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On mild solutions of the generalized nonlinear fractional pseudo-parabolic equation with a nonlocal condition

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Abstract

The aim of this paper is to investigate the existence of mild solutions of the generalized nonlinear fractional pseudo-parabolic equation with a nonlocal condition. Unlike previous papers, in the current paper, we assume that the source function of the problem may have a singularity. The existence results are proven via the Schaefer, nonlinear Leray–Schauder alternatives and Banach fixed point theorems. Further, the continuity of mild solutions in terms of inputs (fractional orders, nonlocal condition, and associated parameters) is extensively discussed.

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References

  1. Aldoghaither, A., Liu, D., Laleg-Kirati, T.: Modulating functions based algorithm for the estimation of the coefficients and differentiation order for a space-fractional advection-dispersion equation. SIAM J. Sci. Comput. 37(6), A2813–A2839 (2015)

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  3. 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(1220), 47–78 (1972)

  4. Boucherif, A.: Semilinear evolution inclusions with nonlocal conditions. Appl. Math. Lett. 22, 1145–1149 (2009)

    Article  MathSciNet  Google Scholar 

  5. Byszewski, L.: Strong maximum and minimum principles for parabolic problems with nonlocal inequalities. Z. Angew. Math. Mech. 70(3), 202–206 (1990)

    Article  MathSciNet  Google Scholar 

  6. Byszewski, L.: Strong maximum principles for parabolic nonlinear problems with nonlocal inequalities together with integrals. J. Appl. Math. Stochas. Anal. 3(5), 65–79 (1990)

    Article  MathSciNet  Google Scholar 

  7. Byszewski, L.: Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem. J. Math. Appl. Anal. 162, 494–505 (1991)

    Article  MathSciNet  Google Scholar 

  8. Byszewski, L.: Application of properties of the right hand sides of evolution equations to an investigation of nonlocal evolution problems. Nonlinear Anal. 33, 413–426 (1998)

    Article  MathSciNet  Google Scholar 

  9. Byszewski, L., Lakshmikantham, V.: Theorem about the existence and uniqueness of solutions of a nonlocal Cauchy problem in a Banach space. Appl. Anal. 40, 11–19 (1990)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  11. Cheng, J., Nakagawa, J., Yamamoto, M., Yamazaki, T.: Uniqueness in an inverse problem for a one-dimension fractional diffusion equation. Inverse Probl. 25(11), Art. 115002 (2009)

  12. Delbosco, D., Rodino, L.: Existence and uniqueness for a nonlinear fractional differential equation. J. Math. Anal. Appl. 204, 609–625 (1996)

    Article  MathSciNet  Google Scholar 

  13. Dien, N.M.: Generalized weakly singular Gronwall-type inequalities and their applications to fractional differential equations. Rocky Mountain J. Math. 51(2), 689–707 (2021). https://doi.org/10.1216/rmj.2021.51.689

    Article  MathSciNet  MATH  Google Scholar 

  14. Dien, N.M.: Existence and continuity results for a nonlinear fractional Langevin equation with a weakly singular source. J. Integral Equ. Appl. 33(3), 349–369 (2021). https://doi.org/10.1216/jie.2021.33.349

    Article  MathSciNet  Google Scholar 

  15. Dien, N.M., Trong, D.D.: On the nonlinear generalized Langevin equation involving \(\psi \)-Caputo fractional derivatives. Fractals 29(06), Art. 2150128 (2021). https://doi.org/10.1142/S0218348X21501280

  16. Dien, N.M., Trong, D.D.: Stability of solutions of a class of nonlinear fractional diffusion equations with respect to a pseudo-differential operator. Math. Methods Appl. Sci. 42(8), 2513–2532 (2019). https://doi.org/10.1002/mma.5528

    Article  MathSciNet  MATH  Google Scholar 

  17. Dien, N.M., Nane, E., Trong, D.D.: The nonlinear fractional diffusion equations with Nagumo-type sources and perturbed orders. arXiv: 2002.06747

  18. Gopala Rao, V.R., Ting, T.W.: Solutions of pseudo-heat equations in the whole space. Arch. Ration. Mech. Anal. 49 (1972/1973)

  19. Granas, A., Dugundji, J.: Fixed Point Theory. Springer, New York (2003)

    Book  Google Scholar 

  20. Kateregga, M., Mataramvura, S., Taylor, D.: Parameter estimation for stable distributions with application to commodity futures log-returns. Cogent Econ. Finance 5(1), Art. 1318813 (2017)

  21. Padron, V.: Effect of aggregation on population recovery modeled by a forward-backward pseudoparabolic equation. Trans. Am. Math. Soc. 356(7), 2739–2756 (2004)

