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doi: 10.3934/eect.2022010
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Controller and asymptotic autonomy of random attractors for stochastic p-Laplace lattice equations

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

* Corresponding author: liyr@swu.edu.cn (Yangrong Li)

Received  October 2021 Revised  December 2021 Early access February 2022

A non-autonomous random dynamical system is called to be controllable if there is a pullback random attractor (PRA) such that each fibre of the PRA converges upper semi-continuously to a nonempty compact set (called a controller) as the time-parameter goes to minus infinity, while the PRA is called to be asymptotically autonomous if there is a random attractor for another (autonomous) random dynamical system as a controller. We establish the criteria for ensuring the existence of the minimal controller and the asymptotic autonomy of a PRA respectively. The abstract results are illustrated in possibly non-autonomous stochastic p-Laplace lattice equations with tempered convergent external forces.

Citation: Li Song, Yangrong Li, Fengling Wang. Controller and asymptotic autonomy of random attractors for stochastic p-Laplace lattice equations. Evolution Equations and Control Theory, doi: 10.3934/eect.2022010
References:
[1]

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

[2]

P. W. BatesK. Lu and B. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differ. Equ., 246 (2009), 845-869.  doi: 10.1016/j.jde.2008.05.017.

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P. W. BatesK. Lu and B. Wang, Attractors of non-autonomous stochastic lattice systems in weighted spaces, Phys. D, 289 (2014), 32-50.  doi: 10.1016/j.physd.2014.08.004.

[4]

T. CaraballoA. N. CarvalhoH. B. da Costa and J. A. Langa, Equi-attraction and continuity of attractors for skew-product semiflows, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 2949-2967.  doi: 10.3934/dcdsb.2016081.

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V. Chepyzhov and M. Vishik, A Hausdorff dimension estimate for kernel sections of nonautonomous evolution equations, Indiana Univ. Math. J., 42 (1993), 1057-1076.  doi: 10.1512/iumj.1993.42.42049.

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

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H. CrauelA. Debussche and F. Flandoli, Random attractors, J. Dyn. Differ. Equ., 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. Cui and P. E. Kloeden, Tail convergences of pullback attractors for asymptotically converging multi-valued dynamical systems, Asymptot. Anal., 112 (2019), 165-184.  doi: 10.3233/ASY-181501.

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H. CuiP. E. Kloeden and F. Wu, Pathwise upper semi-continuity of random pullback attractors along the time axis, Phys. D, 374 (2018), 21-34.  doi: 10.1016/j.physd.2018.03.002.

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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.

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A. Gu and P. E. Kloeden, Asymptotic behavior of a nonautonomous $p$-Laplacian lattice system, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 26 (2016), 1650174, 9 pp. doi: 10.1142/S0218127416501741.

[14]

A. Gu and Y. Li, Dynamic behavior of stochastic $p$-Laplacian-type lattice equations, Stoch. Dyn., 17 (2017), 1750040, 1-19.  doi: 10.1142/S021949371750040X.

[15]

X. HanP. E. Kloeden and S. Sonner, Discretisation of global attractors for lattice dynamical systems, J. Dynam. Differ. Equ., 32 (2020), 1457-1474.  doi: 10.1007/s10884-019-09770-1.

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P. E. Kloeden and T. Lorenz, Construction of nonautonomous forward attractors, Proc. Amer. Math. Soc., 144 (2016), 259-268.  doi: 10.1090/proc/12735.

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P. E. Kloeden and J. Simsen, Attractors of asymptotically autonomous quasi-linear parabolic equation with spatially variable exponents, J. Math. Anal. Appl., 425 (2015), 911-918.  doi: 10.1016/j.jmaa.2014.12.069.

[18]

P. E. KloedenJ. Simsen and M. S. Simsen, Asymptotically autonomous multivalued Cauchy problems with spatially variable exponents, J. Math. Anal. Appl., 445 (2017), 513-531.  doi: 10.1016/j.jmaa.2016.08.004.

[19]

F. LiY. Li and R. Wang, Regular measurable dynamics for reaction-diffusion equations on narrow domains with rough noise, Discrete Contin. Dyn. Syst., 38 (2018), 3663-3685.  doi: 10.3934/dcds.2018158.

[20]

X. Li, Uniform random attractors for 2D non-autonomous stochastic Navier-Stokes equations, J. Differential Equations, 276 (2021), 1-42.  doi: 10.1016/j.jde.2020.12.014.

[21]

Y. LiA. Gu and J. Li, Existence and continuity of bi-spatial random attractors and application to stochastic semilinear Laplacian equations, J. Differ. Equ., 258 (2015), 504-534.  doi: 10.1016/j.jde.2014.09.021.

[22]

Y. LiL. She and R. Wang, Asymptotically autonomous dynamics for parabolic equation, J. Math. Anal. Appl., 459 (2018), 1106-1123.  doi: 10.1016/j.jmaa.2017.11.033.

[23]

Y. LiR. Wang and L. She, Backward controliability of pullback trajectory attractors with applications to multi-valued Jeffreys-Oldroyd equations, Evol. Equ. Control Theory, 7 (2018), 617-637.  doi: 10.3934/eect.2018030.

[24]

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.

[25]

F. Wang and Y. Li, Random attractors for multi-valued multi-stochastic delayed $p$-Laplace lattice equations, J. Difference Equat. Appl., 27 (2021), 1232-1258.  doi: 10.1080/10236198.2021.1976771.

