October  2021, 26(10): 5681-5705. doi: 10.3934/dcdsb.2020376

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

* Corresponding author: Mingji Zhang

Received  June 2020 Published  October 2021 Early access  December 2020

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.

Citation: Kaixuan Zhu, Ji Li, Yongqin Xie, Mingji Zhang. Dynamics of non-autonomous fractional reaction-diffusion equations on $ \mathbb{R}^{N} $ driven by multiplicative noise. Discrete and Continuous Dynamical Systems - B, 2021, 26 (10) : 5681-5705. doi: 10.3934/dcdsb.2020376
References:
[1]

L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.

[2]

A. Adili and B. Wang, Random attractors for stochastic FitzHugh-Nagumo systems driven by deterministic non-autonomous forcing, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 643-666.  doi: 10.3934/dcdsb.2013.18.643.

[3]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland Publishing Co., Amsterdam, 1992.

[4]

P. W. BatesH. Lisei and K. Lu, Attractors for stochastic lattice dynamical system, Stoch. Dyn., 6 (2006), 1-21.  doi: 10.1142/S0219493706001621.

[5]

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. BatesK. 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. BeynB. GessP. 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.

[8]

L. CaffarelliS. 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. CaraballoM. J. Garrido-AtienzaB. 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. CaraballoM. J. Garrido-AtienzaB. 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.  doi: 10.3934/dcdsb.2010.14.439.

[11]

T. CaraballoM. J. Garrido-Atienza and T. Taniguchi, The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion, Nonlinear Anal., 74 (2011), 3671-3684.  doi: 10.1016/j.na.2011.02.047.

[12]

T. Caraballo and J. A. Langa, 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. CaraballoJ. A. LangaV. S. Melnik and J. Valero, Pullback attractors for nonautonomous and stochastic multivalued dynamical systems, Set-Valued Anal., 11 (2003), 153-201.  doi: 10.1023/A:1022902802385.

[14]

T. CaraballoJ. Real and I. D. Chueshov, Pullback attractors for stochastic heat equations in materials with memory, Discrete Contin. Dyn. Syst. B, 9 (2008), 525-539.  doi: 10.3934/dcdsb.2008.9.525.

[15]

D. CaoC. Sun and M. Yang, Dynamics for a stochastic reaction-diffusion equation with additive noise, J. Differential Equations, 259 (2015), 838-872.  doi: 10.1016/j.jde.2015.02.020.

[16]

A. N. Carvalho, J. A. Langa, J. C. Robinson, Attractors for Infinite-Dimensional Nonautonomous Dynamical Systems, Applied Mathematical Sciences, Vol. 182, Springer, New York, 2013. doi: 10.1007/978-1-4614-4581-4.

[17]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Amer. Math. Soc. Colloq. Publ., Vol. 49, AMS, Providence, RI, 2002.

[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.  doi: 10.1017/S0024610700001915.

[21]

H. CrauelA. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341.  doi: 10.1007/BF02219225.

[22]

H. CrauelG. 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. CrauelP. 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. CrauelP. 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. GuD. LiB. 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. GuoY. 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. doi: 10.1063/1.2235026.

[37]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Amer. Math. Soc., Providence, RI, 1988. doi: 10.1090/surv/025.

[38]

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.

[39]

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.

[40]

P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Mathematical Surveys and Monographs, vol. 176, American Mathematical Society, Providence, RI, 2011. doi: 10.1090/surv/176.

[41]

N. Laskin, Fractional Schrödinger equation, Phys. Rev. E, 66 (2002), 056108, 1–7. doi: 10.1103/PhysRevE.66.056108.

[42]

N. Laskin, Fractional quantum mechanics and Levy path integrals, Phys. Lett. A, 268 (2000), 298-305.  doi: 10.1016/S0375-9601(00)00201-2.

[43]

N. Laskin, Fractional Quantum Mechanics, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2018. doi: 10.1142/10541.

[44]

H. LuP. W. BatesS. Lü and M. Zhang, Dynamics of the 3D fractional Ginzburg-Landau equation with multiplicative noise on an unbounded domain, Commun. Math. Sci., 14 (2016), 273-295.  doi: 10.4310/CMS.2016.v14.n1.a11.

[45]

H. LuP. W. BatesJ. 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.

[46]

H. LuJ. QiB. Wang and M. Zhang, Random attractors for non-autonomous fractional stochastic parabolic equations on unbounded domains, Discrete Contin. Dyn. Syst., 39 (2019), 683-706.  doi: 10.3934/dcds.2019028.

