• Previous Article
    On the approximate controllability of neutral integro-differential inclusions of Sobolev-type with infinite delay
  • EECT Home
  • This Issue
  • Next Article
    Optimal control problems for a neutral integro-differential system with infinite delay
doi: 10.3934/eect.2020109

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

* Corresponding author: Nguyen H. Tuan

Received  July 2020 Revised  October 2020 Published  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.

Citation: Nguyen Anh Tuan, Donal O'Regan, Dumitru Baleanu, Nguyen H. Tuan. On time fractional pseudo-parabolic equations with nonlocal integral conditions. Evolution Equations & Control Theory, 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.  Google Scholar

[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.  Google Scholar

[3]

P. N. Belov, The Numerical Methods of Weather Forecasting, Gidrometeoizdat, Leningrad, 1975. Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

V. V. Shelukhin, A non-local (in time) model for radionuclides propagation in a Stokes fluid, Dinamika Sploshn. Sredy, 107 (1993), 180-193.   Google Scholar

[27]

R. E. Showalter and T. W. Ting, Pseudoparabolic partial differential equations, SIAM J. Math. Anal, 1 (1970), 1-26.  doi: 10.1137/0501001.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

P. N. Belov, The Numerical Methods of Weather Forecasting, Gidrometeoizdat, Leningrad, 1975. Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

V. V. Shelukhin, A non-local (in time) model for radionuclides propagation in a Stokes fluid, Dinamika Sploshn. Sredy, 107 (1993), 180-193.   Google Scholar

[27]

R. E. Showalter and T. W. Ting, Pseudoparabolic partial differential equations, SIAM J. Math. Anal, 1 (1970), 1-26.  doi: 10.1137/0501001.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Nguyen Huy Tuan, Vo Van Au, Runzhang Xu. Semilinear Caputo time-fractional pseudo-parabolic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020282

[2]

Vo Van Au, Hossein Jafari, Zakia Hammouch, Nguyen Huy Tuan. On a final value problem for a nonlinear fractional pseudo-parabolic equation. Electronic Research Archive, 2021, 29 (1) : 1709-1734. doi: 10.3934/era.2020088

[3]

Stanislav Nikolaevich Antontsev, Serik Ersultanovich Aitzhanov, Guzel Rashitkhuzhakyzy Ashurova. An inverse problem for the pseudo-parabolic equation with p-Laplacian. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021005

[4]

Biyue Chen, Chunxiang Zhao, Chengkui Zhong. The global attractor for the wave equation with nonlocal strong damping. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021015

[5]

Noufel Frikha, Valentin Konakov, Stéphane Menozzi. Well-posedness of some non-linear stable driven SDEs. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 849-898. doi: 10.3934/dcds.2020302

[6]

Boris Andreianov, Mohamed Maliki. On classes of well-posedness for quasilinear diffusion equations in the whole space. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 505-531. doi: 10.3934/dcdss.2020361

[7]

Jean-Claude Saut, Yuexun Wang. Long time behavior of the fractional Korteweg-de Vries equation with cubic nonlinearity. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1133-1155. doi: 10.3934/dcds.2020312

[8]

Anh Tuan Duong, Phuong Le, Nhu Thang Nguyen. Symmetry and nonexistence results for a fractional Choquard equation with weights. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 489-505. doi: 10.3934/dcds.2020265

[9]

Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020382

[10]

Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248

[11]

Tong Tang, Jianzhu Sun. Local well-posedness for the density-dependent incompressible magneto-micropolar system with vacuum. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020377

[12]

Xiaopeng Zhao, Yong Zhou. Well-posedness and decay of solutions to 3D generalized Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 795-813. doi: 10.3934/dcdsb.2020142

[13]

Dongfen Bian, Yao Xiao. Global well-posedness of non-isothermal inhomogeneous nematic liquid crystal flows. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1243-1272. doi: 10.3934/dcdsb.2020161

[14]

Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020432

[15]

Fuensanta Andrés, Julio Muñoz, Jesús Rosado. Optimal design problems governed by the nonlocal $ p $-Laplacian equation. Mathematical Control & Related Fields, 2021, 11 (1) : 119-141. doi: 10.3934/mcrf.2020030

[16]

Biao Zeng. Existence results for fractional impulsive delay feedback control systems with Caputo fractional derivatives. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021001

[17]

Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319

[18]

S. Sadeghi, H. Jafari, S. Nemati. Solving fractional Advection-diffusion equation using Genocchi operational matrix based on Atangana-Baleanu derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020435

[19]

Lihong Zhang, Wenwen Hou, Bashir Ahmad, Guotao Wang. Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional $ p $-Laplacian. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020445

[20]

Abdollah Borhanifar, Maria Alessandra Ragusa, Sohrab Valizadeh. High-order numerical method for two-dimensional Riesz space fractional advection-dispersion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020355

2019 Impact Factor: 0.953

Article outline

[Back to Top]