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Existence results for fractional differential equations in presence of upper and lower solutions
On the asymptotic behavior of solutions to time-fractional elliptic equations driven by a multiplicative white noise
Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, 10307 Ha Noi, Viet Nam |
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.
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R. Gorenflo, Y. Luchko and M. Yamamoto,
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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.
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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. |
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R. Zacher,
Weak solutions of abstract evolutionary integro-differential equations in Hilbert spaces, Funkcialj Ekvacioj, 52 (2009), 1-18.
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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. |
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [8] |
L. C. Evans, Partial Differential Equations., Graduate Series in Mathematics, 19. American Mathematics Society, 1998. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
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