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Asymptotic behavior of non-autonomous random Ginzburg-Landau equation driven by colored noise
Dynamics of non-autonomous fractional reaction-diffusion equations on $ \mathbb{R}^{N} $ driven by multiplicative noise
1. | Hunan Province Cooperative Innovation Center for the Construction, and Development of Dongting Lake Ecological Economic Zone & College of Mathematics and Physics Science, Hunan University of Arts and Science, Changde, 415000, China |
2. | School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China |
3. | School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha, 410114, China |
4. | Department of Mathematics, New Mexico Institute of Mining and Technology, Socorro, NM 87801, USA |
We examine the asymptotic behavior of the non-autonomous non-local fractional stochastic reaction-diffusion equations on $ \mathbb{R}^{N} $ with the nonlinearity $ f $ satisfying the polynomial growth of arbitrary order $ p-1 $ $ (p\geq2) $. We first establish a Nash-Moser-Alikakos type a priori estimate for the difference of solutions near the initial time. Then, we prove that the solution process is continuous from $ L^{2}(\mathbb{R}^{N}) $ to $ H^{s}(\mathbb{R}^{N}) $ with respect to initial data for $ s\in(0, 1) $. As an application of the higher-order integrability and the continuity, we obtain the pullback $ \mathcal{D} $-random attractors in $ L^{p}(\mathbb{R}^{N}) $ and $ H^{s}(\mathbb{R}^{N}) $, respectively. This is a natural and necessary extension of current existing results to further understand the dynamics of the underlying problem.
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show all references
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A. Adili and B. Wang,
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A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland Publishing Co., Amsterdam, 1992. |
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P. W. Bates, H. Lisei and K. Lu,
Attractors for stochastic lattice dynamical system, Stoch. Dyn., 6 (2006), 1-21.
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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. |
[6] |
P. W. Bates, K. Lu and B. Wang,
Attractors of non-autonomous stochastic lattice systems in weighted spaces, Phys. D, 289 (2014), 32-50.
doi: 10.1016/j.physd.2014.08.004. |
[7] |
W. J. Beyn, B. Gess, P. Lescot and M. Röckner,
The global random attractor for a class of stochastic porous media equations, Comm. Partial Differential Equations, 36 (2011), 446-469.
doi: 10.1080/03605302.2010.523919. |
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L. Caffarelli, S. Salsa and L. Silvestre,
Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Invent. Math., 171 (2008), 425-461.
doi: 10.1007/s00222-007-0086-6. |
[9] |
T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuss and J. Valero,
Non-autonomous and random attractors for delay random semilinear equations without uniqueness, Discrete Contin. Dyn. Syst, 21 (2008), 415-443.
doi: 10.3934/dcds.2008.21.415. |
[10] |
T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuss and J. Valero,
Asymptotic behaviour of a stochastic semilinear dissipative functional equation without uniqueness of solutions, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 439-455.
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[11] |
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On the upper semicontinuity of cocycle attractors for non-autonomous and random dynamical systems, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 10 (2003), 491-513.
|
[13] |
T. Caraballo, J. A. Langa, V. S. Melnik and J. Valero,
Pullback attractors for nonautonomous and stochastic multivalued dynamical systems, Set-Valued Anal., 11 (2003), 153-201.
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[14] |
T. Caraballo, J. Real and I. D. Chueshov,
Pullback attractors for stochastic heat equations in materials with memory, Discrete Contin. Dyn. Syst. B, 9 (2008), 525-539.
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D. Cao, C. Sun and M. Yang,
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[17] |
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[18] |
I. Chueshov and M. Scheutzow,
On the structure of attractors and invariant measures for a class of monotone random systems, Dyn. Syst., 19 (2004), 127-144.
doi: 10.1080/1468936042000207792. |
[19] |
I. Chueshov, Monotone Random Systems Theory and Applications, Lecture Notes in Mathematics, vol. 1779, Springer, Berlin, 2002.
doi: 10.1007/b83277. |
[20] |
H. Crauel,
Random point attractors versus random set attractors, J. London Math. Soc., 63 (2001), 413-427.
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[21] |
H. Crauel, A. Debussche and F. Flandoli,
Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341.
doi: 10.1007/BF02219225. |
[22] |
H. Crauel, G. Dimitroff and M. Scheutzow,
Criteria for strong and weak random attractors, J. Dynam. Differential Equations, 21 (2009), 233-247.
doi: 10.1007/s10884-009-9135-8. |
[23] |
H. Crauel and F. Flandoli,
Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393.
doi: 10.1007/BF01193705. |
[24] |
H. Crauel and F. Flandoli,
Hausdorff dimension of invariant sets for random dynamical systems, J. Dynam. Differential Equations, 10 (1998), 449-474.
doi: 10.1023/A:1022605313961. |
[25] |
H. Crauel, P. E. Kloeden and J. Real,
Stochastic partial differential equations with additive noise on time-varying domains, Bol. Soc. Esp. Mat. Apl. SeMA, 51 (2010), 41-48.
doi: 10.1007/bf03322552. |
[26] |
H. Crauel, P. E. Kloeden and M. Yang,
Random attractors of stochastic reaction-diffusion equations on variable domains, Stoch. Dyn., 11 (2011), 301-314.
doi: 10.1142/S0219493711003292. |
[27] |
H. Crauel and M. Scheutzow,
Minimal random attractors, J. Differential Equations, 265 (2018), 702-718.
doi: 10.1016/j.jde.2018.03.011. |
[28] |
H. Cui and P. E. Kloeden,
Invariant forward attractors of non-autonomous random dynamical systems, J. Differential Equations, 265 (2018), 6166-6186.
doi: 10.1016/j.jde.2018.07.028. |
[29] |
J. Dong and M. Xu,
Space-time fractional Schrödinger equation with time-independent potentials, J. Math. Anal. Appl., 344 (2008), 1005-1017.
doi: 10.1016/j.jmaa.2008.03.061. |
[30] |
J. Droniou and C. Imbert,
Fractal first-order partial differential equations, Arch. Ration. Mech. Anal., 182 (2006), 299-331.
doi: 10.1007/s00205-006-0429-2. |
[31] |
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. |
[32] |
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. |
[33] |
A. Gu and B. Wang,
Asymptotic behavior of random Fitzhugh-Nagumo systems driven by colored noise, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 1689-1720.
doi: 10.3934/dcdsb.2018072. |
[34] |
B. Guo and Z. Huo,
Global well-posedness for the fractional nonlinear Schrödinger equation, Comm. Partial Differential Equations, 36 (2011), 247-255.
doi: 10.1080/03605302.2010.503769. |
[35] |
B. Guo, Y. Han and J. Xin,
Existence of the global smooth solution to the period boundary value problem of fractional nonlinear Schrödinger equation, Appl. Math. Comput., 204 (2008), 468-477.
doi: 10.1016/j.amc.2008.07.003. |
[36] |
X. Guo and M. Xu, Some physical applications of fractional Schrödinger equation, J. Math. Phys., 47 (2006), 082104, 1–9.
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G. Karch, Nonlinear evolution equations with anomalous diffusion: Qualitative properties of solutions to partial differential equations, J. Nečas Cent. Math. Model. Lect. Notes, vol. 5, Matfyzpress, Prague, 2009, 25–68. |
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P. E. Kloeden and J. A. Langa,
Flattening, squeezing and the existence of random attractors, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 163-181.
doi: 10.1098/rspa.2006.1753. |
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