August  2021, 4(3): 193-208. doi: 10.3934/mfc.2021012

Uniform attractors of stochastic three-component Gray-Scott system with multiplicative noise

1. 

School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong 273165, China

2. 

College of Information Science and Engineering, Shandong Agricultural University, Taian, Shandong 271018, China

* Corresponding author: Jie Xin

Received  October 2020 Published  August 2021 Early access  August 2021

Fund Project: The second author is supported by the National Natural Science Foundation of China No. 11901342, the Natural Science Foundation of Shandong Province No. ZR2018QA002, Postdoctoral Innovation Project of Shandong Province No. 202003040 and China Postdoctoral Science Foundation No. 2019M652350. The third author is supported by the NSF of China No. 11371183

In a bounded domain, we study the long time behavior of solutions of the stochastic three-component Gray-Scott system with multiplicative noise. We first show that the stochastic three-component Gray-Scott system can generate a non-autonomous random dynamical system. Then we establish some uniform estimates of solutions for stochastic three-component Gray-Scott system with multiplicative noise. Finally, the existence of uniform and cocycle attractors is proved.

Citation: Junwei Feng, Hui Liu, Jie Xin. Uniform attractors of stochastic three-component Gray-Scott system with multiplicative noise. Mathematical Foundations of Computing, 2021, 4 (3) : 193-208. doi: 10.3934/mfc.2021012
References:
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P. W. BatesH. Lisei and K. Lu, Attractors for stochastic lattice dynamical systems, Stoch. Dyn., 6 (2006), 1-21.  doi: 10.1142/S0219493706001621.  Google Scholar

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

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H. Cui and J. A. Langa, Uniform attractors for non-autonomous random dynamical systems, J. Differential Equations, 263 (2017), 1225-1268.  doi: 10.1016/j.jde.2017.03.018.  Google Scholar

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H. Liu and H. Gao, Ergodicity and dynamics for the stochastic 3D Navier-Stokes equations with damping, Commun. Math. Sci., 16 (2018), 97-122.  doi: 10.4310/CMS.2018.v16.n1.a5.  Google Scholar

[21]

K. Lu and B. Wang, Global attractors for the Klein-Gordon-Schr$$ \rm\ddot{o} $$dinger equation in unbounded domains, J. Differential Equations, 170 (2001), 281-316.  doi: 10.1006/jdeq.2000.3827.  Google Scholar

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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

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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

[27]

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

[28]

B. Wang, Pullback attractors for non-autonomous reaction-diffusion equations on ${\mathbb R}^{n}$, Front. Math. China, 4 (2009), 563-583.  doi: 10.1007/s11464-009-0033-5.  Google Scholar

[29]

Z. Wang and S. Zhou, Random attractor for stochastic reaction-diffusion equation with multiplicative noise on unbounded domains, J. Math. Anal. Appl., 384 (2011), 160-172.  doi: 10.1016/j.jmaa.2011.02.082.  Google Scholar

[30]

Y. You, Dynamics of two-compartment Gray-Scott equations, Nonlin. Anal., 74 (2011), 1969-1986.  doi: 10.1016/j.na.2010.11.004.  Google Scholar

[31]

Y. You, Dynamics of three-compartment reversible Gray-Scott model, Disc. Cont. Dynam. Syst. Ser. B, 14 (2010), 1671-1688.  doi: 10.3934/dcdsb.2010.14.1671.  Google Scholar

[32]

Y. You, Global attractor of the Gray-Scott equations, Commun. Pure. Appl. Anal., 7 (2008), 947-970.  doi: 10.3934/cpaa.2008.7.947.  Google Scholar

[33]

Y. You, Robustness of global attractors for reversible Gray-Scott systems, J. Dynam. Differential Equations, 24 (2012), 495-520.  doi: 10.1007/s10884-012-9252-7.  Google Scholar

show all references

References:
[1]

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

[2]

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

[3]

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

[4]

P. W. BatesK. 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

[5]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society Colloquium Publications, 49, AMS, Providence, RI, 2002.  Google Scholar

[6]

A. Cheskidov and L. Kavlie, Pullback attractors for generalized evolutionary systems, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 749-779.  doi: 10.3934/dcdsb.2015.20.749.  Google Scholar

