Research article

The critical exponents for a semilinear fractional pseudo-parabolic equation with nonlinear memory in a bounded domain

  • Received: 04 January 2023 Revised: 22 February 2023 Accepted: 26 February 2023 Published: 06 March 2023
  • This paper considers blow-up and global existence for a semilinear space-time fractional pseudo-parabolic equation with nonlinear memory in a bounded domain. We determine the critical exponents of the Cauchy problem when $ \alpha < \gamma $ and $ \alpha\ge \gamma, $ respectively. The results obtained in this study are noteworthy extension to the results of time-fractional differential equation. The critical exponent is consistent with the corresponding Cauchy problem for the time-fractional differential equation with nonlinear memory, which illustrates that the diffusion effect of the third order term is not strong enough to change the critical exponents.

    Citation: Yaning Li, Yuting Yang. The critical exponents for a semilinear fractional pseudo-parabolic equation with nonlinear memory in a bounded domain[J]. Electronic Research Archive, 2023, 31(5): 2555-2567. doi: 10.3934/era.2023129

    Related Papers:

  • This paper considers blow-up and global existence for a semilinear space-time fractional pseudo-parabolic equation with nonlinear memory in a bounded domain. We determine the critical exponents of the Cauchy problem when $ \alpha < \gamma $ and $ \alpha\ge \gamma, $ respectively. The results obtained in this study are noteworthy extension to the results of time-fractional differential equation. The critical exponent is consistent with the corresponding Cauchy problem for the time-fractional differential equation with nonlinear memory, which illustrates that the diffusion effect of the third order term is not strong enough to change the critical exponents.



