Random dynamics for a stochastic nonlocal reaction-diffusion equation with an energy functional

Ruonan Liu; Tomás Caraballo; Ruonan Liu; Tomás Caraballo

doi:10.3934/math.2024390

AIMS Mathematics

2024, Volume 9, Issue 4: 8020-8042. doi: 10.3934/math.2024390

Previous Article Next Article

Research article

Random dynamics for a stochastic nonlocal reaction-diffusion equation with an energy functional

Ruonan Liu ¹,
Tomás Caraballo ^{2
,
,}

1.
School of Mathematics and Statistics, Xuzhou University of Technology, Jiangsu, 221008, China
2.
Departamento de Ecuaciones Diferenciales y Análisis Numérico, C/ Tarfia s/n, Facultad de Matemáticas, Universidad de Sevilla, 41012 Sevilla, Spain

Received: 25 January 2024 Revised: 14 February 2024 Accepted: 20 February 2024 Published: 26 February 2024
MSC : 35F05, 60H15

In this paper, the asymptotic behavior of solutions to a fractional stochastic nonlocal reaction-diffusion equation with polynomial drift terms of arbitrary order in an unbounded domain was analysed. First, the stochastic equation was transformed into a random one by using a stationary change of variable. Then, we proved the existence and uniqueness of solutions for the random problem based on pathwise uniform estimates as well as the energy method. Finally, the existence of a unique pullback attractor for the random dynamical system generated by the transformed equation is shown.
- nonlocal reaction-diffusion equations,
- fractional Laplacian,
- unbounded domain,
- random attractors
Citation: Ruonan Liu, Tomás Caraballo. Random dynamics for a stochastic nonlocal reaction-diffusion equation with an energy functional[J]. AIMS Mathematics, 2024, 9(4): 8020-8042. doi: 10.3934/math.2024390

Related Papers:

Abstract

In this paper, the asymptotic behavior of solutions to a fractional stochastic nonlocal reaction-diffusion equation with polynomial drift terms of arbitrary order in an unbounded domain was analysed. First, the stochastic equation was transformed into a random one by using a stationary change of variable. Then, we proved the existence and uniqueness of solutions for the random problem based on pathwise uniform estimates as well as the energy method. Finally, the existence of a unique pullback attractor for the random dynamical system generated by the transformed equation is shown.