[47]

Y. Li and B. Guo, Random attractors for quasi-continuous random dynamical systems and applications to stochastic reaction-diffusion equations, J. Differential Equations, 245 (2008), 1775-1800.  doi: 10.1016/j.jde.2008.06.031.

[48]

J. Li and L. Xia, Well-posedness of fractional Ginzburg-Landau equation in Sobolev spaces, Appl. Anal., 92 (2013), 1074-1084.  doi: 10.1080/00036811.2011.649733.

[49]

A. Miranville and S. V. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, in Handbook of Differential Equations: Evolutionary Equations, Vol. 4, Elsevier/North-Holland, Amsterdam, 2008,103–200. doi: 10.1016/S1874-5717(08)00003-0.

[50]

E. W. Montroll and M. F. Shlesinger, On the wonderful world of random walks, Nonequilibrium Phenomena, II, 1–121, North-Holland, Amsterdam, 1984.

[51]

E. Di NezzaG. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.

[52]

M. Naber, Time fractional Schrödinger equation, J. Math. Phys., 45 (2004), 3339-3352.  doi: 10.1063/1.1769611.

[53]

R. R. Nigmatullin, The realization of the generalized transfer equation in a medium with fractal geometry, Phys. Stat. Sol. (B), 133 (1986), 425-430. 

[54]

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.

[55] J. C. Robinson, Infinite-Dimensional Dynamical Systems. An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge University Press, Cambridge, 2001.  doi: 10.1007/978-94-010-0732-0.
[56]

S. Salsa, Optimal regularity in lower dimensional obstacle problems. Subelliptic PDE's and applications to geometry and finance, Lect. Notes Semin. Interdiscip. Mat, Potenza, 2007,217–226.

[57]

A. I. Saichev and G. M. Zaslavsky, Fractional kinetic equations: Solutions and applications, Chaos, 7 (1997), 753-764.  doi: 10.1063/1.166272.

[58]

M. F. ShlesingerG. M. Zaslavsky and J. Klafter, Strange kinetics, Nature, 363 (1993), 31-37. 

[59]

Y. Sire and E. Valdinoci, Fractional Laplacian phase transitions and boundary reactions: A geometric inequality and a symmetry result, J. Funct. Anal., 256 (2009), 1842-1864.  doi: 10.1016/j.jfa.2009.01.020.

[60]

Z. ShenS. Zhou and W. Shen, One-dimensional random attractor and rotation number of the stochastic damped sine-Gordon equation, J. Differential Equations, 248 (2010), 1432-1457.  doi: 10.1016/j.jde.2009.10.007.

[61]

B. Tang, Regularity of pullback random attractors for stochastic Fitzhugh-Nagumo system on unbounded domains, Discrete Contin. Dyn. Syst., 35 (2015), 441-466.  doi: 10.3934/dcds.2015.35.441.

[62]

W. Tan and C. Sun, Dynamics for a non-autonomous reaction diffusion model with the fractional diffusion, Discrete Contin. Dyn. Syst., 37 (2017), 6035-6067.  doi: 10.3934/dcds.2017260.

[63]

V. E. Tarasov and G. M. Zaslavsky, Fractional Ginzburg-Landau equation for fractal media, Physica A, 354 (2005), 249-261. 

[64]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.

[65]

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.

[66]

B. Wang, Random attractors for the stochastic Benjamin-Bona-Mahony equation on unbounded domains, J. Differential Equations, 246 (2009), 2506-2537.  doi: 10.1016/j.jde.2008.10.012.

[67]

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.

[68]

B. Wang, Attractors for reaction-diffusion equations in unbounded domains, Phys. D, 128 (1999), 41-52.  doi: 10.1016/S0167-2789(98)00304-2.

[69]

G. M. Zaslavsky, Chaos, fractional kinetics, and anomalous transport, Phys. Rep., 371 (2002) 461–580. doi: 10.1016/S0370-1573(02)00331-9.

[70] G. M. Zaslavsky, Hamiltonian Chaos and Fractional Dynamics, Oxford University Press, Oxford, 2008. 
[71]

G. M. Zaslavsky and M. Edelman, Weak mixing and anomalous kinetics along filamented surfaces, Chaos, 11 (2001), 295-305.  doi: 10.1063/1.1355358.

show all references

References:
[1]

L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.

[2]

A. Adili and B. Wang, Random attractors for stochastic FitzHugh-Nagumo systems driven by deterministic non-autonomous forcing, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 643-666.  doi: 10.3934/dcdsb.2013.18.643.

