value problems for degenerating pseudoparabolic equations with Gerasimov–Caputo fractional derivative. Izv. Vyssh. Uchebn. Zaved. Mat. 10, 3–16 (2018)

    MathSciNet  MATH  Google Scholar 

  25. Beshtokov, M.K.: Boundary-value Problems for Loaded Pseudoparabolic Equations of Fractional Order and Difference Methods of Their Solving, Russian Mathematics. Springer, Berlin (2019)

    MATH  Google Scholar 

  26. Beshtokov, M.K.: Boundary value problems for a pseudoparabolic equation with the Caputo fractional derivative. Differ. Uravn. 55(7), 919–928 (2019)

    MathSciNet  MATH  Google Scholar 

  27. Beshtokov, M.K.: Boundary value problems for a pseudoparabolic equation with the Caputo fractional derivative. Differ. Equ. 55(7), 884–893 (2019)

    Article  MathSciNet  Google Scholar 

  28. Sousa, C.V.J., de Oliveira, C.E.: Fractional order pseudoparabolic partial differential equation: Ulam–Hyers stability Bull. Braz. Math. Soc. (N.S.) 50(2), 481–496 (2019)

    Article  MathSciNet  Google Scholar 

  29. Tatara, S., Ulusoy, S.: An inverse source problem for a one-dimensional space-time fractional diffusion equation. Appl. Anal. (2014). https://doi.org/10.1080/00036811.2014.979808

    Article  Google Scholar 

  30. Arendt, W., ter Elst, A.F.M., Warma, M.: Fractional powers of sectorial operators via the Dirichlet-to-Neumann operator. Commun. Partial Differ. Equ. 43(1), 1–24 (2018)

    Article  MathSciNet  Google Scholar 

  31. Arrieta, J.M., Carvalho, A.N.: Abstract parabolic problems with critical nonlinearities and applications to Navier–Stokes and heat equations. Trans. Am. Math. Soc. 352(1), 285–310 (1999)

    Article  MathSciNet  Google Scholar 

  32. Tuan, N.H., Long, L.D., Thinh, V.N., Thanh, T.: On a final value problem for the time-fractional diffusion equation with inhomogeneous source. Inverse Probl. Sci. Eng. (2016). https://doi.org/10.1080/17415977.2016.1259316

    Article  MATH  Google Scholar 

  33. Courant, R., Hilbert, D.: Mehtod of Mathematical Physics. Interscience, New York (NY) (1953)

    MATH  Google Scholar 

  34. Feng, X.L., Elden, L., Fu, C.L.: A quasi-boundary-value method for the Cauchy problem for elliptic equations with nonhomogeneous Neumann data. J. Inverse Ill-Posed Probl. 6, 617–645 (2010)

    MathSciNet  MATH  Google Scholar 

  35. Feng, X.L., Ning, W., Qian, Z.: A quasi-boundary-value method for a Cauchy problem of an elliptic equation in multiple dimensions. Inverse Probl. Sci. Eng. 7, 1045–1061 (2014)

    Article  MathSciNet  Google Scholar 

  36. Zhang, H.W.: Modified quasi-boundary value method for a Cauchy problem of semi-linear elliptic equation. Int. J. Comput. Math. 89(12), 1689–1703 (2012)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work is supported by Thu Dau Mot University.

Author information

Authors and Affiliations

  1. Division of Applied Mathematics, Thu Dau Mot University, Thu Dau Mot, Binh Duong Province, Vietnam

    Le Dinh Long & Nguyen Huy Tuan

  2. Faculty of Information Technology, Macau University of Science and Technology, Macau, 999078, China

    Yong Zhou

  3. Faculty of Mathematics and Computational Science, Xiangtan University, Hunan, 411105, China

    Yong Zhou

  4. Department of Applied Mathematics, Bharathiar University, Coimbatore, 641046, India

    Rathinasamy Sakthivel

  5. Faculty of Mathematics and Computer Science, University of Science, Vietnam National University, 227 Nguyen Van Cu, District 5, Ho Chi Minh City, Vietnam

    Nguyen Huy Tuan

Authors
  1. Le Dinh Long
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yong Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Rathinasamy Sakthivel
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Nguyen Huy Tuan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nguyen Huy Tuan.

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

Long, L.D., Zhou, Y., Sakthivel, R. et al. Well-posedness and ill-posedness results for backward problem for fractional pseudo-parabolic equation. J. Appl. Math. Comput. 67, 175–206 (2021). https://doi.org/10.1007/s12190-020-01488-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12190-020-01488-4

Keywords

Mathematics Subject Classification

Search

Navigation