References

[1]	M. Chipot, Elements of Nonlinear Analysis, Birkhäuser, Basel, 2000.
[2]	M. Chipot, B. Lovat, Some remarks on nonlocal elliptic and parabolic problems, Nonlinear Anal., 30 (1997), 4619–4627. https://doi.org/10.1016/S0362-546X(97)00169-7 doi: 10.1016/S0362-546X(97)00169-7
[3]	M. Chipot, J. F. Rodrigues, On a class of nonlocal nonlinear elliptic problems, RAIRO Modél. Math. Anal. Numér., 26 (1992), 447–467. https://doi.org/10.1051/M2AN/1992260304471 doi: 10.1051/M2AN/1992260304471
[4]	B. Lovat, Etudes de quelques problèmes paraboliques non locaux, PhD Thesis, Uniersité de Metz, 1995.
[5]	Y. Shi, X. H. Yang, Pointwise error estimate of conservative difference scheme for supergeneralized viscous Burgers' equation, Electron. Res. Arch., 32 (2024), 1471–1497. https://doi.org/10.3934/era.2024068 doi: 10.3934/era.2024068
[6]	J. W. Wang, X. X. J, H. X. Zhang, A BDF3 and new nonlinear fourth-order difference scheme for the generalized viscous Burgers' equation, Appl. Math. Lett., 151 (2024), 109002. https://doi.org/10.1016/j.aml.2024.109002 doi: 10.1016/j.aml.2024.109002
[7]	J. W. Wang, X. Xiao, X. H. Yang, H. X. Zhang, A nonlinear compact method based on double reduction order scheme for the nonlocal fourth-order PDEs with Burgers' type nonlinearity, J. Appl. Math. Comput., 70 (2024), 489–511. https://doi.org/10.1007/s12190-023-01975-4 doi: 10.1007/s12190-023-01975-4
[8]	X. Wang, X. H. Yang, Z. Y. Zhou, X. Wang, X. J. Yang, Z. Y. Zhou, Pointwise-in-time-robust error estimate of the ADI difference scheme for three-dimensional fractional subdiffusion equations with variable coefficients, Commun. Anal. Mech., 16 (2024), 53–70. https://doi.org/10.3934/cam.2024003 doi: 10.3934/cam.2024003
[9]	C. J. Li, H. X. Zhang, X. H. Yang, A new-robust nonlinear numerical algorithm for the time fractional nonlinear KdV equation, Commun. Anal. Mech., 16 (2024), 147–168. https://doi.org/10.3934/cam.2024007 doi: 10.3934/cam.2024007
[10]	J. H. Xu, Z. C. Zhang, T. Caraballo, Mild solutions to time fractional stochastic 2D-Stokes equations with bounded and unbounded delay, J. Dynam. Differential Equations, 34 (2022), 583–603. https://doi.org/10.1007/s10884-019-09809-3 doi: 10.1007/s10884-019-09809-3
[11]	H. X. Zhang, X. X. Jiang, F. R. Wang, X. H. Yang, The time two-grid algorithm combined with difference scheme for 2D nonlocal nonlinear wave equation. J. Appl. Math. Comput., (2024). https://doi.org/10.1007/s12190-024-02000-y doi: 10.1007/s12190-024-02000-y
[12]	R. Servadei, E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105–2137. https://doi.org/10.3934/dcds.2013.33.2105 doi: 10.3934/dcds.2013.33.2105
[13]	R. Servadei, E. Valdinoci, On the spectrum of two different fractional operators, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 831–855. https://doi.org/10.1017/S0308210512001783 doi: 10.1017/S0308210512001783
[14]	J. H. Xu, Z. C. Zhang, T. Caraballo, Non-autonomous nonlocal partial differential equations with delay and memory, J. Differential Equations, 270 (2021), 505–546. https://doi.org/10.1016/j.jde.2020.07.037 doi: 10.1016/j.jde.2020.07.037
[15]	M. Fardi, M. A. Zaky, A. S. Hendy, Nonuniform difference schemes for multi-term and distributed-order fractional parabolic equations with fractional Laplacian, Math. Comput. Simulation, 206 (2023), 614–635. https://doi.org/10.1016/j.matcom.2022.12.009 doi: 10.1016/j.matcom.2022.12.009
[16]	M. Fardi, S. K. Q. Al-Omari, S. Araci, A pseudo-spectral method based on reproducing kernel for solving the time-fractional diffusion-wave equation, Adv. Contin. Discrete Models, (2022), Paper No. 54, 14. https://doi.org/10.1186/s13662-022-03726-4
[17]	S. Mohammadi, M. Fardi, M. Ghasemi, A numerical investigation with energy-preservation for nonlinear space-fractional Klein-Gordon-Schrödinger system, Comput. Appl. Math., 42 (2023), Paper No. 356, 27. https://doi.org/10.1007/s40314-023-02495-4 doi: 10.1007/s40314-023-02495-4
[18]	S. Mohammadi, M. Ghasemi, M. Fardi, A fast Fourier spectral exponential time-differencing method for solving the time-fractional mobile-immobile advection-dispersion equation, Comput. Appl. Math., 41 (2022), Paper No. 264, 26. https://doi.org/10.1007/s40314-022-01970-8 doi: 10.1007/s40314-022-01970-8
[19]	H. L, J. G. Qi, B. X. Wang, M. J. Zhang, Random attractors for non-autonomous fractional stochastic parabolic equations on unbounded domains, Discrete Contin. Dyn. Syst., 39 (2019), 683–706. https://doi.org/10.3934/dcds.2019028 doi: 10.3934/dcds.2019028
[20]	B. X. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differential Equations, 253 (2012), 1544–1583. https://doi.org/10.1016/j.jde.2012.05.015 doi: 10.1016/j.jde.2012.05.015
[21]	B. X. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Discrete Contin. Dyn. Syst., 34 (2014), 269–300. https://doi.org/10.3934/dcds.2014.34.269 doi: 10.3934/dcds.2014.34.269
[22]	H. Crauel, A. Debussche, F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307–341. https://doi.org/10.1007/BF02219225 doi: 10.1007/BF02219225
[23]	H. Crauel, F. Flandoli, Attractors for random dynamical systems, Probab. Th. Rel. Fields, 100 (1994), 365–393. https://doi.org/10.1007/BF01193705 doi: 10.1007/BF01193705
[24]	B. Schmalfuß, Backward cocycles and attractors of stochastic differential equations, International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior, 185–192, Dresden, 1992.
[25]	E. Di Nezza, G. Palatucci, E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521–573. https://doi.org/10.1016/j.bulsci.2011.12.004 doi: 10.1016/j.bulsci.2011.12.004
[26]	B. X. Wang, Asymptotic behavior of non-autonomous fractional stochastic reaction-diffusion equations, Nonlinear Anal., 158 (2017), 60–82. https://doi.org/10.1016/j.na.2017.04.006 doi: 10.1016/j.na.2017.04.006
[27]	J. H. Xu, T. Caraballo, Dynamics of stochastic nonlocal reaction-diffusion equations driven by multiplicative noise, Anal. Appl., 21 (2023), 597–633. https://doi.org/10.1142/S0219530522500075 doi: 10.1142/S0219530522500075
[28]	J. L. Lions, Quelques Methodes de Resolution des Problemes aux Limites Non Lineaires, Dunod, Paris, 1969.
[29]	J. H. Xu, T. Caraballo, Long time behavior of stochastic nonlocal partial differential equations and Wong-Zakai approximations, SIAM J. Math. Anal., 54 (2022), 2792–2844. https://doi.org/10.1137/21M1412645 doi: 10.1137/21M1412645

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)