February  2022, 15(2): 481-499. doi: 10.3934/dcdss.2021118

Existence and regularity results for stochastic fractional pseudo-parabolic equations driven by white noise

1. 

Applied Analysis Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

2. 

Department of Mathematics, University of Rajasthan, Jaipur 302004, Rajasthan, India

3. 

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

4. 

Division of Applied Mathematics, Science and Technology Advanced Institute, Van Lang University, Ho Chi Minh City, Vietnam

5. 

Faculty of Technology, Van Lang University, Ho Chi Minh City, Vietnam

*Corresponding authors: Nguyen Huy Tuan (nguyenhuytuan@vlu.edu.vn) and Devendra Kumar (devendra.maths@gmail.com)

Received  January 2021 Revised  June 2021 Published  February 2022 Early access  November 2021

Solutions of a direct problem for a stochastic pseudo-parabolic equation with fractional Caputo derivative are investigated, in which the non-linear space-time-noise is assumed to satisfy distinct Lipshitz conditions including globally and locally assumptions. The main aim of this work is to establish some existence, uniqueness, regularity, and continuity results for mild solutions.

Citation: Tran Ngoc Thach, Devendra Kumar, Nguyen Hoang Luc, Nguyen Huy Tuan. Existence and regularity results for stochastic fractional pseudo-parabolic equations driven by white noise. Discrete and Continuous Dynamical Systems - S, 2022, 15 (2) : 481-499. doi: 10.3934/dcdss.2021118
References:
[1]

W. ArendtA. F. Ter Elst and M. Warma, Fractional powers of sectorial operators via the Dirichlet-to-Neumann operator, Comm. Partial Differential Equations, 43 (2018), 1-24.  doi: 10.1080/03605302.2017.1363229.

[2]

S. A. AsogwaM. FoondunJ. B. Mijena and E. Nane, Critical parameters for reaction-diffusion equations involving space-time fractional derivatives, NoDEA Nonlinear Differential Equations Appl., 27 (2020), 1-22.  doi: 10.1007/s00030-020-00629-9.

[3]

V. Van AuH. JafariZ. HammouchN. H. Tuan and (2 019), On a final value problem for a nonlinear fractional pseudo-parabolic equation, Electron. Res. Arch., 29 (2021), 1709-1734.  doi: 10.3934/era.2020088.

[4]

B. BaeumerM. Geissert and M. Kovács, Existence, uniqueness and regularity for a class of semilinear stochastic Volterra equations with multiplicative noise, J. Differential Equations, 258 (2015), 535-554.  doi: 10.1016/j.jde.2014.09.020.

[5]

L. Bai and F. H. Zhang, Existence of random attractors for 2D-stochastic nonclassical diffusion equations on unbounded domains, Results Math., 69 (2016), 129-160.  doi: 10.1007/s00025-015-0505-8.

[6]

G. I. BarenblattY. P. Zheltov and I. N. Kochina, On basic ideas of the theory of filtration of homogeneous fluids in fractured rocks, Prikl. Mat. Mekh., 24 (1960), 852-864. 

[7]

T. B. BenjaminJ. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philos. Trans. Roy. Soc. London Ser., 272 (1972), 47-78.  doi: 10.1098/rsta.1972.0032.

[8]

M. K. Beshtokov, Toward boundary-value problems for degenerating pseudoparabolic equations with a Gerasimov-Caputo fractional derivative, Russian Math., 62 (2018), 1-14. 

[9]

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. 

[10]

M. K. Beshtokov, Boundary value problems for a pseudoparabolic equation with the Caputo fractional derivative, Differe. Equ., 55 (2019), 884-893.  doi: 10.1134/S0012266119070024.

[11]

C. BurgosJ. C. CortésA. DebboucheL. Villafuerte and R. J. Villanueva, Random fractional generalized Airy differential equations: A probabilistic analysis using mean square calculus, Appl. Math. Comput., 352 (2019), 15-29.  doi: 10.1016/j.amc.2019.01.039.

