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March  2021, 26(3): 1749-1762. doi: 10.3934/dcdsb.2020318

On the asymptotic behavior of solutions to time-fractional elliptic equations driven by a multiplicative white noise

Hoang The Tuan ,

Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, 10307 Ha Noi, Viet Nam

 

Received  February 2020 Revised  July 2020 Published  March 2021 Early access  November 2020

Fund Project: This research is supported by a grant from the Vietnam Academy of Science and Technology under the grant number DLTE00.01–20/21

This paper is devoted to study of time-fractional elliptic equations driven by a multiplicative noise. By combining the eigenfunction expansion method for symmetry elliptic operators, the variation of constant formula for strong solutions to scalar stochastic fractional differential equations, Ito's formula and establishing a new weighted norm associated with a Lyapunov–Perron operator defined from this representation of solutions, we show the asymptotic behaviour of solutions to these systems in the mean square sense. As a consequence, we also prove existence, uniqueness and the convergence rate of their solutions.

Keywords: Stochastic partial differential equations, time fractional derivatives, mild solution, existence and uniqueness, asymptotic behavior.
Mathematics Subject Classification: Primary: 34A08, 35A01, 35B20; Secondary: 35B40, 60H15, 35R60.
Citation: Hoang The Tuan. On the asymptotic behavior of solutions to time-fractional elliptic equations driven by a multiplicative white noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1749-1762. doi: 10.3934/dcdsb.2020318
References:
[1]

M. Allen, L. Caffarelli and A. Vasseur, A parabolic problem with a fractional time derivative, Archive for Rational Mechanics and Analysis, 221 (2016), 603-630.  doi: 10.1007/s00205-016-0969-z.  Google Scholar

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P. T. Anh, T. S. Doan and P. T. Huong, A variation of constant formula for Caputo fractional stochastic differential equations, Statistics and Probability Letters, 145 (2019), 351-358.  doi: 10.1016/j.spl.2018.10.010.  Google Scholar

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L. Chen, Y. Hu and D. Nualart, Nonlinear stochastic time-fractional slow and fast diffusion equations on $ \mathbb{R}^d$, Stochastic Processes and their Applications, 129 (2019), 5073-5112.  doi: 10.1016/j.spa.2019.01.003.  Google Scholar

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Z.-Q. Chen, K.-H. Kim and P. Kim, Fractional time stochastic partial differential equations, Stochastic Processes and their Applications, 125 (2015), 1470-1499.  doi: 10.1016/j.spa.2014.11.005.  Google Scholar

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N. D. Cong, T. S. Doan, S. Siegmund and H. T. Tuan, On stable manifolds for planar fractional differential equations, Applied Mathematics and Computation, 226 (2014), 157-168.  doi: 10.1016/j.amc.2013.10.010.  Google Scholar

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S. D. Eidelman and A. N. Kochubei, Cauchy problem for fractional diffusion equations, Journal of Differential Equations, 199 (2004), 211-255.  doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[8]

L. C. Evans, Partial Differential Equations., Graduate Series in Mathematics, 19. American Mathematics Society, 1998.  Google Scholar

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M. Ginoa, S. Cerbelli and H. E. Roman, Fractional diffusion equation and relaxation in complex viscoelastic material, Physica A: Statistical Mechanics and its Applications, 191 (1992), 449-453.  doi: 10.1016/0378-4371(92)90566-9.  Google Scholar

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R. Gorenflo, A. A. Kilbas, F. Mainardi and S. V. Rogosin, Mittag-Leffler Functions, Related Topics and Applications, Springer Monographs in Mathematics, Springer, Heidelberg, 2014. doi: 10.1007/978-3-662-43930-2.  Google Scholar

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R. Gorenflo, Y. Luchko and M. Yamamoto, Time-fractional diffusion equation in the fractional Sobolev spaces, Fractional Calculus and Applied Analysis, 18 (2015), 799-820.  doi: 10.1515/fca-2015-0048.  Google Scholar