[7]

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

[8]

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

[9]

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

[10]

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

[11]

H. CuiM. M. Freitas and J. A. Langa, On random cocycle attractors with autonomous attraction universes, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 3379-3407.  doi: 10.3934/dcdsb.2017142.  Google Scholar

[12]

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

[13]

H. Cui and J. A. Langa, Uniform attractors for non-autonomous random dynamical systems, J. Differential Equations, 263 (2017), 1225-1268.  doi: 10.1016/j.jde.2017.03.018.  Google Scholar

[14]

X. Ding and J. Jiang, Random attractors for stochastic retarded reaction-diffusion equations on unbounded domains, Abstr. Appl. Anal., 1 (2013), 16pp. doi: 10.1155/2013/981576.  Google Scholar

[15]

X. Ding and J. Jiang, Randoms attractors for stochastic retarded lattice dynamical systems, Abstr. Appl. Anal., 2 (2012), 27pp. Google Scholar

[16]

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

[17]

A. Gu and H. Xiang, Upper semicontinuity of random attractors for stochastic three-component reversible Gray-Scott system, Appl. Math. Comput., 225 (2013), 387-400.  doi: 10.1016/j.amc.2013.09.041.  Google Scholar

[18]

A. GuS. Zhou and Z. Wang, Uniform attractor of non-autonomous three-component reversible Gray-Scott system, Appl. Math. Comput., 219 (2013), 8718-8729.  doi: 10.1016/j.amc.2013.02.056.  Google Scholar

[19]

X. JiaJ. Gao and X. Ding, Random attractors for stochastic two-compartment Gray-Scott equations with a multiplicative noise, Open. Math., 14 (2016), 586-602.  doi: 10.1515/math-2016-0052.  Google Scholar

[20]

H. Liu and H. Gao, Ergodicity and dynamics for the stochastic 3D Navier-Stokes equations with damping, Commun. Math. Sci., 16 (2018), 97-122.  doi: 10.4310/CMS.2018.v16.n1.a5.  Google Scholar

[21]

K. Lu and B. Wang, Global attractors for the Klein-Gordon-Schr$$ \rm\ddot{o} $$dinger equation in unbounded domains, J. Differential Equations, 170 (2001), 281-316.  doi: 10.1006/jdeq.2000.3827.  Google Scholar

[22]

H. MharaN. J. SuematsuT. YamaguchiK. OhganeY. Nishiura and M. Shimomura, Three-variable reversible Gray-Scott model, J. Chem. Phys., 121 (2004), 8968-8972.   Google Scholar

[23]

G. Ochs, Weak Random Attractors, Citeseer, 1999. Google Scholar

[24]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2$^{nd}$ edition, Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[25]

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

[26]

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

[27]

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

[28]

B. Wang, Pullback attractors for non-autonomous reaction-diffusion equations on ${\mathbb R}^{n}$, Front. Math. China, 4 (2009), 563-583.  doi: 10.1007/s11464-009-0033-5.  Google Scholar

[29]

Z. Wang and S. Zhou, Random attractor for stochastic reaction-diffusion equation with multiplicative noise on unbounded domains, J. Math. Anal. Appl., 384 (2011), 160-172.  doi: 10.1016/j.jmaa.2011.02.082.  Google Scholar

[30]

Y. You, Dynamics of two-compartment Gray-Scott equations, Nonlin. Anal., 74 (2011), 1969-1986.  doi: 10.1016/j.na.2010.11.004.  Google Scholar

[31]

Y. You, Dynamics of three-compartment reversible Gray-Scott model, Disc. Cont. Dynam. Syst. Ser. B, 14 (2010), 1671-1688.  doi: 10.3934/dcdsb.2010.14.1671.  Google Scholar

[32]

Y. You, Global attractor of the Gray-Scott equations, Commun. Pure. Appl. Anal., 7 (2008), 947-970.  doi: 10.3934/cpaa.2008.7.947.  Google Scholar

[33]

Y. You, Robustness of global attractors for reversible Gray-Scott systems, J. Dynam. Differential Equations, 24 (2012), 495-520.  doi: 10.1007/s10884-012-9252-7.  Google Scholar

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