First order necessary conditions of optimality for the two dimensional tidal dynamics system
December 2021, 11(4): 715-737. doi: 10.3934/mcrf.2020044
Limiting behavior of fractional stochastic integro-Differential equations on unbounded domains
Ji Shu 1, , Linyan Li 2, , Xin Huang 3, and Jian Zhang 4,,
School of Mathematical Sciences and V.C. & V.R. Key Lab, Sichuan Normal University, Chengdu, Sichuan 610068, China
School of Mathematical Sciences, Sichuan Normal University, Chengdu, Sichuan 610068, China
Department of Basic Courses, Sichuan Vocational College of Finance and Economics, Chengdu, Sichuan 610101, China
School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, China
* Corresponding author: Jian Zhang
Received May 2020 Revised July 2020 Published December 2021 Early access October 2020
Under a Creative Commons license
We consider the dynamical behavior of fractional stochastic integro-differential equations with additive noise on unbounded domains. The existence and uniqueness of tempered random attractors for the equation in $ \mathbb{R}^{3} $ are proved. The upper semicontinuity of random attractors is also obtained when the intensity of noise approaches zero. The main difficulty is to show the pullback asymptotic compactness due to the lack of compactness on unbounded domains and the fact that the memory term includes the whole past history of the phenomenon. We establish such compactness by the tail-estimate method and the splitting method.
Keywords: Random attractor, stochastic integro-differential equation with memory, unbounded domains, asymptotic compactness, upper semicontinuity.
Mathematics Subject Classification: Primary: 37L55, 60H15; Secondary: 35Q56.
Citation: Ji Shu, Linyan Li, Xin Huang, Jian Zhang. Limiting behavior of fractional stochastic integro-Differential equations on unbounded domains. Mathematical Control & Related Fields, 2021, 11 (4) : 715-737. doi: 10.3934/mcrf.2020044
L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7. Google Scholar
Q. Bai, J. Shu, L. Li and H. Li, Dynamical behavior of non-autonomous fractional stochastic reaction-diffusion equations, J. Math. Anal. Appl., 485 (2020), 123833. doi: 10.1016/j.jmaa.2019.123833. Google Scholar
P. W. Bates, H. Lisei and K. Lu, Attractors for stochastic lattice dynamical systems, Stoch. Dyn., 6 (2006), 1-21. doi: 10.1142/S0219493706001621. Google Scholar
P. W. Bates, K. Lu and B. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 246 (2009), 845-869. doi: 10.1016/j.jde.2008.05.017. Google Scholar
T. Caraballo, J. Real and I. D. Chueshov, Pullback attractors for stochastic heat equations in materials with memory, Discrete Contin. Dyn. Syst. Ser. B, 9 (2008), 525-539. doi: 10.3934/dcdsb.2008.9.525. Google Scholar
T. Caraballo, I. D. Chueshov, P. Marin-Rubio and J. Real, Existence and asymptotic behaviour for stochastic heat equations with multiplicative noise in materials with memory, Discrete Contin. Dyn. Syst., 18 (2007), 253-270. doi: 10.3934/dcds.2007.18.253. Google Scholar
T. Caraballo, J. A. Langa, V. S. Melnik and J. Valero, Pullback attractors of nonautonomous and stochastic multivalued dynamical systems, Set–Valued Anal., 11 (2003), 153-201. doi: 10.1023/A:1022902802385. Google Scholar
T. Caraballo, G. Lukaszewicz and J. Real, Pullback attractors for asymptotically compact nonautonomous dynamical systems, Nonlinear Anal., 64 (2006), 484-498. doi: 10.1016/j.na.2005.03.111. Google Scholar
M. Conti, V. Pata and M. Squassina, Singular limit of differential systems with memory, Indiana Univ. Math. J., 55 (2006), 169-215. doi: 10.1512/iumj.2006.55.2661. Google Scholar
H. Crauel and P. E. Kloeden, Nonautonomous and random attractors, Jahresber. Dtsch. Math.-Ver., 117 (2015), 173-206. doi: 10.1365/s13291-015-0115-0. Google Scholar
H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393. doi: 10.1007/BF01193705. Google Scholar
H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341. doi: 10.1007/BF02219225. Google Scholar
C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Ration. Mech. Anal., 37 (1970), 297-308. doi: 10.1007/BF00251609. Google Scholar
