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February  2021, 20(2): 583-621. doi: 10.3934/cpaa.2020282

Semilinear Caputo time-fractional pseudo-parabolic equations

1. 

Department of Mathematics and Computer Science, University of Science Ho Chi Minh City, Vietnam

2. 

Vietnam National University, Ho Chi Minh City, Vietnam

3. 

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

4. 

Institute of Fundamental and Applied Sciences, Duy Tan University Ho Chi Minh City, 700000, Vietnam

5. 

Faculty of Natural Sciences, Duy Tan University, Da Nang, 550000, Vietnam

6. 

College of Mathematical Sciences, Harbin Engineering University, 150001, China

*Corresponding author

Received  July 2020 Revised  September 2020 Published  February 2021 Early access  December 2020

Fund Project: The first and the second author were supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.02-2019.09. The third author was supported by National Natural Science Foundation of China (11871017)

This paper considers two problems: the initial boundary value problem of nonlinear Caputo time-fractional pseudo-parabolic equations with fractional Laplacian, and the Cauchy problem (initial value problem) of Caputo time-fractional pseudo-parabolic equations. For the first problem with the source term satisfying the globally Lipschitz condition, we establish the local well-posedness theory including existence, uniqueness and regularity of the local solution, and the further local existence theory related to the finite time blow-up are also obtained for the problem with logarithmic nonlinearity. For the second problem with the source term satisfying the globally Lipschitz condition, we prove the global existence theorem.

Citation: Nguyen Huy Tuan, Vo Van Au, Runzhang Xu. Semilinear Caputo time-fractional pseudo-parabolic equations. Communications on Pure and Applied Analysis, 2021, 20 (2) : 583-621. doi: 10.3934/cpaa.2020282
References:
[1]

J. M. Arrieta and A. N. Carvalho, Abstract parabolic problems with critical nonlinearities and applications to Navier-Stokes and heat equations, T. Am. Math. Soc., 352 (1999), 285-310.  doi: 10.1090/S0002-9947-99-02528-3.

[2] R. P. AgarwalM. Meehan and D. O'Regan, Fixed Point Theory and Applications, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511543005.
[3]

G. Akagi, Fractional flows driven by subdifferentials in Hilbert spaces, Israel J. Math., 234 (2019), 809-862.  doi: 10.1007/s11856-019-1936-9.

[4]

B. de Andrade and A. Viana, Abstract Volterra integro-differential equations with applications to parabolic models with memory, Math. Ann., 369 (2017), 1131-1175.  doi: 10.1007/s00208-016-1469-z.

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B. Andrade and A. Viana, On a fractional reaction-diffusion equation, Z. Angew. Math. Phys., 68 (2017), 11 pp. doi: 10.1007/s00033-017-0801-0.

[6]

S. Antontsev and S. Shmarev, On a class of fully nonlinear parabolic equations, Adv. Nonlinear Anal., 8 (2019), 79-100.  doi: 10.1515/anona-2016-0055.

[7]

H. Allouba and W. Zheng, Brownian-time processes: the PDE connection and the half-derivative generator, Ann. Probab., 29 (2001), 1780-1795.  doi: 10.1214/aop/1015345772.

[8]

J. Barrow and P. Parsons, Inflationary models with logarithmic potentials, Phys. Rev. D, 52 (1995), 5576-5587. 

[9]

E. D. Benedetto and M. Pierre, On the maximum principle for pseudoparabolic equations, Indiana Univ. Math. J., 30 (1981), 821-854.  doi: 10.1512/iumj.1981.30.30062.

[10]

M. Caputo, Linear models of dissipation whose $Q$ is almost frequency independent, Geophys. J. Int., 13 (1967), 529-539. 

[11]

Y. CaoJ. Yin and C. Wang, Cauchy problems of semilinear pseudo-parabolic equations, J. Differ. Equ., 246 (2009), 4568-4590.  doi: 10.1016/j.jde.2009.03.021.

[12]

Y. Cao and J. X. Yin, Small perturbation of a semilinear pseudo-parabolic equation, Discrete Contin. Dyn. Syst. Ser. A, 36 (2016), 631-642.  doi: 10.3934/dcds.2016.36.631.

