Finite fractal dimension of random attractors for non-autonomous fractional stochastic reaction-diffusion equations in \(\mathbb{R}\). (English) Zbl 1497.37098
Summary: This paper investigates the asymptotic behavior of solutions for non-autonomous fractional stochastic reaction-diffusion equations with multiplicative noise in \(\mathbb{R}\) We prove the existence and uniqueness of the tempered pullback random attractor for the equations in \(L^2(\mathbb{R})\) and obtain the finite fractal dimension for the pullback random attractor. Two main difficulties here are that the fractional Laplacian operator is non-local and the Sobolev embedding is not compact on unbounded domains. To solve this, we derive the tail-estimates of solutions of the equation and decompose the solutions into a sum of three parts, which one part is finite-dimensional and other two parts are quickly decay in mean sense.
MSC:
| 37L55 | Infinite-dimensional random dynamical systems; stochastic equations |
| 37L30 | Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems |
| 35R11 | Fractional partial differential equations |
| 35R60 | PDEs with randomness, stochastic partial differential equations |
| 35Q56 | Ginzburg-Landau equations |
| 35K57 | Reaction-diffusion equations |
| 60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
| 26A33 | Fractional derivatives and integrals |
Keywords:
random dynamical system; fractional reaction-diffusion equation; multiplicative noise; random attractor; fractal dimensionReferences:
| [1] | Dong, J.; Xu, M., Space-time fractional Schrödinger equation with time-independent potentials, J Math Anal Appl, 344, 1005-1017 (2008) · Zbl 1140.81357 |
| [2] | Guan, Q.; Ma, Z., Boundary problems for fractional laplacians, Stoch Dyn, 5, 385-424 (2005) · Zbl 1077.60036 |
| [3] | Lan, Y.; Shu, J., Fractal dimension of random attractors for non-autonomous fractional stochastic Ginzburg-Landau equations with multiplicative noise, Dyn Syst, 34, 274-300 (2019) · Zbl 1415.37100 |
| [4] | Lu, H.; Bates, PW; Lu, S., Dynamics of the 3D fractional Ginzburg-Landau equation with multiplicative noise on a unbounded domain, Comm Math Sci, 14, 273-295 (2016) · Zbl 1331.37070 |
| [5] | Pu, X.; Guo, B., Well-posedness and dynamics for the fractional Ginzburg-Laudau equation, Appl Anal, 92, 318-334 (2013) · Zbl 1293.35310 |
| [6] | Shu, J.; Li, P.; Zhang, J., Random attractors for the stochastic coupled fractional Ginzburg-Landau equation with additive noise, J Math Phys, 56 (2015) · Zbl 1341.35153 |
| [7] | Guo, B.; Zeng, M., Soltuions for the fractional Landau-Lifshitz equation, J Math Anal Appl, 361, 131-138 (2010) · Zbl 1179.35320 |
| [8] | Gu, A.; Li, D.; Wang, B., Regularity of random attractors for fractional stochastic reaction-diffusion equations on \(####\), J Differ Equ, 264, 7094-7137 (2018) · Zbl 1388.35231 |
| [9] | Wang, B., Asymptotic behavior of non-autonomous fractional stochastic reaction-diffusion equations, Nonlinear Anal, 158, 60-82 (2017) · Zbl 1366.35229 |
| [10] | Zhou, S.; Tian, Y.; Wang, W., Fractal dimension of random attractors for stochastic non-autonomous reaction-diffusion equations, Appl Math Comput, 276, 80-95 (2016) · Zbl 1410.37068 |
| [11] | Servadei, R.; Valdinoci, E., On the spectrum of two different fractional operators, Proc Roy Soc Edinburgh Sect A, 144, 831-855 (2014) · Zbl 1304.35752 |
| [12] | Cao, D.; Sun, C.; Yang, M., Dynamics for a stochastic reaction-diffusion equation with additive noise, J Differ Equ, 259, 838-872 (2015) · Zbl 1323.35226 |