E. DiNezza, G. Palatucci and E. Valdinoci, Hitchhikers guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004. Google Scholar
J. Dong and M. Xu, Space-time fractional Schr$\ddot{o}$dinger equation with time-independent potentials, J. Math. Anal. Appl., 344 (2008), 1005-1017. doi: 10.1016/j.jmaa.2008.03.061. Google Scholar
F. Flandoli and B. Schmalfuss, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative white noise, Stochastics Stochastics Rep., 59 (1996), 21-45. doi: 10.1080/17442509608834083. Google Scholar
C. Gal and M. Warma, Reaction-diffusion equations with fractional diffusion on non-smooth domains with various boundary conditions, Discrete Contin. Dyn. Syst., 36 (2016), 1279-1319. doi: 10.3934/dcds.2016.36.1279. Google Scholar
C. Giorgi, V. Pata and A. Marzocchi, Asymptotic behavior of a semilinear problem in heat conduction with memory, NoDEA Nonlinear Differential Equations Appl., 5 (1998), 333-354. doi: 10.1007/s000300050049. Google Scholar
M. Grasselli and V. Pata, Uniform attractors of nonautonomous dynamical systems with memory, in Progress in Nonlinear Differential Equations and Their Applications, Vol. 50, Birkh$\ddot{a}$user, Basel, 2002,155–178. Google Scholar
A. Gu, D. Li, B. Wang and H. Yang, Regularity of random attractors for fractional stochastic reaction-diffusion equations on $\mathbb{R}^{n}$, J. Differential Equations, 264 (2018), 7094-7137. doi: 10.1016/j.jde.2018.02.011. Google Scholar
B. Guo, Y. Han and J. Xin, Existence of the global smooth solution to the period boundary value problem of fractional nonlinear Schr$\ddot{o}$dinger equation, Appl. Math. Comput., 204 (2008), 468-477. doi: 10.1016/j.amc.2008.07.003. Google Scholar
B. Guo and Z. Huo, Global well-posedness for the fractional nonlinear Schr$\ddot{o}$dinger equation, Comm. Partial Differential Equations, 36 (2011), 247-255. doi: 10.1080/03605302.2010.503769. Google Scholar
B. Guo and M. Zeng, Solutions for the fractional Landau-Lifshitz equation, J. Math. Anal. Appl., 361 (2010), 131-138. doi: 10.1016/j.jmaa.2009.09.009. Google Scholar
B. Guo and G. Zhou, Ergodicity of the stochastic fractional reaction diffusion equation, Nonlinear Anal., 109 (2014), 1-22. doi: 10.1016/j.na.2014.06.008. Google Scholar
C. Guo, J. Shu and X. Wang, Fractal dimension of random attractors for non-autonomous fractional stochastic Ginzburg-Landau equations, Acta Math. Sin.(Engl. Ser.), 36 (2020), 318-336. doi: 10.1007/s10114-020-8407-4. Google Scholar
Y. Lan and J. Shu, Fractal dimension of random attractors for non-autonomous fractional stochastic Ginzburg-Landau equations with multiplicative noise, Dyn. Syst., 34 (2019), 274-300. doi: 10.1080/14689367.2018.1523368. Google Scholar
Y. Lan and J. Shu, Dynamics of non-autonomous fractional stochastic Ginzburg-Landau equations with multiplicative noise, Commun. Pure Appl. Anal., 18 (2019), 2409-2431. doi: 10.3934/cpaa.2019109. Google Scholar
D. Li, B. Wang and X. Wang, Limiting behavior of non-autonomous stochastic reaction-diffusion equations on thin domains, J. Differential Equtions, 262 (2017), 1575-1602. doi: 10.1016/j.jde.2016.10.024. Google Scholar
L. Li, J. Shu, Q. Bai and H. Li, Asymptotic behavior of fractional stochastic heat equations in materials with memory, Appl. Anal. (2019). doi: 10.1080/00036811.2019.1597057. Google Scholar
J.-L. Lions, Quelques methodes de resolution des problemes aux limites non lineaires, Dunod, Gauthier-Villars, Paris, 1969. Google Scholar
L. Liu and T. Caraballo, Well-posedness and dynamics of a fractional stochastic integro-differential equation, Phys. D, 355 (2017), 45-57. doi: 10.1016/j.physd.2017.05.006. Google Scholar
H. Lu, P. W. Bates, J. Xin and M. Zhang, Asymptotic behavior of stochastic fractional power dissipative equations on $\mathbb{R}^{n}$, Nonlinear Anal., 128 (2015), 176-198. doi: 10.1016/j.na.2015.06.033. Google Scholar
H. Lu, P. W. Bates, S. Lu and M. Zhang, Dynamics of 3-D fractional complex Ginzburg-Landau equation, J. Differential Equations, 259 (2015), 5276-5301. doi: 10.1016/j.jde.2015.06.028. Google Scholar
