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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

Junwei Feng 1, , Hui Liu 1, and  Jie Xin 1,2,,

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  December 2020 Early access  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.

Keywords: Gray-Scott system, uniform attractor, cocycle attractor, multiplicative noise.
Mathematics Subject Classification: 35K51, 37C70, 35B41.
Citation: Junwei Feng, Hui Liu, Jie Xin. Uniform attractors of stochastic two-compartment Gray-Scott system with multiplicative noise. Discrete and Continuous Dynamical Systems - B, 2020, 25 (12) : 4617-4640. doi: 10.3934/dcdsb.2020116
References:
[1]

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

[2]

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

[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.

[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.

[5]

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

[6]

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

[7]

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

[8]

H. Crauel, P. 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.

[9]

H. Cui, M. 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.

[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.

[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.

[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.

[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.

[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.

[15]

A. Gu, S. 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.

[16]

X. Jia, J. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[25]

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

[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.

[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.

show all references

References:
[1]

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

[2]

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

[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.

[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.

[5]

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

[6]

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

[7]

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

[8]

H. Crauel, P. 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.

[9]

H. Cui, M. 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.

[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.

[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.

[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.

[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.

[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.

[15]

A. Gu, S. 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.

[16]

X. Jia, J. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[25]

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

[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.

[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.

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