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  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 & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020376
References:
[1]

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

[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.  Google Scholar

[3]

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

[4]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

I. Chueshov, Monotone Random Systems Theory and Applications, Lecture Notes in Mathematics, vol. 1779, Springer, Berlin, 2002. doi: 10.1007/b83277.  Google Scholar

[20]

H. Crauel, Random point attractors versus random set attractors, J. London Math. Soc., 63 (2001), 413-427.  doi: 10.1017/S0024610700001915.  Google Scholar

[21]

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

[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.  Google Scholar

[23]

H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393.  doi: 10.1007/BF01193705.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

H. Crauel and M. Scheutzow, Minimal random attractors, J. Differential Equations, 265 (2018), 702-718.  doi: 10.1016/j.jde.2018.03.011.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[37]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[41]

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

[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.  Google Scholar

[43]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[50]

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

[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.  Google Scholar

[52]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar
[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.  Google Scholar

[57]

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

[58]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[63]

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

[64]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[69]

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

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

show all references

References:
[1]

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

[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.  Google Scholar

[3]

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

[4]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

I. Chueshov, Monotone Random Systems Theory and Applications, Lecture Notes in Mathematics, vol. 1779, Springer, Berlin, 2002. doi: 10.1007/b83277.  Google Scholar

[20]

H. Crauel, Random point attractors versus random set attractors, J. London Math. Soc., 63 (2001), 413-427.  doi: 10.1017/S0024610700001915.  Google Scholar

[21]

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

[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.  Google Scholar

[23]

H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393.  doi: 10.1007/BF01193705.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

H. Crauel and M. Scheutzow, Minimal random attractors, J. Differential Equations, 265 (2018), 702-718.  doi: 10.1016/j.jde.2018.03.011.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[37]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[41]

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

[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.  Google Scholar

[43]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[50]

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

[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.  Google Scholar

[52]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar
[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.  Google Scholar

[57]

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

[58]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[63]

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

[64]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[69]

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

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

[1]

Lin Shi, Xuemin Wang, Dingshi Li. Limiting behavior of non-autonomous stochastic reaction-diffusion equations with colored noise on unbounded thin domains. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5367-5386. doi: 10.3934/cpaa.2020242

[2]

Chungang Shi, Wei Wang, Dafeng Chen. Weak time discretization for slow-fast stochastic reaction-diffusion equations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021019

[3]

Klemens Fellner, Jeff Morgan, Bao Quoc Tang. Uniform-in-time bounds for quadratic reaction-diffusion systems with mass dissipation in higher dimensions. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 635-651. doi: 10.3934/dcdss.2020334

[4]

Abdelghafour Atlas, Mostafa Bendahmane, Fahd Karami, Driss Meskine, Omar Oubbih. A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020321

[5]

Guillaume Cantin, M. A. Aziz-Alaoui. Dimension estimate of attractors for complex networks of reaction-diffusion systems applied to an ecological model. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020283

[6]

Masaharu Taniguchi. Axisymmetric traveling fronts in balanced bistable reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3981-3995. doi: 10.3934/dcds.2020126

[7]

Yangrong Li, Shuang Yang, Qiangheng Zhang. Odd random attractors for stochastic non-autonomous Kuramoto-Sivashinsky equations without dissipation. Electronic Research Archive, 2020, 28 (4) : 1529-1544. doi: 10.3934/era.2020080

[8]

Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319

[9]

Hideki Murakawa. Fast reaction limit of reaction-diffusion systems. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1047-1062. doi: 10.3934/dcdss.2020405

[10]

Nabahats Dib-Baghdadli, Rabah Labbas, Tewfik Mahdjoub, Ahmed Medeghri. On some reaction-diffusion equations generated by non-domiciliated triatominae, vectors of Chagas disease. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021004

[11]

Weiwei Liu, Jinliang Wang, Yuming Chen. Threshold dynamics of a delayed nonlocal reaction-diffusion cholera model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020316

[12]

Maho Endo, Yuki Kaneko, Yoshio Yamada. Free boundary problem for a reaction-diffusion equation with positive bistable nonlinearity. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3375-3394. doi: 10.3934/dcds.2020033

[13]

Shin-Ichiro Ei, Shyuh-Yaur Tzeng. Spike solutions for a mass conservation reaction-diffusion system. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3357-3374. doi: 10.3934/dcds.2020049

[14]

Chihiro Aida, Chao-Nien Chen, Kousuke Kuto, Hirokazu Ninomiya. Bifurcation from infinity with applications to reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3031-3055. doi: 10.3934/dcds.2020053

[15]

Leanne Dong. Random attractors for stochastic Navier-Stokes equation on a 2D rotating sphere with stable Lévy noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020352

[16]

Wenlong Sun, Jiaqi Cheng, Xiaoying Han. Random attractors for 2D stochastic micropolar fluid flows on unbounded domains. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 693-716. doi: 10.3934/dcdsb.2020189

[17]

Bixiang Wang. Mean-square random invariant manifolds for stochastic differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1449-1468. doi: 10.3934/dcds.2020324

[18]

Zaihui Gan, Fanghua Lin, Jiajun Tong. On the viscous Camassa-Holm equations with fractional diffusion. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3427-3450. doi: 10.3934/dcds.2020029

[19]

H. M. Srivastava, H. I. Abdel-Gawad, Khaled Mohammed Saad. Oscillatory states and patterns formation in a two-cell cubic autocatalytic reaction-diffusion model subjected to the Dirichlet conditions. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020433

[20]

Shin-Ichiro Ei, Hiroshi Ishii. The motion of weakly interacting localized patterns for reaction-diffusion systems with nonlocal effect. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 173-190. doi: 10.3934/dcdsb.2020329

2019 Impact Factor: 1.27

Article outline

[Back to Top]