[13]

D. del Castillo-Negrete, B. A. Carreras and V. E. Lynch, Fractional diffusion in plasma turbulence, Phys. Plasmas, 11, 3854 (2004).

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H. Chen and S. Tian, Initial boundary value problem for a class of semilinear pseudo-parabolic equations with logarithmic nonlinearity, J. Differ. Equ., 258 (2015), 4424-4442.  doi: 10.1016/j.jde.2015.01.038.

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Y. Chen and R. Xu, Global well-posedness of solutions for fourth order dispersive wave equation with nonlinear weak damping, linear strong damping and logarithmic nonlinearity, Nonlinear Anal., 192 (2020), 39pp. doi: 10.1016/j.na.2019.111664.

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Y. ChenH. GaoM. Garrido-Atienza and B. Schmalfuß, Pathwise solutions of SPDEs driven by Hölder-continuous integrators with exponent larger than 1/2 and random dynamical systems, Discrete Cont. Dyn. Syst. Ser. A, 34 (2014), 79-98.  doi: 10.3934/dcds.2014.34.79.

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W. Chen and C. Li, Maximum principles for the fractional $p$-Laplacian and symmetry of solutions, Adv. Math., 335 (2018), 735-758.  doi: 10.1016/j.aim.2018.07.016.

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B. D. Coleman and W. Noll, Foundations of linear viscoelasticity, Rev. Mod. Phys., 33 (1961), 239-249.  doi: 10.1103/RevModPhys.33.239.

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P. Clément and J. A. Nohel, Asymptotic behavior of solutions of nonlinear volterra equations with completely positive kernels, SIAM J. Math. Anal., 12 (1981), 514-535.  doi: 10.1137/0512045.

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H. Dong and D. Kim, $L_p$-estimates for time fractional parabolic equations with coefficients measurable in time, Adv. Math., 345 (2019), 289-345.  doi: 10.1016/j.aim.2019.01.016.

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M. Fila and J. Lankeit, Lack of smoothing for bounded solutions of a semilinear parabolic equation, Adv. Nonlinear Anal., 9 (2020), 1437-1452.  doi: 10.1515/anona-2020-0059.

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C. G. Gal and M. Warma, Reaction-diffusion equations with fractional diffusion on non-smooth domains with various boundary conditions, Discrete Cont. Dyn. Syst. Ser. A, 36 (2016), 1279-1319.  doi: 10.3934/dcds.2016.36.1279.

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T-E. GhoulN. V. Tien and H. Zaag, Construction of type I blowup solutions for a higher order semilinear parabolic equation, Adv. Nonlinear Anal., 9 (2020), 388-412.  doi: 10.1515/anona-2020-0006.

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V. R. Gopala Rao and T. W. Ting, Solutions of pseudo-heat equations in the whole space, Arch. Ration. Mech. Anal., 49 (1972), 57-78.  doi: 10.1007/BF00281474.

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Y. Giga and T. Namba, Well-posedness of Hamilton-Jacobi equations with Caputo's time fractional derivative, Commun. Partial Differ. Equ., 42 (2017), 1088-1120.  doi: 10.1080/03605302.2017.1324880.

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R. GorenfloY. Luchko and F. Mainardi, Analytical properties and applications of the Wright function, Fract. Calc. Appl. Anal., 2 (1999), 383-414. 

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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.

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E. Hewitt and K. Stromberg, Real and abstract analysis. A modern treatment of the theory of functions of a real variable, Second printing corrected, Springer-Verlag, Berlin, (1969).

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L. JinL. Li and S. Fang, The global existence and time-decay for the solutions of the fractional pseudo-parabolic equation, Comput. Math. Appl., 73 (2017), 2221-2232.  doi: 10.1016/j.camwa.2017.03.005.

[30]

V. Keyantuo and M. Warma, On the interior approximate controllability for fractional wave equations, Discrete Cont. Dyn. Syst. Ser. A, 36 (2016), 3719-3739.  doi: 10.3934/dcds.2016.36.3719.

[31]

S. Khomrutai, Global well-posedness and grow-up rate of solutions for a sublinear pseudoparabolic equation, J. Differ. Equ., 260 (2015), 3598-3657.  doi: 10.1016/j.jde.2015.10.043.

