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On time fractional pseudo-parabolic equations with nonlocal integral conditions

  • * Corresponding author: Nguyen H. Tuan

    * Corresponding author: Nguyen H. Tuan
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  • In this paper, we study the nonlocal problem for pseudo-parabolic equation with time and space fractional derivatives. The time derivative is of Caputo type and of order $ \sigma,\; \; 0<\sigma<1 $ and the space fractional derivative is of order $ \alpha,\beta >0 $. In the first part, we obtain some results of the existence and uniqueness of our problem with suitably chosen $ \alpha, \beta $. The technique uses a Sobolev embedding and is based on constructing a Mittag-Leffler operator. In the second part, we give the ill-posedness of our problem and give a regularized solution. An error estimate in $ L^p $ between the regularized solution and the sought solution is obtained.

    Mathematics Subject Classification: 26A33, 35B65, 35R11.


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  • [1] J. M. Arrieta and A. N. Carvalho, Abstract parabolic problems with critical nonlinearities and applications to Navier-Stokes and heat equations, Trans. Amer. Math. Soc, 352 (2000), 285-310.  doi: 10.1090/S0002-9947-99-02528-3.
    [2] A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Therm Sci, 20 (2016), 763-769.  doi: 10.2298/TSCI160111018A.
    [3] P. N. Belov, The Numerical Methods of Weather Forecasting, Gidrometeoizdat, Leningrad, 1975.
    [4] T. B. BenjaminJ. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philos. Trans. Roy. Soc. London Ser.A, 272 (1972), 47-78.  doi: 10.1098/rsta.1972.0032.
    [5] M. K. Beshtokov, Boundary-value problems for loaded pseudoparabolic equations of fractional order and difference methods of their solving, Russian Mathematics, 63 (2019), 1-10. 
    [6] M. K. Beshtokov, Boundary value problems for a pseudoparabolic equation with the Caputo fractional derivative, Differ. Equ, 55 (2019), 884-893.  doi: 10.1134/S0012266119070024.
    [7] M. K. Beshtokov, Toward boundary-value problems for degenerating pseudoparabolic equations with Gerasimov-Caputo fractional derivative, Izv. Vyssh. Uchebn. Zaved. Mat, 62 (2018), 3-16. 
    [8] P. J. Chen and M. E. Gurtin, On a theory of heat conduction involving two temperatures, Z. Angew. Math. Phys, 19 (1968), 614-627.  doi: 10.1007/BF01594969.
    [9] H. Chen and S. Tian, Initial boundary value problem for a class of semilinear pseudo-parabolic equations with logarithmic nonlinearity, J. Differential Equations, 258 (2015), 4424-4442.  doi: 10.1016/j.jde.2015.01.038.
    [10] E. Di NezzaG. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math, 1360 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.
    [11] J.-D. DjidaA. Fernandez and I. Area, Well-posedness results for fractional semi-linear wave equations, Discrete Contin. Dyn. Syst. Ser B, 25 (2020), 569-597.  doi: 10.3934/dcdsb.2019255.
    [12] N. Dokuchaev, On recovering parabolic diffusions from their time-averages, Calc. Var. Partial Differential Equations, 58 (2019), 14 pp. doi: 10.1007/s00526-018-1464-1.
    [13] R. E. EwingR. D. Lazarov and Y. Lin, Finite volume element approximations of nonlocal in time one-dimensional flows in porous media, Computing, 64 (2000), 157-182.  doi: 10.1007/s006070050007.
    [14] R. Gorenflo, A. A. Kilbas and F. Mainardi, Mittag-Leffler Functions, Related Topics and Applications, Springer, Berlin, (2014). doi: 10.1007/978-3-662-43930-2.