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T. D. Ke, N. N. Thang and L. T. P. Thuy, Regularity and stability analysis fro a class of semilinear nonlocal differential equations in Hilbert spaces, Journal of Mathematical Analysis and Applications, 483 (2020), 123655. doi: 10.1016/j.jmaa.2019.123655.  Google Scholar

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P. E. Kloeden and E. Platen, Numerical Solutions of Stochastic Differential Equations, Stochastic Modelling and Applied Probability. Springer-Verlag Berlin Heidelberg, New York, 1992. doi: 10.1007/978-3-662-12616-5.  Google Scholar

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W. Liu, M. Röckner and J. L. da Silva., Quasi-linear (stochastic) partial differential equations with time-fractional derivatives, SIAM Journal on Mathematical Analysis, 50 (2018), 2588-2607.  doi: 10.1137/17M1144593.  Google Scholar

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R. Metzler and J. Klafter, Boundary value problems for fractional diffusion equations, Physica A: Statistical Mechanics and its Applications, 278 (2000), 107-125.  doi: 10.1016/S0378-4371(99)00503-8.  Google Scholar

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R. R. Nigmatullin, The realization of the generalized transfer equation in a medium with fractal geometry, Physica Status Solidi (b), 133 (1986), 425-430.  doi: 10.1002/pssb.2221330150.  Google Scholar

[17]

I. Podlubny, Fractional Differential Equations, An Introduction to Fractional Derivatives, Fractional Differential Equations, To Methods of Their Solution and Some of Their Applications, Academic Press, Inc., San Diego, CA, 1999.  Google Scholar

[18]

H. E. Roman and P. A. Alemany, Continuous-time random walks and the fractional diffusion equation, Journal of Physics A: Mathematical and General, 27 (1994), 3407-3410.  doi: 10.1088/0305-4470/27/10/017.  Google Scholar

[19]

K. Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, Journal of Mathematical Analysis and Applications, 382 (2011), 426-447.  doi: 10.1016/j.jmaa.2011.04.058.  Google Scholar

[20]

D. T. Son, P. T. Huong, P. E. Kloeden and H. T. Tuan, Asymptotic separation between solutions of Caputo fractional stochastic differential equations, Stochastic Analysis and Applications, 36 (2018), 654-664.  doi: 10.1080/07362994.2018.1440243.  Google Scholar

[21]

R. Zacher, Weak solutions of abstract evolutionary integro-differential equations in Hilbert spaces, Funkcialj Ekvacioj, 52 (2009), 1-18.  doi: 10.1619/fesi.52.1.  Google Scholar

[22]

R. Zacher, A De Giorgi–Nash type theorem for time fractional diffusion equations, Mathematische Annalen, 356 (2013), 99-146.  doi: 10.1007/s00208-012-0834-9.  Google Scholar

show all references

References:
[1]

M. Allen, L. Caffarelli and A. Vasseur, A parabolic problem with a fractional time derivative, Archive for Rational Mechanics and Analysis, 221 (2016), 603-630.  doi: 10.1007/s00205-016-0969-z.  Google Scholar

[2]

P. T. Anh, T. S. Doan and P. T. Huong, A variation of constant formula for Caputo fractional stochastic differential equations, Statistics and Probability Letters, 145 (2019), 351-358.  doi: 10.1016/j.spl.2018.10.010.  Google Scholar

[3]

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

[4]

L. Chen, Y. Hu and D. Nualart, Nonlinear stochastic time-fractional slow and fast diffusion equations on $ \mathbb{R}^d$, Stochastic Processes and their Applications, 129 (2019), 5073-5112.  doi: 10.1016/j.spa.2019.01.003.  Google Scholar

[5]

Z.-Q. Chen, K.-H. Kim and P. Kim, Fractional time stochastic partial differential equations, Stochastic Processes and their Applications, 125 (2015), 1470-1499.  doi: 10.1016/j.spa.2014.11.005.  Google Scholar

[6]

N. D. Cong, T. S. Doan, S. Siegmund and H. T. Tuan, On stable manifolds for planar fractional differential equations, Applied Mathematics and Computation, 226 (2014), 157-168.  doi: 10.1016/j.amc.2013.10.010.  Google Scholar