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    [1] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, London, 2010. https://doi.org/10.1142/p614
    [2] R. Metzler, J. Klafter, The random walks guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), 1–77. https://doi.org/10.1016/S0370-1573(00)00070-3 doi: 10.1016/S0370-1573(00)00070-3
    [3] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Preface, North-Holland Math. Stud., 204 (2006).
    [4] E. Orsingher, L. Beghin, Fractional diffusion equations and processes with randomly varying time, Ann. Probab., 37 (2009), 206–249. https://doi.org/10.1214/08-AOP401 doi: 10.1214/08-AOP401
    [5] B. B. Mandelbrot, J. W. V. Ness, Fractional Brownian motions, fractional noises and applications, SIAM Rev., 10 (1968), 422–437. https://doi.org/10.1137/1010093 doi: 10.1137/1010093
    [6] W. Chen, C. Li, Maximum principles for the fractional p-Laplacian and symmetry of solutions, Adv. Math., 335 (2018), 735–758. https://doi.org/10.1016/j.aim.2018.07.016 doi: 10.1016/j.aim.2018.07.016
    [7] L. Li, J. G. Liu, L. Wang, Cauchy problems for Keller-Segel type time-space fractional diffusion equation, J. Differ. Equations, 265 (2018), 1044–1096. https://doi.org/10.1016/j.jde.2018.03.025 doi: 10.1016/j.jde.2018.03.025
    [8] H. Dong, D. Kim, Lp-estimates for time fractional parabolic equations with coefficients measurable in time, Adv. Math., 345 (2019), 289–345. https://doi.org/10.1016/j.aim.2019.01.016 doi: 10.1016/j.aim.2019.01.016
    [9] E. C. Aifantis, On the problem of diffusion in solids, Acta Mech., 37 (1980), 265–296. https://doi.org/10.1007/BF01202949 doi: 10.1007/BF01202949
    [10] K. Kuttler, E. C. Aifantis, Quasilinear evolution equations in nonclassical diffusion, SIAM J. Math. Anal., 19 (1988), 110–120. https://doi.org/10.1137/0519008 doi: 10.1137/0519008
    [11] Y. Giga, T. Namba, Well-posedness of Hamilton-Jacobi equations with Caputo's time fractional derivative, Commun. Partial Differ. Equations, 42 (2017), 1088–1120. https://doi.org/10.1080/03605302.2017.1324880 doi: 10.1080/03605302.2017.1324880
    [12] R. H. Nochetto, E. Otarola, A. J. Salgado, A PDE approach to space-time fractional parabolic problems, SIAM J. Numer. Anal., 54 (2016), 848–873. https://doi.org/10.1137/14096308X doi: 10.1137/14096308X
    [13] A. Carbotti, S. Dipierro, E. Valdinoci, Local Density of Solutions to Fractional Equations, Berlin, 2019. https://doi.org/10.1515/9783110664355
    [14] Y. Cao, J. Yin, C. Wang, Cauchy problems of semilinear pseudo-parabolic equations, J. Differ. Equations, 246 (2009), 4568–4590. https://doi.org/10.1016/j.jde.2009.03.021 doi: 10.1016/j.jde.2009.03.021
    [15] L. Jin, L. Li, S. Fang, The global existence and time-decay for the solutions of the fractional pseudo-parabolic equation, Comput. Math. Appl., 73 (2017), 2221–2232. https://doi.org/10.1016/j.camwa.2017.03.005 doi: 10.1016/j.camwa.2017.03.005
    [16] R. Wang, Y. Li, B. Wang, Random dynamics of fractional nonclassical diffusion equations driven by colored noise, Discrete Contin. Dynam. Syst. Series A, 39 (2019), 4091–4126. https://doi.org/10.3934/dcds.2019165 doi: 10.3934/dcds.2019165
    [17] R. Wang, Y. Li, B. Wang, Bi-spatial pullback attractors of fractional non-classical diffusion equations on unbounded domains with (p, q)-growth nonlinearities, Appl. Math. Optim., 84 (2021), 425–461. https://doi.org/10.1007/s00245-019-09650-6 doi: 10.1007/s00245-019-09650-6
    [18] T. Q. Bao, C. T. Anh, Dynamics of non-autonomous nonclassical diffusion equations on $ R^n $, Commun. Pure Appl. Anal., 11 (2012), 1231–1252. https://doi.org/10.3934/cpaa.2012.11.1231 doi: 10.3934/cpaa.2012.11.1231
    [19] R. Wang, L. Shi, B. Wang, Asymptotic behavior of fractional nonclassical diffusion equations driven by nonlinear colored noise on $\mathbb{R^N}$, Nonlinearity, 32 (2019), 4524–4556. https://doi.org/10.1088/1361-6544/ab32d7 doi: 10.1088/1361-6544/ab32d7
    [20] Q. Zhang, Y. N. Li, The critical exponent for a time fractional diffusion equation with nonlinear memory, Math. Methods Appl. Sci., 41 (2018), 6443–6456. https://doi.org/10.1002/mma.5169 doi: 10.1002/mma.5169
    [21] Q. Zhang, Y. N. Li, The critical exponents for a time fractional diffusion equation with nonlinear memory in a bounded domain, Appl. Math. Lett., 92 (2019), 1–7. https://doi.org/10.1016/j.aml.2018.12.021 doi: 10.1016/j.aml.2018.12.021
    [22] N. H. Tuan, V. V. Au, R. Xu, Semilinear Caputo time-fractional pseudo-parabolic equations, Commun. Pure Appl. Anal., 20 (2021), 583–621. https://doi.org/10.3934/cpaa.2020282 doi: 10.3934/cpaa.2020282
    [23] R. Gorenflo, A. A. Kilbas, F. Mainardi, S. V. Rogosin, Mittag-Leffler Functions, Related Topics and Applications, Springer, Berlin, 2014. https://doi.org/10.1007/978-3-662-61550-8
    [24] F. Mainardi, On the initial value problem for the fractional diffusion-wave equation, in Waves and Stability in Continuous Media, World Scientific, Singapore, 1994,246–251.
    [25] R. Servadei, E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105–2137. https://doi.org/10.3934/dcds.2013.33.2105 doi: 10.3934/dcds.2013.33.2105
    [26] Q. Zhang, H. Sun, The blow-up and global existence of solutions of Cauchy problems for a time fractional diffusion equation, Topol. Methods Nonlinear Anal., 46 (2015), 69–92. https://doi.org/10.12775/TMNA.2015.038 doi: 10.12775/TMNA.2015.038
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