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doi: 10.3934/dcdss.2021118
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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 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 & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2021118
References:
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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[8]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[26]

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

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

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

[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.  Google Scholar

[34]

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

[35]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[39]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar
[43]

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

[44]

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

[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.  Google Scholar

[46]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[50]

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

[51]

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

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