[3]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland Publishing Co., Amsterdam, 1992.

[4]

P. W. BatesH. Lisei and K. Lu, Attractors for stochastic lattice dynamical system, Stoch. Dyn., 6 (2006), 1-21.  doi: 10.1142/S0219493706001621.

[5]

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. BatesK. 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. BeynB. GessP. 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.

[8]

L. CaffarelliS. 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. CaraballoM. J. Garrido-AtienzaB. 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. CaraballoM. J. Garrido-AtienzaB. 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.  doi: 10.3934/dcdsb.2010.14.439.

[11]

T. CaraballoM. J. Garrido-Atienza and T. Taniguchi, The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion, Nonlinear Anal., 74 (2011), 3671-3684.  doi: 10.1016/j.na.2011.02.047.

[12]

T. Caraballo and J. A. Langa, 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. CaraballoJ. A. LangaV. S. Melnik and J. Valero, Pullback attractors for nonautonomous and stochastic multivalued dynamical systems, Set-Valued Anal., 11 (2003), 153-201.  doi: 10.1023/A:1022902802385.

[14]

T. CaraballoJ. Real and I. D. Chueshov, Pullback attractors for stochastic heat equations in materials with memory, Discrete Contin. Dyn. Syst. B, 9 (2008), 525-539.  doi: 10.3934/dcdsb.2008.9.525.

[15]

D. CaoC. Sun and M. Yang, Dynamics for a stochastic reaction-diffusion equation with additive noise, J. Differential Equations, 259 (2015), 838-872.  doi: 10.1016/j.jde.2015.02.020.

[16]

A. N. Carvalho, J. A. Langa, J. C. Robinson, Attractors for Infinite-Dimensional Nonautonomous Dynamical Systems, Applied Mathematical Sciences, Vol. 182, Springer, New York, 2013. doi: 10.1007/978-1-4614-4581-4.

[17]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Amer. Math. Soc. Colloq. Publ., Vol. 49, AMS, Providence, RI, 2002.

[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.  doi: 10.1017/S0024610700001915.

[21]

H. CrauelA. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341.  doi: 10.1007/BF02219225.

[22]

H. CrauelG. 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. CrauelP. 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. CrauelP. 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. GuD. LiB. 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. GuoY. 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. doi: 10.1063/1.2235026.

[37]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Amer. Math. Soc., Providence, RI, 1988. doi: 10.1090/surv/025.

[38]

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.

[39]

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.

[40]

P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Mathematical Surveys and Monographs, vol. 176, American Mathematical Society, Providence, RI, 2011. doi: 10.1090/surv/176.

[41]

N. Laskin, Fractional Schrödinger equation, Phys. Rev. E, 66 (2002), 056108, 1–7. doi: 10.1103/PhysRevE.66.056108.

[42]

N. Laskin, Fractional quantum mechanics and Levy path integrals, Phys. Lett. A, 268 (2000), 298-305.  doi: 10.1016/S0375-9601(00)00201-2.

[43]

N. Laskin, Fractional Quantum Mechanics, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2018. doi: 10.1142/10541.

[44]

H. LuP. W. BatesS. Lü and M. Zhang, Dynamics of the 3D fractional Ginzburg-Landau equation with multiplicative noise on an unbounded domain, Commun. Math. Sci., 14 (2016), 273-295.  doi: 10.4310/CMS.2016.v14.n1.a11.

[45]

H. LuP. W. BatesJ. 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.

[46]

H. LuJ. QiB. Wang and M. Zhang, Random attractors for non-autonomous fractional stochastic parabolic equations on unbounded domains, Discrete Contin. Dyn. Syst., 39 (2019), 683-706.  doi: 10.3934/dcds.2019028.

[47]

Y. Li and B. Guo, Random attractors for quasi-continuous random dynamical systems and applications to stochastic reaction-diffusion equations, J. Differential Equations, 245 (2008), 1775-1800.  doi: 10.1016/j.jde.2008.06.031.

[48]

J. Li and L. Xia, Well-posedness of fractional Ginzburg-Landau equation in Sobolev spaces, Appl. Anal., 92 (2013), 1074-1084.  doi: 10.1080/00036811.2011.649733.

[49]

A. Miranville and S. V. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, in Handbook of Differential Equations: Evolutionary Equations, Vol. 4, Elsevier/North-Holland, Amsterdam, 2008,103–200. doi: 10.1016/S1874-5717(08)00003-0.