    Article  MathSciNet  Google Scholar 

  22. Sakamoto, K., Yamamoto, M.: Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems. J. Math. Appl. Anal. 382(1), 426–447 (2011)

    Article  MathSciNet  Google Scholar 

  23. Schdmügen, K.: Unbounded Self-adjoint Operators on Hilbert Space. Springer, Dordrecht (2012)

    Book  Google Scholar 

  24. Schneider, W.R.: Completely monotone generalized Mittag–Leffler functions. Expo. Math. 14(1), 3–16 (1996)

    MathSciNet  MATH  Google Scholar 

  25. Showalter, R.E., Ting, T.W.: Pseudoparabolic partial differential equations. SIAM J. Math. Anal. 1(1), 1–26 (1970)

  26. Sin, C.S., Zheng, L.: Existence and uniqueness of global solutions of Caputo-type fractional differential equations. Fract. Calc. Appl. Anal. 19(3), 765–774 (2016). https://doi.org/10.1515/fca-2016-0040

    Article  MathSciNet  MATH  Google Scholar 

  27. Thanh, B.L.T.: On the well-posedness of a spectral fractional forward-backward pseudo-parabolic equation. Comput. Math. Appl. 77(3), 323–333 (2019)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  29. Trong, D.D., Nane, E., Minh, N.D., Tuan, N.H.: Continuity of solutions of a class of fractional equations. Potential Anal. 49(3), 423–478 (2018)

    Article  MathSciNet  Google Scholar 

  30. Trong, D.D., Hai, D.N.D., Dien, N.M.: On a time-space fractional backward diffusion problem with inexact orders. Comput. Math. Appl. 78(5), 1572–1593 (2019)

    Article  MathSciNet  Google Scholar 

  31. Trong, D.D., Dien, N.M., Viet, T.Q.: Global solution of space-fractional diffusion equations with nonlinear reaction source terms. Appl. Anal. 99(15), 2709–2739 (2020). https://doi.org/10.1080/00036811.2019.1582030

    Article  MathSciNet  MATH  Google Scholar 

  32. Tuan, N.H., Ngoc, T.B., Huynh, L.N., Kirane, M.: Existence and uniqueness of mild solution of time-fractional semilinear differential equations with a nonlocal final condition. Comput. Math. Appl. 78(5), 1651–1668 (2019)

    Article  MathSciNet  Google Scholar 

  33. 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 

  34. Tuan, N.H., Au, V.V., Xu, R.: Semilinear Caputo time-fractional pseudo-parabolic equations. Commun. Pure Appl. Anal. 20(2), 583–621 (2021). https://doi.org/10.3934/cpaa.2020282

    Article  MathSciNet  MATH  Google Scholar 

  35. Viet, T.Q., Dien, N.M., Trong, D.D.: Stability of solutions of a class of nonlinear fractional Laplacian parabolic problems. J. Comput. Appl. Math. 355, 51–76 (2019)

    Article  MathSciNet  Google Scholar 

  36. Webb, J.R.L.: Weakly singular Gronwall inequalities and applications to fractional differential equations. J. Math. Anal. Appl. 471, 692–711 (2019)

    Article  MathSciNet  Google Scholar 

  37. Xiao, T.J., Liang, J.: Existence of classical solutions to nonautonomous nonlocal parabolic problems. Nonlinear Anal. 63, e225–e232 (2005)

    Article  Google Scholar 

  38. Xie, M., Tan, Z., Wu, Z.: Local existence and uniqueness of weak solutions to fractional pseudo-parabolic equation with singular potential. Appl. Math. Lett. 114, Art. 106898 (2021)

  39. Yang, M., Wang, Q.: Existence of mild solutions for a class of Hiffer fractional evolution equations with nonlocal conditions. Fract. Calc. Appl. Anal. 20(3), 679–705 (2017). https://doi.org/10.1515/fca-2017-0036

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

We are very grateful to Editor-in-Chief and the anonymous reviewers for their constructive comments and valuable suggestions, which helped us to improve the manuscript.

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  1. Faculty of Education, Thu Dau Mot University, Thu Dau Mot City, Binh Duong Province, Vietnam

    Nguyen Minh Dien

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  1. Nguyen Minh Dien
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Dien, N.M. On mild solutions of the generalized nonlinear fractional pseudo-parabolic equation with a nonlocal condition. Fract Calc Appl Anal 25, 559–583 (2022). https://doi.org/10.1007/s13540-022-00024-4

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  • DOI: https://doi.org/10.1007/s13540-022-00024-4

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