[7]

S. D. Eidelman and A. N. Kochubei, Cauchy problem for fractional diffusion equations, Journal of Differential Equations, 199 (2004), 211-255.  doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[8]

L. C. Evans, Partial Differential Equations., Graduate Series in Mathematics, 19. American Mathematics Society, 1998.  Google Scholar

[9]

M. Ginoa, S. Cerbelli and H. E. Roman, Fractional diffusion equation and relaxation in complex viscoelastic material, Physica A: Statistical Mechanics and its Applications, 191 (1992), 449-453.  doi: 10.1016/0378-4371(92)90566-9.  Google Scholar

[10]

R. Gorenflo, A. A. Kilbas, F. Mainardi and S. V. Rogosin, Mittag-Leffler Functions, Related Topics and Applications, Springer Monographs in Mathematics, Springer, Heidelberg, 2014. doi: 10.1007/978-3-662-43930-2.  Google Scholar

[11]

R. Gorenflo, Y. Luchko and M. Yamamoto, Time-fractional diffusion equation in the fractional Sobolev spaces, Fractional Calculus and Applied Analysis, 18 (2015), 799-820.  doi: 10.1515/fca-2015-0048.  Google Scholar

[12]

T. D. Ke, N. N. Thang and L. T. P. Thuy, Regularity and stability analysis fro a class of semilinear nonlocal differential equations in Hilbert spaces, Journal of Mathematical Analysis and Applications, 483 (2020), 123655. doi: 10.1016/j.jmaa.2019.123655.  Google Scholar

[13]

P. E. Kloeden and E. Platen, Numerical Solutions of Stochastic Differential Equations, Stochastic Modelling and Applied Probability. Springer-Verlag Berlin Heidelberg, New York, 1992. doi: 10.1007/978-3-662-12616-5.  Google Scholar

[14]

W. Liu, M. Röckner and J. L. da Silva., Quasi-linear (stochastic) partial differential equations with time-fractional derivatives, SIAM Journal on Mathematical Analysis, 50 (2018), 2588-2607.  doi: 10.1137/17M1144593.  Google Scholar

[15]

R. Metzler and J. Klafter, Boundary value problems for fractional diffusion equations, Physica A: Statistical Mechanics and its Applications, 278 (2000), 107-125.  doi: 10.1016/S0378-4371(99)00503-8.  Google Scholar

[16]

R. R. Nigmatullin, The realization of the generalized transfer equation in a medium with fractal geometry, Physica Status Solidi (b), 133 (1986), 425-430.  doi: 10.1002/pssb.2221330150.  Google Scholar

[17]

I. Podlubny, Fractional Differential Equations, An Introduction to Fractional Derivatives, Fractional Differential Equations, To Methods of Their Solution and Some of Their Applications, Academic Press, Inc., San Diego, CA, 1999.  Google Scholar

[18]

H. E. Roman and P. A. Alemany, Continuous-time random walks and the fractional diffusion equation, Journal of Physics A: Mathematical and General, 27 (1994), 3407-3410.  doi: 10.1088/0305-4470/27/10/017.  Google Scholar

[19]

K. Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, Journal of Mathematical Analysis and Applications, 382 (2011), 426-447.  doi: 10.1016/j.jmaa.2011.04.058.  Google Scholar

[20]

D. T. Son, P. T. Huong, P. E. Kloeden and H. T. Tuan, Asymptotic separation between solutions of Caputo fractional stochastic differential equations, Stochastic Analysis and Applications, 36 (2018), 654-664.  doi: 10.1080/07362994.2018.1440243.  Google Scholar

[21]

R. Zacher, Weak solutions of abstract evolutionary integro-differential equations in Hilbert spaces, Funkcialj Ekvacioj, 52 (2009), 1-18.  doi: 10.1619/fesi.52.1.  Google Scholar

[22]

R. Zacher, A De Giorgi–Nash type theorem for time fractional diffusion equations, Mathematische Annalen, 356 (2013), 99-146.  doi: 10.1007/s00208-012-0834-9.  Google Scholar

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