[50]

E. W. Montroll and M. F. Shlesinger, On the wonderful world of random walks, Nonequilibrium Phenomena, II, 1–121, North-Holland, Amsterdam, 1984.

[51]

E. Di NezzaG. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.

[52]

M. Naber, Time fractional Schrödinger equation, J. Math. Phys., 45 (2004), 3339-3352.  doi: 10.1063/1.1769611.

[53]

R. R. Nigmatullin, The realization of the generalized transfer equation in a medium with fractal geometry, Phys. Stat. Sol. (B), 133 (1986), 425-430. 

[54]

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.

[55] J. C. Robinson, Infinite-Dimensional Dynamical Systems. An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge University Press, Cambridge, 2001.  doi: 10.1007/978-94-010-0732-0.
[56]

S. Salsa, Optimal regularity in lower dimensional obstacle problems. Subelliptic PDE's and applications to geometry and finance, Lect. Notes Semin. Interdiscip. Mat, Potenza, 2007,217–226.

[57]

A. I. Saichev and G. M. Zaslavsky, Fractional kinetic equations: Solutions and applications, Chaos, 7 (1997), 753-764.  doi: 10.1063/1.166272.

[58]

M. F. ShlesingerG. M. Zaslavsky and J. Klafter, Strange kinetics, Nature, 363 (1993), 31-37. 

[59]

Y. Sire and E. Valdinoci, Fractional Laplacian phase transitions and boundary reactions: A geometric inequality and a symmetry result, J. Funct. Anal., 256 (2009), 1842-1864.  doi: 10.1016/j.jfa.2009.01.020.

[60]

Z. ShenS. Zhou and W. Shen, One-dimensional random attractor and rotation number of the stochastic damped sine-Gordon equation, J. Differential Equations, 248 (2010), 1432-1457.  doi: 10.1016/j.jde.2009.10.007.

[61]

B. Tang, Regularity of pullback random attractors for stochastic Fitzhugh-Nagumo system on unbounded domains, Discrete Contin. Dyn. Syst., 35 (2015), 441-466.  doi: 10.3934/dcds.2015.35.441.

[62]

W. Tan and C. Sun, Dynamics for a non-autonomous reaction diffusion model with the fractional diffusion, Discrete Contin. Dyn. Syst., 37 (2017), 6035-6067.  doi: 10.3934/dcds.2017260.

[63]

V. E. Tarasov and G. M. Zaslavsky, Fractional Ginzburg-Landau equation for fractal media, Physica A, 354 (2005), 249-261. 

[64]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.

[65]

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.

[66]

B. Wang, Random attractors for the stochastic Benjamin-Bona-Mahony equation on unbounded domains, J. Differential Equations, 246 (2009), 2506-2537.  doi: 10.1016/j.jde.2008.10.012.

[67]

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.

[68]

B. Wang, Attractors for reaction-diffusion equations in unbounded domains, Phys. D, 128 (1999), 41-52.  doi: 10.1016/S0167-2789(98)00304-2.

[69]

G. M. Zaslavsky, Chaos, fractional kinetics, and anomalous transport, Phys. Rep., 371 (2002) 461–580. doi: 10.1016/S0370-1573(02)00331-9.

[70] G. M. Zaslavsky, Hamiltonian Chaos and Fractional Dynamics, Oxford University Press, Oxford, 2008. 
[71]

G. M. Zaslavsky and M. Edelman, Weak mixing and anomalous kinetics along filamented surfaces, Chaos, 11 (2001), 295-305.  doi: 10.1063/1.1355358.

[1]

Yuncheng You. Random attractors and robustness for stochastic reversible reaction-diffusion systems. Discrete and Continuous Dynamical Systems, 2014, 34 (1) : 301-333. doi: 10.3934/dcds.2014.34.301

[2]

Dingshi Li, Xuemin Wang. Regular random attractors for non-autonomous stochastic reaction-diffusion equations on thin domains. Electronic Research Archive, 2021, 29 (2) : 1969-1990. doi: 10.3934/era.2020100

[3]

Peter E. Kloeden, Thomas Lorenz. Pullback attractors of reaction-diffusion inclusions with space-dependent delay. Discrete and Continuous Dynamical Systems - B, 2017, 22 (5) : 1909-1964. doi: 10.3934/dcdsb.2017114

[4]

Linfang Liu, Xianlong Fu, Yuncheng You. Pullback attractor in $H^{1}$ for nonautonomous stochastic reaction-diffusion equations on $\mathbb{R}^n$. Discrete and Continuous Dynamical Systems - B, 2017, 22 (10) : 3629-3651. doi: 10.3934/dcdsb.2017143