    [15] R. Ikehata and T. Suzuki, Stable and unstable sets for evolution equations of parabolic and hyperbolic type, Hiroshima Math. J, 26 (1996), 475-491.  doi: 10.32917/hmj/1206127254.
    [16] M. Kwaśnicki, Ten equivalent definitions of the fractional Laplace operator, Fract. Calc. Appl. Anal, 20 (2017), 7-51.  doi: 10.1515/fca-2017-0002.
    [17] B. Kaltenbacher and W. Rundell, Regularization of a backward parabolic equation by fractional operators, Inverse Probl. Imaging, 13 (2019), 401-430.  doi: 10.3934/ipi.2019020.
    [18] N. H. Luc, L. N. Huynh, D. Baleanu and N. H. Can, Identifying the space source term problem for a generalization of the fractional diffusion equation with hyper-Bessel operator, Adv. Difference Equ, 2020 (2020), 23 pp. doi: 10.1186/s13662-020-02712-y.
    [19] Y. LiuR. Xu and T. Yu, Global existence, nonexistence and asymptotic behavior of solutions for the Cauchy problem of semilinear heat equations, Nonlinear Anal, 68 (2008), 3332-3348.  doi: 10.1016/j.na.2007.03.029.
    [20] T. B. NgocD. BaleanuL. T. M. Duc and N. H. Tuan, Regularity results for fractional diffusion equations involving fractional derivative with Mittag-Leffler kernel, Math. Methods Appl. Sci, 43 (2020), 7208-7226.  doi: 10.1002/mma.6459.
    [21] E. Otárola and A. J. Salgado, Regularity of solutions to space-time fractional wave equations: A PDE approach, Fract. Calc. Appl. Anal, 21 (2018), 1262-1293.  doi: 10.1515/fca-2018-0067.
    [22] C. V. Pao, Reaction diffusion equations with nonlocal boundary and nonlocal initial conditions, J. Math. Anal. Appl., 195 (1995), 702-718.  doi: 10.1006/jmaa.1995.1384.
    [23] Q. Pavol and P. Souplet, Superlinear Parabolic Problems, Blow-up, Global Existence and Steady States, Birkhäuser Advanced Texts, Birkhäuser Verlag, Basel, 2007.
    [24] V. Padron, Effect of aggregation on population recovery modeled by a forward-backward pseudo parabolic equation, Trans. Amer. Math. Soc, 356 (2004), 2739-2756.  doi: 10.1090/S0002-9947-03-03340-3.
    [25] K. Sakamoto and M. Yamamoto, Initial value/boudary value problems for fractional diffusion - wave equations and applications to some inverse problems, J. Math. Anal. Appl, 382 (2011), 426-447.  doi: 10.1016/j.jmaa.2011.04.058.
    [26] V. V. Shelukhin, A non-local (in time) model for radionuclides propagation in a Stokes fluid, Dinamika Sploshn. Sredy, 107 (1993), 180-193. 
    [27] R. E. Showalter and T. W. Ting, Pseudoparabolic partial differential equations, SIAM J. Math. Anal, 1 (1970), 1-26.  doi: 10.1137/0501001.
    [28] N. H. Tuan, D. Baleanu, T. N. Thach, D. O'Regan and N. H. Can, Final value problem for nonlinear time fractional reaction-diffusion equation with discrete data, J. Comput. Appl. Math, 376 (2020), 25 pp. doi: 10.1016/j.cam.2020.112883.
    [29] N. H. Tuan, T. B. Ngoc, Y. Zhou and D. O'Regan, On existence and regularity of a terminal value problem for the time fractional diffusion equation, Inverse Problems, 36, (2020), 41 pp. doi: 10.1088/1361-6420/ab730d.
    [30] J. M. Vaquero and S. Sajavicius, The two-level finite difference schemes for the heat equation with nonlocal initial condition, Appl. Math. Comput., 342 (2019), 166-177.  doi: 10.1016/j.amc.2018.09.025.
    [31] R. Xu and J. Su, Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations, J. Functional Analysis, 264 (2013), 2732-2763.  doi: 10.1016/j.jfa.2013.03.010.
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