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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 |
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.
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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