[12]

C. BurgosJ. C. CortésL. Villafuerte and R. J. Villanueva, Solving random mean square fractional linear differential equations by generalized power series: Analysis and computing, J. Compu. Appl. Math., 339 (2018), 94-110.  doi: 10.1016/j.cam.2017.12.042.

[13]

C. BurgosJ.-C. CortésL. Villafuerte and R.-J. Villanueva, Extending the deterministic Riemann-Liouville and Caputo operators to the random framework: A mean square approach with applications to solve random fractional differential equations, Chaos Solitons Fractals, 102 (2017), 305-318.  doi: 10.1016/j.chaos.2017.02.008.

[14]

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

[15]

Y. Cao and C. Liu, Initial boundary value problem for a mixed pseudo-parabolic p-Laplacian type equation with logarithmic nonlinearity. Electron, J. Differential Equations, (2018), 1–19.

[16]

M. Caputo, Linear models of dissipation whose Q is almost frequency independent-II, Geophysical J. Inter., 13 (1967), 529-539. 

[17]

T. CaraballoM. J. Garrido-Atienza and J. Real, Stochastic stabilization of differential systems with general decay rate, Systems Control Lett., 48 (2003), 397-406.  doi: 10.1016/S0167-6911(02)00293-1.

[18]

T. CaraballoM. J. Garrido-Atienza and T. Taniguchi, The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion, Nonlinear Anal: Theory, Methods and Applications, 74 (2011), 3671-3684.  doi: 10.1016/j.na.2011.02.047.

[19]

H. Chen and H. Xu, Global existence, exponential decay and blow-up in finite time for a class of finitely degenerate semilinear parabolic equations, Acta Math. Sci. Ser., 39 (2019), 1290-1308.  doi: 10.1007/s10473-019-0508-8.

[20]

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.

[21] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, 2$^{nd}$ edition, Cambridge University Press, Cambridge, 2014.  doi: 10.1017/CBO9781107295513.
[22]

L. Debbi, Well-posedness of the multidimensional fractional stochastic Navier-Stokes equations on the torus and on bounded domains, J. Math. Fluid Mech., 18 (2016), 25-69.  doi: 10.1007/s00021-015-0234-5.

[23]

H. DiY. Shang and X. Zheng, Global well-posedness for a fourth order pseudo-parabolic equation with memory and source terms, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 781-801.  doi: 10.3934/dcdsb.2016.21.781.

[24]

H. Ding and J. Zhou, Global existence and blow-up for a mixed pseudo-parabolic p-Laplacian type equation with logarithmic nonlinearity, J. Math. Anal. Appl., 478 (2019), 393-420.  doi: 10.1016/j.jmaa.2019.05.018.

[25]

E. S. Dzektser, Equations of motion of ground water with a free surface in multilayer media, Dokl. Akad. Nauk SSSR, 220 (1975), 540-543. 

[26]

M. Foondun, Remarks on a fractional-time stochastic equation, Proc. Amer. Math. Soc., 149 (2018), 2235-2247.  doi: 10.1090/proc/14644.

[27]

M. Foondun and E. Nualart, On the behaviour of stochastic heat equations on bounded domains, ALEA Lat. Am. J. Probab. Math. Stat., 12 (2015), 551-571. 

[28]

Y. HeH. Gao and H. Wang, Blow-up and decay for a class of pseudo-parabolic p-Laplacian equation with logarithmic nonlinearity, Comput. Math. Appl., 75 (2018), 459-469.  doi: 10.1016/j.camwa.2017.09.027.

[29]

G. HuY. Lou and P. D. Christofides, Dynamic output feedback covariance control of stochastic dissipative partial differential equations, Chemical Engineering Science, 63 (2008), 4531-4542. 

[30]

Y. JiangT. Wei and X. Zhou, Stochastic generalized Burgers equations driven by fractional noises, J. Differential Equations, 252 (2012), 1934-1961.  doi: 10.1016/j.jde.2011.07.032.

[31]

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.