[32]

L. LiJ. G. Liu and L. Wang, Cauchy problems for Keller-Segel type time-space fractional diffusion equation, J. Differ. Equ., 265 (2018), 1044-1096.  doi: 10.1016/j.jde.2018.03.025.

[33]

Z. P. Li and W. J. Du, Cauchy problems of pseudo-parabolic equations with inhomogeneous terms, Z. Angew. Math. Phys., 66 (2015), 3181-3203.  doi: 10.1007/s00033-015-0558-2.

[34]

G. Liu, The existence, general decay and blow-up for a plate equation with nonlinear damping and a logarithmic source term, Electron. Res. Arch., 28 (2020), 263-289.  doi: 10.3934/era.2020016.

[35]

S. JiJ. Yin and Y. Cao, Instability of positive periodic solutions for semilinear pseudo-parabolic equations with logarithmic nonlinearity, J. Differ. Equ., 261 (2016), 5446-5464.  doi: 10.1016/j.jde.2016.08.017.

[36]

W. LianJ. Wang and R. Xu, Global existence and blow up of solutions for pseudo-parabolic equation with singular potential, J. Differ. Equ., 269 (2020), 4914-4959.  doi: 10.1016/j.jde.2020.03.047.

[37]

A. MaganaA. Miranville and R. Quintanilla, On the time decay in phase-lag thermoelasticity with two temperatures, Electron. Res. Arch., 27 (2019), 7-19.  doi: 10.3934/era.2019007.

[38]

B. B. Mandelbrot and J. W. V. Ness, Fractional Brownian motions, fractional noises and applications, SIAM Rev., 10 (1968), 422-437.  doi: 10.1137/1010093.

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E. Di NezzaG. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.

[40]

R. H. NochettoE. Otárola and A. J. Salgado, A PDE approach to space-time fractional parabolic problems, SIAM J. Numer. Anal., 54 (2016), 848-873.  doi: 10.1137/14096308X.

[41]

S. Pan, Asymptotic spreading in a delayed dispersal predator-prey system without comparison principle, Electronic Research Archive, 27 (2019), 89-99.  doi: 10.3934/era.2019011.

[42]

N. PanP. PucciR. Xu and B. Zhang, Degenerate Kirchhoff-type wave problems involving the fractional Laplacian with nonlinear damping and source terms, J. Evol. Equ., 19 (2019), 615-643.  doi: 10.1007/s00028-019-00489-6.

[43]

N. S. PapageorgiouaeV. D. Rădulescu and D. D. Repovă, Positive solutions for nonlinear Neumann problems with singular terms and convection, J. Math. Pures Appl., 136 (2020), 1-21.  doi: 10.1016/j.matpur.2020.02.004.

[44]

V. Padron, Effect of aggregation on population recovery modeled by a forward-backward pseudoparabolic equation, T. Am. Math. Soc., 356 (2004), 2739-2756.  doi: 10.1090/S0002-9947-03-03340-3.

[45]

I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Vol. 198 (1998), Elsevier, Amsterdam.

[46]

M. Ralf and K. Joseph, The random walk's guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep., 339 (2000), 1-77.  doi: 10.1016/S0370-1573(00)00070-3.

[47]

T. Saanouni, Global and non global solutions for a class of coupled parabolic systems, Adv. Nonlinear Anal., 9 (2020), 1383-1401.  doi: 10.1515/anona-2020-0073.

[48]

E. Shivanian and A. Jafarabadi, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.

[49]

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

[50]

Y. F. SunZ. Zeng and J. Song, Quasilinear iterative method for the boundary value problem of nonlinear fractional differential equation, Numer. Algebra, Control. Optim., 10 (2020), 157-164.  doi: 10.3934/naco.2019045.

[51]

E. Orsingher and L. Beghin, Fractional diffusion equations and processes with randomly varying time, Ann. Probab., 37 (2009), 206-249.  doi: 10.1214/08-AOP401.

[52]

E. Topp and M. Yangari, Existence and uniqueness for parabolic problems with Caputo time derivative, J. Differ. Equ., 262 (2017), 6018-6046.  doi: 10.1016/j.jde.2017.02.024.

