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February  2022, 11(1): 225-238. doi: 10.3934/eect.2020109

On time fractional pseudo-parabolic equations with nonlocal integral conditions

Nguyen Anh Tuan 1, , Donal O'Regan 2, , Dumitru Baleanu 3, and  Nguyen H. Tuan 1,,

1. 

Division of Applied Mathematics, Thu Dau Mot University, Binh Duong Province, Vietnam
2. 

School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland
3. 

Department of Mathematics, Faculty of Arts and Sciences, Cankaya University, 06530 Ankara, Turkey, Institute of Space Sciences, P.O.Box, MG-23, R 76900, Magurele-Bucharest, Romania

* Corresponding author: Nguyen H. Tuan

Received  July 2020 Revised  October 2020 Published  February 2022 Early access  December 2020

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.

Keywords: Fractional partial differential equation, Caputo fractional, well-posedness, pseudo-parabolic equation, nonlocal conditions, nonlocal in time.
Mathematics Subject Classification: 26A33, 35B65, 35R11.
Citation: Nguyen Anh Tuan, Donal O'Regan, Dumitru Baleanu, Nguyen H. Tuan. On time fractional pseudo-parabolic equations with nonlocal integral conditions. Evolution Equations and Control Theory, 2022, 11 (1) : 225-238. doi: 10.3934/eect.2020109
References:
[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. Benjamin, J. 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 Nezza, G. 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. Djida, A. 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. Ewing, R. 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. Liu, R. 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. Ngoc, D. Baleanu, L. 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.

show all references

References:
[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. Benjamin, J. 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 Nezza, G. 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. Djida, A. 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. Ewing, R. 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. Liu, R. 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. Ngoc, D. Baleanu, L. 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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