G. Lv and J. Duan, Martingale and weak solutions for a stochastic nonlocal Burgers equation on finite intervals, J. Math. Anal. Appl., 449 (2017), 176-194. doi: 10.1016/j.jmaa.2016.12.011. Google Scholar
G. Lv, H. Gao, J. Wei and J. Wu, BMO and Morrey Campanato estimates for stochastic convolutions and Schauder estimates for stochastic parabolic equations, J. Differential Equations, 266 (2019), 2666-2717. doi: 10.1016/j.jde.2018.08.042. Google Scholar
F. Morillas and J. Valero, Attractors for reaction-diffusion equations in $\mathbb{R}^{n}$ with continuous nonlinearity, Asymptot. Anal., 44 (2005), 111-130. Google Scholar
L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 13 (1959), 115-162. Google Scholar
V. Pata and A. Zucchi, Attractors for a damped hyperbolic equation with linear memory, Adv. Math. Sci. Appl., 11 (2001), 505-529. Google Scholar
X. Pu and B. Guo, Global weak solutions of the fractional Landau-Lifshitz-Maxwell equation, J. Math. Anal. Appl., 372 (2010), 86-98. doi: 10.1016/j.jmaa.2010.06.035. Google Scholar
X. Pu and B. Guo, Well-posedness and dynamics for the fractional Ginzburg-Laudau equation, Appl. Anal., 92 (2013), 318-334. doi: 10.1080/00036811.2011.614601. Google Scholar
B. Schmalfuss, Backward cocycle and atttractors of stochastic differential equations, in International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior, Technische Universit$\ddot{a}$t, (1992), 185–192. Google Scholar
T. Shen and J. Huang, Well-posedness and dynamics of stochastic fractional model for nonlinear optical fiber materials, Nonlinear Anal., 110 (2014), 33-46. doi: 10.1016/j.na.2014.06.018. Google Scholar
J. Shu, Random attractors for stochastic discrete Klein-Gordon-Schrödinger equations driven by fractional Brownian motions, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1587-1599. doi: 10.3934/dcdsb.2017077. Google Scholar
J. Shu, P. Li, J. Zhang and O. Liao, Random attractors for the stochastic coupled fractional Ginzburg-Landau equation with additive noise, J. Math. Phys., 56 (2015), 102702. doi: 10.1063/1.4934724. Google Scholar
R. Temam, Infinite Dimension Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3. Google Scholar
B. Wang, Asymptotic behavior of non-autonomous fractional stochastic reaction-diffusion equations, Nonlinear Anal., 158 (2017), 60-82. doi: 10.1016/j.na.2017.04.006. Google Scholar
B. Wang, Upper semicontinuity of random attractors for non-compact random systems, Electron J. Differential Equations, 139 (2009), 1-18. Google Scholar
B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differential Equations, 253 (2012), 1544-1583. doi: 10.1016/j.jde.2012.05.015. Google Scholar
B. Wang, Existence and upper-semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms, Stoch. Dyn., 14 (2014), 1450009. 1–31. doi: 10.1142/S0219493714500099. Google Scholar
B. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Discrete Contin. Dyn. Syst., 34 (2014), 269-300. doi: 10.3934/dcds.2014.34.269. Google Scholar
X. Wang, S. Li and D. Xu, Random attractors for second-order stochastic lattice dynamical systems, Nonlinear Anal., 72 (2010), 483-494. doi: 10.1016/j.na.2009.06.094. Google Scholar
X. Wang, K. Lu and B. Wang, Wong-Zakai approximations and attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 264 (2018), 378-424. doi: 10.1016/j.jde.2017.09.006. Google Scholar
S. Zhou, Random exponential attractor for stochastic reaction-diffusion equation with multiplicative noise in $\mathbb{R}^{3}$, J. Differential Equations, 263 (2017), 6347-6383. doi: 10.1016/j.jde.2017.07.013. Google Scholar
Ling Xu, Jianhua Huang, Qiaozhen Ma. Upper semicontinuity of random attractors for the stochastic non-autonomous suspension bridge equation with memory. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 5959-5979. doi: 10.3934/dcdsb.2019115
Shihchung Chiang. Numerical optimal unbounded control with a singular integro-differential equation as a constraint. Conference Publications, 2013, 2013 (special) : 129-137. doi: 10.3934/proc.2013.2013.129