[53]

Tokinaga and P. Rybka, On viscosity solutions of space-fractional diffusion equations of Caputo type, SIAM J. Math. Anal., 52 (2020), 653-681.  doi: 10.1137/19M1259316.

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D. D. TrongE. NaneN. D. Minh and N. H. Tuan, Continuity of solutions of a class of fractional equations, Potential Anal., 49 (2018), 423-478.  doi: 10.1007/s11118-017-9663-5.

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Y. Wang and Y. Feng, theta scheme with two dimensional wavelet-like incremental unknowns for a class of porous medium diffusion-type equations, Numer. Algebra, Control. Optim., 9 (2019), 461-481.  doi: 10.3934/naco.2019027.

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X. WangY. ChenY. YangJ. Li and R. Xu, Kirchhoff-type system with linear weak damping and logarithmic nonlinearities, Nonlinear Anal., 188 (2019), 475-499.  doi: 10.1016/j.na.2019.06.019.

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R. Xu and J. Su, Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations, J. Funct. Anal., 264 (2013), 2732-2763.  doi: 10.1016/j.jfa.2013.03.010.

[60]

R. XuW. Lian and Y. Niu, Global well-posedness of coupled parabolic systems, Sci. China Math., 63 (2020), 321-356.  doi: 10.1007/s11425-017-9280-x.

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J. Zhou, Initial boundary value problem for a inhomogeneous pseudo-parabolic equation, Electron. Res. Arch., 28 (2020), 67-90.  doi: 10.3934/era.2020005.

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, T. Am. Math. Soc., 352 (1999), 285-310.  doi: 10.1090/S0002-9947-99-02528-3.

[2] R. P. AgarwalM. Meehan and D. O'Regan, Fixed Point Theory and Applications, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511543005.
[3]

G. Akagi, Fractional flows driven by subdifferentials in Hilbert spaces, Israel J. Math., 234 (2019), 809-862.  doi: 10.1007/s11856-019-1936-9.

[4]

B. de Andrade and A. Viana, Abstract Volterra integro-differential equations with applications to parabolic models with memory, Math. Ann., 369 (2017), 1131-1175.  doi: 10.1007/s00208-016-1469-z.

[5]

B. Andrade and A. Viana, On a fractional reaction-diffusion equation, Z. Angew. Math. Phys., 68 (2017), 11 pp. doi: 10.1007/s00033-017-0801-0.

[6]

S. Antontsev and S. Shmarev, On a class of fully nonlinear parabolic equations, Adv. Nonlinear Anal., 8 (2019), 79-100.  doi: 10.1515/anona-2016-0055.

[7]

H. Allouba and W. Zheng, Brownian-time processes: the PDE connection and the half-derivative generator, Ann. Probab., 29 (2001), 1780-1795.  doi: 10.1214/aop/1015345772.

[8]

J. Barrow and P. Parsons, Inflationary models with logarithmic potentials, Phys. Rev. D, 52 (1995), 5576-5587. 

[9]

E. D. Benedetto and M. Pierre, On the maximum principle for pseudoparabolic equations, Indiana Univ. Math. J., 30 (1981), 821-854.  doi: 10.1512/iumj.1981.30.30062.

[10]

M. Caputo, Linear models of dissipation whose $Q$ is almost frequency independent, Geophys. J. Int., 13 (1967), 529-539. 

[11]

Y. CaoJ. Yin and C. Wang, Cauchy problems of semilinear pseudo-parabolic equations, J. Differ. Equ., 246 (2009), 4568-4590.  doi: 10.1016/j.jde.2009.03.021.

[12]

Y. Cao and J. X. Yin, Small perturbation of a semilinear pseudo-parabolic equation, Discrete Contin. Dyn. Syst. Ser. A, 36 (2016), 631-642.  doi: 10.3934/dcds.2016.36.631.

[13]

D. del Castillo-Negrete, B. A. Carreras and V. E. Lynch, Fractional diffusion in plasma turbulence, Phys. Plasmas, 11, 3854 (2004).