[32]

T. Kato, Perturbation theory for linear operators, Springer Science & Business Media, 132 (2013).

[33]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006.

[34]

M. Kovács and S. Larsson, Introduction to stochastic partial differential equations, In Publications of the ICMCS, 4 (2008), 159-232. 

[35]

P. D. Lax, Functional Analysis, Wiley Interscience, New York, 2002.

[36]

F. LiY. Li and R. Wang, Regular measurable dynamics for reaction-diffusion equations on narrow domains with rough noise, Discrete Contin. Dyn. Syst., 38 (2018), 3663-3685.  doi: 10.3934/dcds.2018158.

[37]

Y. Li and Y. Wang, The existence and asymptotic behavior of solutions to fractional stochastic evolution equations with infinite delay, J. Differential Equations, 266 (2019), 3514-3558.  doi: 10.1016/j.jde.2018.09.009.

[38]

Y. Lu and L. Fei, Bounds for blow-up time in a semilinear pseudo-parabolic equation with nonlocal source, J. Inequal. Appl., 2016 (2016), 1-11.  doi: 10.1186/s13660-016-1171-4.

[39]

F. Mainardi, Applications of integral transforms in fractional diffusion processes, Integral Transforms Spec. Funct., 15 (2004), 477-484.  doi: 10.1080/10652460412331270652.

[40]

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

[41]

J. C. Pedjeu and G. S. Ladde, Stochastic fractional differential equations: Modeling, method and analysis, Chaos Solitons Fractals, 45 (2012), 279-293.  doi: 10.1016/j.chaos.2011.12.009.

[42] I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Mathematics in Science and Engineering, 198. Academic Press, Inc., San Diego, CA, 1999. 
[43]

C. Prévôt and M. Röckner, A Concise Course on Stochastic Partial Differential Equations, Lecture Notes in Mathematics, 1905. Springer, Berlin, 2007.

[44]

L. I. Rubinshtein, Heat propagation process in heterogeneous media, Izv. Akad. Nauk SSSR Ser. Geogr., 12 (1948), 27-45. 

[45]

K. Sakamoto and M. Yamamoto, Initial value/boundary 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.

[46]

S. L. Sobolev, On a new problem of mathematical physics, Izv. Akad. Nauk SSSR Ser. Mat., 18 (1954), 3-50. 

[47]

J. V. D. C. Sousa and E. C. de Oliveira, Fractional order pseudoparabolic partial differential equation: Ulam-Hyers stability, Bull. Braz. Math. Soc., 50 (2019), 481-496.  doi: 10.1007/s00574-018-0112-x.

[48]

X. Su and M. Li, The regularity of fractional stochastic evolution equations in Hilbert space, Stoch. Anal. Appl., 36 (2018), 639-653.  doi: 10.1080/07362994.2018.1436973.

[49]

F. I. Taukenova and M. K. Shkhanukov-Lafishev, Difference methods for solving boundary value problems for fractional differential equations, Comput. Math. Math. Phys., 46 (2006), 1785-1795.  doi: 10.1134/S0965542506100149.

[50]

T. W. Ting, Certain non-steady flows of second-order fluids, Arch. Rational Mech. Anal., 14 (1963), 1-26.  doi: 10.1007/BF00250690.

[51]

T. W. Ting, Parabolic and pseudo-parabolic partial differential equations, J. Math. Soc. Japan, 21 (1969), 440-453.  doi: 10.2969/jmsj/02130440.

[52]

R. WangY. Li and B. Wang, Random dynamics of fractional nonclassical diffusion equations driven by colored noise, Discrete Contin. Dyn. Syst., 39 (2019), 4091-4126.  doi: 10.3934/dcds.2019165.

[53]

J. Weidmann, Linear Operators in Hilbert Spaces, Translated from the German by Joseph Szücs. Graduate Texts in Mathematics, 68. Springer-Verlag, New York-Berlin, 1980.