| [13] | Caraballo, T.; Langa, J.; Robinson, JC., Staility and random attractors for a reaction-diffusion equation with multiplicative noise, Discrete Contin Dyn Syst, 6, 875-892 (2000) · Zbl 1011.37031 |
| [14] | Crauel, H.; Flandoli, F., Attractors for random dynamical systems, Probab Theory Related Fields, 100, 365-393 (1994) · Zbl 0819.58023 |
| [15] | Debussche, A., Hausdorff dimension of a random invariant set, J Math Pures Appl, 77, 967-988 (1998) · Zbl 0919.58044 |
| [16] | Guo, C.; Shu, J.; Wang, X., Fractal dimension of random attractors for non-autonomous fractional stochastic Ginzburg-Landau equations, Acta Math Sin (Engl Ser), 36, 3, 318-336 (2020) · Zbl 1440.37075 |
| [17] | Li, D.; Wang, B.; Wang, X., Limiting behavior of non-autonomous stochastic reaction-diffusion equations on thin domains, J Differ Equ, 262, 1575-1602 (2017) · Zbl 1360.60125 |
| [18] | Li, J.; Li, Y.; Wang, B., Random attractors of reaction-diffusion equations with multiplicative noise in \(####\), Appl Math Comput, 215, 3399-3407 (2010) · Zbl 1190.37061 |
| [19] | Li, Y.; Guo, B., Random attractors for quasi-continuous random dynamical systems and applications to stochastic reaction-diffusion equations, J Differ Equ, 245, 1775-1800 (2008) · Zbl 1188.37076 |
| [20] | Ma, D.; Shu, J.; Qin, L., Wong-Zakai approximations and asymptotic behavior of stochastic Ginzburg-Landau equations, Discrete Contin Dyn Syst Ser B (2020) · Zbl 1457.37098 · doi:10.3934/dcdsb.2020100 |
| [21] | Wang, G.; Tang, Y., Random attractors for stochastic reaction-diffusion equations with multiplicative noise in \(####\), Math Nachr, 287, 1774-1791 (2014) · Zbl 1332.37058 |
| [22] | Zhou, S.; Zhao, M., Fractal dimension of random attractor for stochastic damped wave equation with multiplicative noise, Discrete Contin Dyn Syst, 263, 2247-2279 (2017) · Zbl 1364.37158 |
| [23] | Bates, PW; Lu, K.; Wang, B., Random attractors for stochastic reaction-diffusion equations on unbounded domains, J Differ Equ, 246, 845-869 (2009) · Zbl 1155.35112 |
| [24] | Chueshov, ID., Gevrey ragularity of random attractors for stochastic reaction-diffusion equations, Random Oper Stoch Equ, 8, 143-162 (2000) · Zbl 0959.60045 |
| [25] | Chepyzhov, VV; Efendiev, MA., Hausdorff dimension estimation for attractors of nonautonomous dynamical systems in unbounded domains: an example, Comm Pure Appl Math, 53, 647-665 (2000) · Zbl 1022.37048 |
| [26] | Wang, B., Attractors for reaction-diffusion equations in unbounded domains, Phys D, 128, 41-52 (1999) · Zbl 0953.35022 |
| [27] | Wang, B., Upper semicontinuity of random attractors for non-compact random systems, J Differ Equ, 139, 1-18 (2009) · Zbl 1181.37111 |
| [28] | Wang, B., Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differ Equ, 253, 1544-1583 (2012) · Zbl 1252.35081 |
| [29] | Wang, X.; Lu, K.; Wang, B., Wong-Zakai approximations and attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differ Equ, 264, 378-424 (2018) · Zbl 1379.60074 |
| [30] | Wang, Z.; Zhou, S., Random attractor for stochastic reaction-diffusion equation with multiplicative noise on unbounded domains, J Math Anal Appl, 384, 160-172 (2011) · Zbl 1227.60087 |
| [31] | Zhao, W.; Li, Y., \(####\)-random attractors for stochastic reaction-diffusion equation on unbounded domains, Nonlinear Anal, 75, 485-502 (2012) · Zbl 1229.60081 |
| [32] | Zhou, S., Random exponential attractor for stochastic reaction-diffusion equation with multiplicative noise in \(####\), J Differ Equ, 263, 6347-6383 (2017) · Zbl 1373.37173 |