[14]

H. Chen and S. Tian, Initial boundary value problem for a class of semilinear pseudo-parabolic equations with logarithmic nonlinearity, J. Differ. Equ., 258 (2015), 4424-4442.  doi: 10.1016/j.jde.2015.01.038.

[15]

Y. Chen and R. Xu, Global well-posedness of solutions for fourth order dispersive wave equation with nonlinear weak damping, linear strong damping and logarithmic nonlinearity, Nonlinear Anal., 192 (2020), 39pp. doi: 10.1016/j.na.2019.111664.

[16]

Y. ChenH. GaoM. Garrido-Atienza and B. Schmalfuß, Pathwise solutions of SPDEs driven by Hölder-continuous integrators with exponent larger than 1/2 and random dynamical systems, Discrete Cont. Dyn. Syst. Ser. A, 34 (2014), 79-98.  doi: 10.3934/dcds.2014.34.79.

[17]

W. Chen and C. Li, Maximum principles for the fractional $p$-Laplacian and symmetry of solutions, Adv. Math., 335 (2018), 735-758.  doi: 10.1016/j.aim.2018.07.016.

[18]

B. D. Coleman and W. Noll, Foundations of linear viscoelasticity, Rev. Mod. Phys., 33 (1961), 239-249.  doi: 10.1103/RevModPhys.33.239.

[19]

P. Clément and J. A. Nohel, Asymptotic behavior of solutions of nonlinear volterra equations with completely positive kernels, SIAM J. Math. Anal., 12 (1981), 514-535.  doi: 10.1137/0512045.

[20]

H. Dong and D. Kim, $L_p$-estimates for time fractional parabolic equations with coefficients measurable in time, Adv. Math., 345 (2019), 289-345.  doi: 10.1016/j.aim.2019.01.016.

[21]

M. Fila and J. Lankeit, Lack of smoothing for bounded solutions of a semilinear parabolic equation, Adv. Nonlinear Anal., 9 (2020), 1437-1452.  doi: 10.1515/anona-2020-0059.

[22]

C. G. Gal and M. Warma, Reaction-diffusion equations with fractional diffusion on non-smooth domains with various boundary conditions, Discrete Cont. Dyn. Syst. Ser. A, 36 (2016), 1279-1319.  doi: 10.3934/dcds.2016.36.1279.

[23]

T-E. GhoulN. V. Tien and H. Zaag, Construction of type I blowup solutions for a higher order semilinear parabolic equation, Adv. Nonlinear Anal., 9 (2020), 388-412.  doi: 10.1515/anona-2020-0006.

[24]

V. R. Gopala Rao and T. W. Ting, Solutions of pseudo-heat equations in the whole space, Arch. Ration. Mech. Anal., 49 (1972), 57-78.  doi: 10.1007/BF00281474.

[25]

Y. Giga and T. Namba, Well-posedness of Hamilton-Jacobi equations with Caputo's time fractional derivative, Commun. Partial Differ. Equ., 42 (2017), 1088-1120.  doi: 10.1080/03605302.2017.1324880.

[26]

R. GorenfloY. Luchko and F. Mainardi, Analytical properties and applications of the Wright function, Fract. Calc. Appl. Anal., 2 (1999), 383-414. 

[27]

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.

[28]

E. Hewitt and K. Stromberg, Real and abstract analysis. A modern treatment of the theory of functions of a real variable, Second printing corrected, Springer-Verlag, Berlin, (1969).

[29]

L. JinL. Li and S. Fang, The global existence and time-decay for the solutions of the fractional pseudo-parabolic equation, Comput. Math. Appl., 73 (2017), 2221-2232.  doi: 10.1016/j.camwa.2017.03.005.

[30]

V. Keyantuo and M. Warma, On the interior approximate controllability for fractional wave equations, Discrete Cont. Dyn. Syst. Ser. A, 36 (2016), 3719-3739.  doi: 10.3934/dcds.2016.36.3719.

[31]

S. Khomrutai, Global well-posedness and grow-up rate of solutions for a sublinear pseudoparabolic equation, J. Differ. Equ., 260 (2015), 3598-3657.  doi: 10.1016/j.jde.2015.10.043.