[54]

W. Wyss, The fractional diffusion equation, J. Math. Physics, 27 (1986), 2782-2785.  doi: 10.1063/1.527251.

[55]

H. YeJ. Gao and Y. Ding, A generalized Gronwall inequality and its application to a fractional differential equation, J. Math. Analysis and Appl., 328 (2007), 1075-1081.  doi: 10.1016/j.jmaa.2006.05.061.

[56]

G. A. Zou and B. Wang, Stochastic Burgers' equation with fractional derivative driven by multiplicative noise, Comput. Math. Appl., 74 (2017), 3195-3208.  doi: 10.1016/j.camwa.2017.08.023.

[57]

Y. Zhou, J. Wang and L. Zhang, Basic Theory of Fractional Differential Equations, 2$^{nd}$ edition, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2017.

show all references

References:
[1]

W. ArendtA. F. Ter Elst and M. Warma, Fractional powers of sectorial operators via the Dirichlet-to-Neumann operator, Comm. Partial Differential Equations, 43 (2018), 1-24.  doi: 10.1080/03605302.2017.1363229.

[2]

S. A. AsogwaM. FoondunJ. B. Mijena and E. Nane, Critical parameters for reaction-diffusion equations involving space-time fractional derivatives, NoDEA Nonlinear Differential Equations Appl., 27 (2020), 1-22.  doi: 10.1007/s00030-020-00629-9.

[3]

V. Van AuH. JafariZ. HammouchN. H. Tuan and (2 019), On a final value problem for a nonlinear fractional pseudo-parabolic equation, Electron. Res. Arch., 29 (2021), 1709-1734.  doi: 10.3934/era.2020088.

[4]

B. BaeumerM. Geissert and M. Kovács, Existence, uniqueness and regularity for a class of semilinear stochastic Volterra equations with multiplicative noise, J. Differential Equations, 258 (2015), 535-554.  doi: 10.1016/j.jde.2014.09.020.

[5]

L. Bai and F. H. Zhang, Existence of random attractors for 2D-stochastic nonclassical diffusion equations on unbounded domains, Results Math., 69 (2016), 129-160.  doi: 10.1007/s00025-015-0505-8.

[6]

G. I. BarenblattY. P. Zheltov and I. N. Kochina, On basic ideas of the theory of filtration of homogeneous fluids in fractured rocks, Prikl. Mat. Mekh., 24 (1960), 852-864. 

[7]

T. B. BenjaminJ. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philos. Trans. Roy. Soc. London Ser., 272 (1972), 47-78.  doi: 10.1098/rsta.1972.0032.

[8]

M. K. Beshtokov, Toward boundary-value problems for degenerating pseudoparabolic equations with a Gerasimov-Caputo fractional derivative, Russian Math., 62 (2018), 1-14. 

[9]

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. 

[10]

M. K. Beshtokov, Boundary value problems for a pseudoparabolic equation with the Caputo fractional derivative, Differe. Equ., 55 (2019), 884-893.  doi: 10.1134/S0012266119070024.

[11]

C. BurgosJ. C. CortésA. DebboucheL. Villafuerte and R. J. Villanueva, Random fractional generalized Airy differential equations: A probabilistic analysis using mean square calculus, Appl. Math. Comput., 352 (2019), 15-29.  doi: 10.1016/j.amc.2019.01.039.

[12]

C. BurgosJ. C. CortésL. Villafuerte and R. J. Villanueva, Solving random mean square fractional linear differential equations by generalized power series: Analysis and computing, J. Compu. Appl. Math., 339 (2018), 94-110.  doi: 10.1016/j.cam.2017.12.042.

[13]

C. BurgosJ.-C. CortésL. Villafuerte and R.-J. Villanueva, Extending the deterministic Riemann-Liouville and Caputo operators to the random framework: A mean square approach with applications to solve random fractional differential equations, Chaos Solitons Fractals, 102 (2017), 305-318.  doi: 10.1016/j.chaos.2017.02.008.

