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December  2020, 25(12): 4617-4640. doi: 10.3934/dcdsb.2020116

Uniform attractors of stochastic two-compartment Gray-Scott system with multiplicative noise

1. 

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

2. 

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

* Corresponding author: Jie Xin

Received  September 2019 Revised  January 2020 Published  March 2020

Fund Project: The second author is supported by the Natural Science Foundation of Shandong under Grant No. ZR2018QA002, the National Natural Science Foundation of China No. 11901342 and China Postdoctoral Science Foundation No. 2019M652350. The third author is supported by the NSF of China (No. 11371183) and the NSF of Shandong Province (No. ZR2013AM004)

We first show that the stochastic two-compartment Gray-Scott system generates a non-autonomous random dynamical system. Then we establish some uniform estimates of solutions for stochastic two-compartment 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 two-compartment Gray-Scott system with multiplicative noise. Discrete & Continuous Dynamical Systems - B, 2020, 25 (12) : 4617-4640. doi: 10.3934/dcdsb.2020116
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L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.  Google Scholar

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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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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

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I. Chueshov, Monotone Random Systems Theory and Applications, Springer-Verlag, Berlin, 2002. doi: 10.1007/b83277.  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. 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

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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

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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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F. Flandoli and B. Schmalfuss, Random attractors for the $3$D stochastic Navier-Stokes equation with multiplicative white noise, Stochastics Stochastics Rep., 59 (1996), 21-45.  doi: 10.1080/17442509608834083.  Google Scholar

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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

[15]

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

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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

[17]

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

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K. Lu and B. Wang, Global attractors for the Klein-Gordon-Schrödinger equation in unbounded domains, J. Differential Equations, 170 (2001), 281-316.  doi: 10.1006/jdeq.2000.3827.  Google Scholar

[19]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Second edition, Applied Mathematical Sciences, Vol. 68, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[20]

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

[21]

B. Wang, Sufficient and necessary criteria for exitence 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

[22]

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

[23]

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

[24]

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.   Google Scholar

[25]

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

[26]

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

[27]

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

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]

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

[3]

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

[4]

V. V. Chepyzhov and M. I. Vishik, Attractors for equations of mathematical physics, in American Mathematical Society Colloquium Publications, Vol. 49, American Mathematical Society, Providence, RI, 2002.  Google Scholar

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

X. Fan, Attractors for a damped stochastic wave equation of sine-Gordon type with sublinear multiplicative noise, Stoc. Anal. Appl., 24 (2006), 767-793.  doi: 10.1080/07362990600751860.  Google Scholar

[13]

F. Flandoli and B. Schmalfuss, Random attractors for the $3$D stochastic Navier-Stokes equation with multiplicative white noise, Stochastics Stochastics Rep., 59 (1996), 21-45.  doi: 10.1080/17442509608834083.  Google Scholar

[14]

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

[15]

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

[16]

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

[17]

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

[18]

K. Lu and B. Wang, Global attractors for the Klein-Gordon-Schrödinger equation in unbounded domains, J. Differential Equations, 170 (2001), 281-316.  doi: 10.1006/jdeq.2000.3827.  Google Scholar

[19]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Second edition, Applied Mathematical Sciences, Vol. 68, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[20]

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

[21]

B. Wang, Sufficient and necessary criteria for exitence 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

[22]

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

[23]

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

[24]

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.   Google Scholar

[25]

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

[26]

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

[27]

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

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