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On time fractional pseudo-parabolic equations with nonlocal integral conditions
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 |
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.
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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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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