[14]

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

[15]

Y. Cao and C. Liu, Initial boundary value problem for a mixed pseudo-parabolic p-Laplacian type equation with logarithmic nonlinearity. Electron, J. Differential Equations, (2018), 1–19.

[16]

M. Caputo, Linear models of dissipation whose Q is almost frequency independent-II, Geophysical J. Inter., 13 (1967), 529-539. 

[17]

T. CaraballoM. J. Garrido-Atienza and J. Real, Stochastic stabilization of differential systems with general decay rate, Systems Control Lett., 48 (2003), 397-406.  doi: 10.1016/S0167-6911(02)00293-1.

[18]

T. CaraballoM. J. Garrido-Atienza and T. Taniguchi, The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion, Nonlinear Anal: Theory, Methods and Applications, 74 (2011), 3671-3684.  doi: 10.1016/j.na.2011.02.047.

[19]

H. Chen and H. Xu, Global existence, exponential decay and blow-up in finite time for a class of finitely degenerate semilinear parabolic equations, Acta Math. Sci. Ser., 39 (2019), 1290-1308.  doi: 10.1007/s10473-019-0508-8.

[20]

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.

[21] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, 2$^{nd}$ edition, Cambridge University Press, Cambridge, 2014.  doi: 10.1017/CBO9781107295513.
[22]

L. Debbi, Well-posedness of the multidimensional fractional stochastic Navier-Stokes equations on the torus and on bounded domains, J. Math. Fluid Mech., 18 (2016), 25-69.  doi: 10.1007/s00021-015-0234-5.

[23]

H. DiY. Shang and X. Zheng, Global well-posedness for a fourth order pseudo-parabolic equation with memory and source terms, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 781-801.  doi: 10.3934/dcdsb.2016.21.781.

[24]

H. Ding and J. Zhou, Global existence and blow-up for a mixed pseudo-parabolic p-Laplacian type equation with logarithmic nonlinearity, J. Math. Anal. Appl., 478 (2019), 393-420.  doi: 10.1016/j.jmaa.2019.05.018.

[25]

E. S. Dzektser, Equations of motion of ground water with a free surface in multilayer media, Dokl. Akad. Nauk SSSR, 220 (1975), 540-543. 

[26]

M. Foondun, Remarks on a fractional-time stochastic equation, Proc. Amer. Math. Soc., 149 (2018), 2235-2247.  doi: 10.1090/proc/14644.

[27]

M. Foondun and E. Nualart, On the behaviour of stochastic heat equations on bounded domains, ALEA Lat. Am. J. Probab. Math. Stat., 12 (2015), 551-571. 

[28]

Y. HeH. Gao and H. Wang, Blow-up and decay for a class of pseudo-parabolic p-Laplacian equation with logarithmic nonlinearity, Comput. Math. Appl., 75 (2018), 459-469.  doi: 10.1016/j.camwa.2017.09.027.

[29]

G. HuY. Lou and P. D. Christofides, Dynamic output feedback covariance control of stochastic dissipative partial differential equations, Chemical Engineering Science, 63 (2008), 4531-4542. 

[30]

Y. JiangT. Wei and X. Zhou, Stochastic generalized Burgers equations driven by fractional noises, J. Differential Equations, 252 (2012), 1934-1961.  doi: 10.1016/j.jde.2011.07.032.

[31]

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.

[32]

T. Kato, Perturbation theory for linear operators, Springer Science & Business Media, 132 (2013).

[33]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006.

[34]

M. Kovács and S. Larsson, Introduction to stochastic partial differential equations, In Publications of the ICMCS, 4 (2008), 159-232. 

[35]

P. D. Lax, Functional Analysis, Wiley Interscience, New York, 2002.

[36]

F. LiY. Li and R. Wang, Regular measurable dynamics for reaction-diffusion equations on narrow domains with rough noise, Discrete Contin. Dyn. Syst., 38 (2018), 3663-3685.  doi: 10.3934/dcds.2018158.

