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

[2]

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

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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.

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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.

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

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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.

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

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

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

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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.

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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.

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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.

[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.

[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.

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

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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.

[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.

[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.

[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.

[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.

[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.

[22]

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

[23]

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

[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.

[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.

[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.

[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.

[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.

[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.

[30]

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

[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.

[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.

[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.

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.

[2]

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

[3]

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

[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.

[5]

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

[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.

[7]

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

[8]

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

[9]

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

[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.

[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.

[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.

[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.

[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.

[15]

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

[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.

[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.

[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.

[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.

[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.

[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.

[22]

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

[23]

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

[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.

[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.

[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.

[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.

[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.

[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.

[30]

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

[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.

[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.

[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.

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