Xiaobin Yao. Asymptotic behavior for stochastic plate equations with memory and additive noise on unbounded domains. Discrete & Continuous Dynamical Systems - B, 2022, 27 (1) : 443-468. doi: 10.3934/dcdsb.2021050
Zhaojuan Wang, Shengfan Zhou. Existence and upper semicontinuity of random attractors for non-autonomous stochastic strongly damped wave equation with multiplicative noise. Discrete & Continuous Dynamical Systems, 2017, 37 (5) : 2787-2812. doi: 10.3934/dcds.2017120
Shuang Yang, Yangrong Li. Forward controllability of a random attractor for the non-autonomous stochastic sine-Gordon equation on an unbounded domain. Evolution Equations & Control Theory, 2020, 9 (3) : 581-604. doi: 10.3934/eect.2020025
Walter Allegretto, John R. Cannon, Yanping Lin. A parabolic integro-differential equation arising from thermoelastic contact. Discrete & Continuous Dynamical Systems, 1997, 3 (2) : 217-234. doi: 10.3934/dcds.1997.3.217
Narcisa Apreutesei, Nikolai Bessonov, Vitaly Volpert, Vitali Vougalter. Spatial structures and generalized travelling waves for an integro-differential equation. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 537-557. doi: 10.3934/dcdsb.2010.13.537
Frederic Abergel, Remi Tachet. A nonlinear partial integro-differential equation from mathematical finance. Discrete & Continuous Dynamical Systems, 2010, 27 (3) : 907-917. doi: 10.3934/dcds.2010.27.907
Samir K. Bhowmik, Dugald B. Duncan, Michael Grinfeld, Gabriel J. Lord. Finite to infinite steady state solutions, bifurcations of an integro-differential equation. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 57-71. doi: 10.3934/dcdsb.2011.16.57
Zhaojuan Wang, Shengfan Zhou. Existence and upper semicontinuity of attractors for non-autonomous stochastic lattice systems with random coupled coefficients. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2221-2245. doi: 10.3934/cpaa.2016035
Michel Chipot, Senoussi Guesmia. On a class of integro-differential problems. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1249-1262. doi: 10.3934/cpaa.2010.9.1249
Eduardo Cuesta. Asymptotic behaviour of the solutions of fractional integro-differential equations and some time discretizations. Conference Publications, 2007, 2007 (Special) : 277-285. doi: 10.3934/proc.2007.2007.277
Linfang Liu, Xianlong Fu. Existence and upper semicontinuity of (L2, Lq) pullback attractors for a stochastic p-laplacian equation. Communications on Pure & Applied Analysis, 2017, 6 (2) : 443-474. doi: 10.3934/cpaa.2017023
Ahmed Y. Abdallah. Upper semicontinuity of the attractor for a second order lattice dynamical system. Discrete & Continuous Dynamical Systems - B, 2005, 5 (4) : 899-916. doi: 10.3934/dcdsb.2005.5.899
María Astudillo, Marcelo M. Cavalcanti. On the upper semicontinuity of the global attractor for a porous medium type problem with large diffusion. Evolution Equations & Control Theory, 2017, 6 (1) : 1-13. doi: 10.3934/eect.2017001
Xinjie Dai, Aiguo Xiao, Weiping Bu. Stochastic fractional integro-differential equations with weakly singular kernels: Well-posedness and Euler–Maruyama approximation. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021225
Bahareh Akhtari, Esmail Babolian, Andreas Neuenkirch. An Euler scheme for stochastic delay differential equations on unbounded domains: Pathwise convergence. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 23-38. doi: 10.3934/dcdsb.2015.20.23
Olivier Bonnefon, Jérôme Coville, Jimmy Garnier, Lionel Roques. Inside dynamics of solutions of integro-differential equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3057-3085. doi: 10.3934/dcdsb.2014.19.3057
Nestor Guillen, Russell W. Schwab. Neumann homogenization via integro-differential operators. Discrete & Continuous Dynamical Systems, 2016, 36 (7) : 3677-3703. doi: 10.3934/dcds.2016.36.3677
Paola Loreti, Daniela Sforza. Observability of $N$-dimensional integro-differential systems. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 745-757. doi: 10.3934/dcdss.2016026
Ji Shu Linyan Li Xin Huang Jian Zhang