[32]

L. LiJ. G. Liu and L. Wang, Cauchy problems for Keller-Segel type time-space fractional diffusion equation, J. Differ. Equ., 265 (2018), 1044-1096.  doi: 10.1016/j.jde.2018.03.025.

[33]

Z. P. Li and W. J. Du, Cauchy problems of pseudo-parabolic equations with inhomogeneous terms, Z. Angew. Math. Phys., 66 (2015), 3181-3203.  doi: 10.1007/s00033-015-0558-2.

[34]

G. Liu, The existence, general decay and blow-up for a plate equation with nonlinear damping and a logarithmic source term, Electron. Res. Arch., 28 (2020), 263-289.  doi: 10.3934/era.2020016.

[35]

S. JiJ. Yin and Y. Cao, Instability of positive periodic solutions for semilinear pseudo-parabolic equations with logarithmic nonlinearity, J. Differ. Equ., 261 (2016), 5446-5464.  doi: 10.1016/j.jde.2016.08.017.

[36]

W. LianJ. Wang and R. Xu, Global existence and blow up of solutions for pseudo-parabolic equation with singular potential, J. Differ. Equ., 269 (2020), 4914-4959.  doi: 10.1016/j.jde.2020.03.047.

[37]

A. MaganaA. Miranville and R. Quintanilla, On the time decay in phase-lag thermoelasticity with two temperatures, Electron. Res. Arch., 27 (2019), 7-19.  doi: 10.3934/era.2019007.

[38]

B. B. Mandelbrot and J. W. V. Ness, Fractional Brownian motions, fractional noises and applications, SIAM Rev., 10 (1968), 422-437.  doi: 10.1137/1010093.

[39]

E. Di NezzaG. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.

[40]

R. H. NochettoE. Otárola and A. J. Salgado, A PDE approach to space-time fractional parabolic problems, SIAM J. Numer. Anal., 54 (2016), 848-873.  doi: 10.1137/14096308X.

[41]

S. Pan, Asymptotic spreading in a delayed dispersal predator-prey system without comparison principle, Electronic Research Archive, 27 (2019), 89-99.  doi: 10.3934/era.2019011.

[42]

N. PanP. PucciR. Xu and B. Zhang, Degenerate Kirchhoff-type wave problems involving the fractional Laplacian with nonlinear damping and source terms, J. Evol. Equ., 19 (2019), 615-643.  doi: 10.1007/s00028-019-00489-6.

[43]

N. S. PapageorgiouaeV. D. Rădulescu and D. D. Repovă, Positive solutions for nonlinear Neumann problems with singular terms and convection, J. Math. Pures Appl., 136 (2020), 1-21.  doi: 10.1016/j.matpur.2020.02.004.

[44]

V. Padron, Effect of aggregation on population recovery modeled by a forward-backward pseudoparabolic equation, T. Am. Math. Soc., 356 (2004), 2739-2756.  doi: 10.1090/S0002-9947-03-03340-3.

[45]

I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Vol. 198 (1998), Elsevier, Amsterdam.

[46]

M. Ralf and K. Joseph, The random walk's guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep., 339 (2000), 1-77.  doi: 10.1016/S0370-1573(00)00070-3.

[47]

T. Saanouni, Global and non global solutions for a class of coupled parabolic systems, Adv. Nonlinear Anal., 9 (2020), 1383-1401.  doi: 10.1515/anona-2020-0073.

[48]

E. Shivanian and A. Jafarabadi, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.

[49]

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

[50]

Y. F. SunZ. Zeng and J. Song, Quasilinear iterative method for the boundary value problem of nonlinear fractional differential equation, Numer. Algebra, Control. Optim., 10 (2020), 157-164.  doi: 10.3934/naco.2019045.

[51]

E. Orsingher and L. Beghin, Fractional diffusion equations and processes with randomly varying time, Ann. Probab., 37 (2009), 206-249.  doi: 10.1214/08-AOP401.

[52]

E. Topp and M. Yangari, Existence and uniqueness for parabolic problems with Caputo time derivative, J. Differ. Equ., 262 (2017), 6018-6046.  doi: 10.1016/j.jde.2017.02.024.

[53]

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