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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  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:
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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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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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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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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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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

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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

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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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