[5]

Gaocheng Yue. Attractors for non-autonomous reaction-diffusion equations with fractional diffusion in locally uniform spaces. Discrete and Continuous Dynamical Systems - B, 2017, 22 (4) : 1645-1671. doi: 10.3934/dcdsb.2017079

[6]

Tomás Caraballo, José A. Langa, James C. Robinson. Stability and random attractors for a reaction-diffusion equation with multiplicative noise. Discrete and Continuous Dynamical Systems, 2000, 6 (4) : 875-892. doi: 10.3934/dcds.2000.6.875

[7]

María Anguiano, Tomás Caraballo, José Real, José Valero. Pullback attractors for reaction-diffusion equations in some unbounded domains with an $H^{-1}$-valued non-autonomous forcing term and without uniqueness of solutions. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 307-326. doi: 10.3934/dcdsb.2010.14.307

[8]

Piermarco Cannarsa, Giuseppe Da Prato. Invariance for stochastic reaction-diffusion equations. Evolution Equations and Control Theory, 2012, 1 (1) : 43-56. doi: 10.3934/eect.2012.1.43

[9]

Wilhelm Stannat, Lukas Wessels. Deterministic control of stochastic reaction-diffusion equations. Evolution Equations and Control Theory, 2021, 10 (4) : 701-722. doi: 10.3934/eect.2020087

[10]

Xuping Zhang. Pullback random attractors for fractional stochastic $ p $-Laplacian equation with delay and multiplicative noise. Discrete and Continuous Dynamical Systems - B, 2022, 27 (3) : 1695-1724. doi: 10.3934/dcdsb.2021107

[11]

Jihoon Lee, Vu Manh Toi. Attractors for a class of delayed reaction-diffusion equations with dynamic boundary conditions. Discrete and Continuous Dynamical Systems - B, 2020, 25 (8) : 3135-3152. doi: 10.3934/dcdsb.2020054

[12]

Gaocheng Yue. Limiting behavior of trajectory attractors of perturbed reaction-diffusion equations. Discrete and Continuous Dynamical Systems - B, 2019, 24 (10) : 5673-5694. doi: 10.3934/dcdsb.2019101

[13]

Yangrong Li, Shuang Yang, Guangqing Long. Continuity of random attractors on a topological space and fractional delayed FitzHugh-Nagumo equations with WZ-noise. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021303

[14]

Daomin Cao, Guolin Qin. Liouville type theorems for fractional and higher-order fractional systems. Discrete and Continuous Dynamical Systems, 2021, 41 (5) : 2269-2283. doi: 10.3934/dcds.2020361

[15]

Sven Jarohs, Tobias Weth. Asymptotic symmetry for a class of nonlinear fractional reaction-diffusion equations. Discrete and Continuous Dynamical Systems, 2014, 34 (6) : 2581-2615. doi: 10.3934/dcds.2014.34.2581

[16]

Ricardo Enguiça, Andrea Gavioli, Luís Sanchez. A class of singular first order differential equations with applications in reaction-diffusion. Discrete and Continuous Dynamical Systems, 2013, 33 (1) : 173-191. doi: 10.3934/dcds.2013.33.173

[17]

Wei Wang, Anthony Roberts. Macroscopic discrete modelling of stochastic reaction-diffusion equations on a periodic domain. Discrete and Continuous Dynamical Systems, 2011, 31 (1) : 253-273. doi: 10.3934/dcds.2011.31.253

[18]

Yangyang Shi, Hongjun Gao. Homogenization for stochastic reaction-diffusion equations with singular perturbation term. Discrete and Continuous Dynamical Systems - B, 2022, 27 (4) : 2401-2426. doi: 10.3934/dcdsb.2021137

[19]

Yuzo Hosono. Phase plane analysis of travelling waves for higher order autocatalytic reaction-diffusion systems. Discrete and Continuous Dynamical Systems - B, 2007, 8 (1) : 115-125. doi: 10.3934/dcdsb.2007.8.115

[20]

Pengyu Chen, Xuping Zhang. Upper semi-continuity of attractors for non-autonomous fractional stochastic parabolic equations with delay. Discrete and Continuous Dynamical Systems - B, 2021, 26 (8) : 4325-4357. doi: 10.3934/dcdsb.2020290

2021 Impact Factor: 1.497

Metrics

  • PDF downloads (339)
  • HTML views (368)
  • Cited by (0)

Other articles
by authors

[Back to Top]