[37]

Y. Li and Y. Wang, The existence and asymptotic behavior of solutions to fractional stochastic evolution equations with infinite delay, J. Differential Equations, 266 (2019), 3514-3558.  doi: 10.1016/j.jde.2018.09.009.

[38]

Y. Lu and L. Fei, Bounds for blow-up time in a semilinear pseudo-parabolic equation with nonlocal source, J. Inequal. Appl., 2016 (2016), 1-11.  doi: 10.1186/s13660-016-1171-4.

[39]

F. Mainardi, Applications of integral transforms in fractional diffusion processes, Integral Transforms Spec. Funct., 15 (2004), 477-484.  doi: 10.1080/10652460412331270652.

[40]

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

[41]

J. C. Pedjeu and G. S. Ladde, Stochastic fractional differential equations: Modeling, method and analysis, Chaos Solitons Fractals, 45 (2012), 279-293.  doi: 10.1016/j.chaos.2011.12.009.

[42] I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Mathematics in Science and Engineering, 198. Academic Press, Inc., San Diego, CA, 1999. 
[43]

C. Prévôt and M. Röckner, A Concise Course on Stochastic Partial Differential Equations, Lecture Notes in Mathematics, 1905. Springer, Berlin, 2007.

[44]

L. I. Rubinshtein, Heat propagation process in heterogeneous media, Izv. Akad. Nauk SSSR Ser. Geogr., 12 (1948), 27-45. 

[45]

K. Sakamoto and M. Yamamoto, Initial value/boundary 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.

[46]

S. L. Sobolev, On a new problem of mathematical physics, Izv. Akad. Nauk SSSR Ser. Mat., 18 (1954), 3-50. 

[47]

J. V. D. C. Sousa and E. C. de Oliveira, Fractional order pseudoparabolic partial differential equation: Ulam-Hyers stability, Bull. Braz. Math. Soc., 50 (2019), 481-496.  doi: 10.1007/s00574-018-0112-x.

[48]

X. Su and M. Li, The regularity of fractional stochastic evolution equations in Hilbert space, Stoch. Anal. Appl., 36 (2018), 639-653.  doi: 10.1080/07362994.2018.1436973.

[49]

F. I. Taukenova and M. K. Shkhanukov-Lafishev, Difference methods for solving boundary value problems for fractional differential equations, Comput. Math. Math. Phys., 46 (2006), 1785-1795.  doi: 10.1134/S0965542506100149.

[50]

T. W. Ting, Certain non-steady flows of second-order fluids, Arch. Rational Mech. Anal., 14 (1963), 1-26.  doi: 10.1007/BF00250690.

[51]

T. W. Ting, Parabolic and pseudo-parabolic partial differential equations, J. Math. Soc. Japan, 21 (1969), 440-453.  doi: 10.2969/jmsj/02130440.

[52]

R. WangY. Li and B. Wang, Random dynamics of fractional nonclassical diffusion equations driven by colored noise, Discrete Contin. Dyn. Syst., 39 (2019), 4091-4126.  doi: 10.3934/dcds.2019165.

[53]

J. Weidmann, Linear Operators in Hilbert Spaces, Translated from the German by Joseph Szücs. Graduate Texts in Mathematics, 68. Springer-Verlag, New York-Berlin, 1980.

[54]

W. Wyss, The fractional diffusion equation, J. Math. Physics, 27 (1986), 2782-2785.  doi: 10.1063/1.527251.

[55]

H. YeJ. Gao and Y. Ding, A generalized Gronwall inequality and its application to a fractional differential equation, J. Math. Analysis and Appl., 328 (2007), 1075-1081.  doi: 10.1016/j.jmaa.2006.05.061.

[56]

G. A. Zou and B. Wang, Stochastic Burgers' equation with fractional derivative driven by multiplicative noise, Comput. Math. Appl., 74 (2017), 3195-3208.  doi: 10.1016/j.camwa.2017.08.023.

[57]

Y. Zhou, J. Wang and L. Zhang, Basic Theory of Fractional Differential Equations, 2$^{nd}$ edition, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2017.

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