| [33] | Bekmaganbetov, KA; Chechkin, GA; Chepyzhov, VV., Strong convergence of trajectory attractors for reaction-diffusion systems with random rapidly oscillating terms, Commun Pure Appl Anal, 19, 5, 2419-2443 (2020) · Zbl 1435.35072 |
| [34] | Bekmaganbetov, KA; Chechkin, GA; Chepyzhov, VV., Weak convergence of attractors of reaction-diffusion systems with randomly oscillating coefficients, Appl Anal, 98, 1-2, 256-271 (2019) · Zbl 1411.35043 |
| [35] | Bekmaganbetov, KA; Chechkin, GA; Chepyzhov, VV, Homogenization of trajectory attractors of 3D Navier-Stokes system with randomly oscillating force, Discrete Contin Dyn Syst, 37, 5, 2375-2393 (2017) · Zbl 1357.35045 |
| [36] | Chechkin, GA; Chepyzhov, VV; Pankratov, LS., Homogenization of trajectory attractors of Ginzburg-Landau equations with randomly oscillating terms, Discrete Contin Dyn Syst Ser B, 23, 3, 1133-1154 (2018) · Zbl 1404.35023 |
| [37] | Bai, Q.; Shu, J.; Li, L., Dynamical behavior of non-autonomous fractional stochastic reaction-diffusion equations, J Math Anal Appl, 485 (2020) · Zbl 1431.60052 |
| [38] | Lan, Y.; Shu, J., Dynamics of non-autonomous fractional stochastic Ginzburg-Landau equations with multiplicative noise, Commun Pure Appl Anal, 18, 2409-2431 (2019) · Zbl 1489.37092 |
| [39] | Li, L.; Shu, J.; Bai, Q., Asymptotic behavior of fractional stochastic heat equations in materials with memory, Appl Anal (2019) · Zbl 1458.37076 · doi:10.1080/00036811.2019.1597057 |
| [40] | Liu, L.; Fu, X., Dynamics of a stochastic fractional reaction-diffusion equation, Taiwanese J Math, 22, 95-124 (2018) · Zbl 1398.35274 |
| [41] | Lu, H.; Bates, PW; Xin, J., Asymptotic behavior of stochastic fractional power dissipative equations on \(####\), Nonlinear Anal, 128, 176-198 (2015) · Zbl 1337.37062 |
| [42] | Wang, B., Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Discrete Contin Dyn Syst, 34, 269-300 (2014) · Zbl 1277.35068 |
| [43] | Crauel, H.; Debussche, A.; Flandoli, F., Random attractors, J Dynam Differ Equ, 9, 307-341 (1997) · Zbl 0884.58064 |
| [44] | Flandoli, F.; Schmalfuss, B., Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative white noise, Stoch Stoch Rep, 59, 21-45 (1996) · Zbl 0870.60057 |
| [45] | Schmalfuss, B.Backward cocycle and atttractors of stochastic differential equations. In: Reitmann V, Riedrich T, Koksch N, editors. International semilar on applied mathematics-nonlinear dynamics: attractor approximation and global behavior. Dresden: Technische Universität; 1992. p. 185-192. |
| [46] | Arnold, L., Random dynamical systems (1998), Berlin: Springer, Berlin · Zbl 0906.34001 |
| [47] | Caraballo, T.; Kloeden, PE; Schmalfuß, B., Exponentially stable stationary solutions for stochastic evolution equations and their perturbation, Appl Math Optim, 50, 183-207 (2004) · Zbl 1066.60058 |
| [48] | Debussche, A., On the finite dimensionality of random attractors, Stoch Anal Appl, 15, 473-491 (1997) · Zbl 0888.60051 |
| [49] | Di Nezza, E.; Palatucci, G.; Valdinoci, E., Hitchhiker’s guide to the fractional Sobolev spaces, Bull Sci Math, 136, 521-573 (2012) · Zbl 1252.46023 |
| [50] | Tang, B., Regularity of random attractors for stochastic reaction-diffusion equations on unbounded domains, Stoch Dyn, 16 (2016) · Zbl 1331.35409 |
| [51] | Morosi, C.; Pizzocchero, L., On the constants for some fractional Gagliardo-Nirenberg and Sobolev inequalities, Expo Math, 36, 32-77 (2018) · Zbl 1396.46030 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