[26]

R. Wang and Y. Li, Asymptotic autonomy of random attractors for BBM equations with Laplace-multiplier noise, J. Appl. Anal. Comput., 10 (2020), 1199-1222.  doi: 10.11948/20180145.

[27]

R. Wang and B. Wang, Random dynamics of $p$-Laplacian lattice systems driven by infinite-dimensional nonlinear noise, Stoch. Proc. Appl., 130 (2020), 7431-7462.  doi: 10.1016/j.spa.2020.08.002.

[28]

S. Wang and Y. Li, Longtime robustness of pullback random attractors for stochastic magneto-hydrodynamics equations, Physica D, 382 (2018), 46-57.  doi: 10.1016/j.physd.2018.07.003.

[29]

X. WangK. Lu and B. Wang, Exponential stability of non-autonomous stochastic delay lattice systems with multiplicative noise, J. Dynam. Differential Equations, 28 (2016), 1309-1335.  doi: 10.1007/s10884-015-9448-8.

[30]

S. Yang and Y. Li, Forward controllability of a random attractor for the non-autonomous stochastic sine-Gordon equation on an unbounded domain, Evol. Equ. Control Theory, 9 (2020), 581-604.  doi: 10.3934/eect.2020025.

[31]

S. Zhou, Random exponential attractor for cocycle and application to non-autonomous stochastic lattice systems with multiplicative white noise, J. Differential Equations, 263 (2017), 2247-2279.  doi: 10.1016/j.jde.2017.03.044.

show all references

References:
[1]

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

[2]

P. W. BatesK. Lu and B. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differ. Equ., 246 (2009), 845-869.  doi: 10.1016/j.jde.2008.05.017.

[3]

P. W. BatesK. Lu and B. Wang, Attractors of non-autonomous stochastic lattice systems in weighted spaces, Phys. D, 289 (2014), 32-50.  doi: 10.1016/j.physd.2014.08.004.

[4]

T. CaraballoA. N. CarvalhoH. B. da Costa and J. A. Langa, Equi-attraction and continuity of attractors for skew-product semiflows, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 2949-2967.  doi: 10.3934/dcdsb.2016081.

[5]

V. Chepyzhov and M. Vishik, A Hausdorff dimension estimate for kernel sections of nonautonomous evolution equations, Indiana Univ. Math. J., 42 (1993), 1057-1076.  doi: 10.1512/iumj.1993.42.42049.

[6]

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

[7]

H. CrauelA. Debussche and F. Flandoli, Random attractors, J. Dyn. Differ. Equ., 9 (1997), 307-341.  doi: 10.1007/BF02219225.

[8]

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

[9]

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.

[10]

H. Cui and P. E. Kloeden, Tail convergences of pullback attractors for asymptotically converging multi-valued dynamical systems, Asymptot. Anal., 112 (2019), 165-184.  doi: 10.3233/ASY-181501.

[11]

H. CuiP. E. Kloeden and F. Wu, Pathwise upper semi-continuity of random pullback attractors along the time axis, Phys. D, 374 (2018), 21-34.  doi: 10.1016/j.physd.2018.03.002.

[12]

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.

[13]

A. Gu and P. E. Kloeden, Asymptotic behavior of a nonautonomous $p$-Laplacian lattice system, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 26 (2016), 1650174, 9 pp. doi: 10.1142/S0218127416501741.

[14]

A. Gu and Y. Li, Dynamic behavior of stochastic $p$-Laplacian-type lattice equations, Stoch. Dyn., 17 (2017), 1750040, 1-19.  doi: 10.1142/S021949371750040X.

[15]

X. HanP. E. Kloeden and S. Sonner, Discretisation of global attractors for lattice dynamical systems, J. Dynam. Differ. Equ., 32 (2020), 1457-1474.  doi: 10.1007/s10884-019-09770-1.

[16]

P. E. Kloeden and T. Lorenz, Construction of nonautonomous forward attractors, Proc. Amer. Math. Soc., 144 (2016), 259-268.  doi: 10.1090/proc/12735.

[17]

P. E. Kloeden and J. Simsen, Attractors of asymptotically autonomous quasi-linear parabolic equation with spatially variable exponents, J. Math. Anal. Appl., 425 (2015), 911-918.  doi: 10.1016/j.jmaa.2014.12.069.

[18]

P. E. KloedenJ. Simsen and M. S. Simsen, Asymptotically autonomous multivalued Cauchy problems with spatially variable exponents, J. Math. Anal. Appl., 445 (2017), 513-531.  doi: 10.1016/j.jmaa.2016.08.004.

[19]

F. LiY. Li and R. Wang, Regular measurable dynamics for reaction-diffusion equations on narrow domains with rough noise, Discrete Contin. Dyn. Syst., 38 (2018), 3663-3685.  doi: 10.3934/dcds.2018158.

[20]

X. Li, Uniform random attractors for 2D non-autonomous stochastic Navier-Stokes equations, J. Differential Equations, 276 (2021), 1-42.  doi: 10.1016/j.jde.2020.12.014.

[21]

Y. LiA. Gu and J. Li, Existence and continuity of bi-spatial random attractors and application to stochastic semilinear Laplacian equations, J. Differ. Equ., 258 (2015), 504-534.  doi: 10.1016/j.jde.2014.09.021.

[22]

Y. LiL. She and R. Wang, Asymptotically autonomous dynamics for parabolic equation, J. Math. Anal. Appl., 459 (2018), 1106-1123.  doi: 10.1016/j.jmaa.2017.11.033.

[23]

Y. LiR. Wang and L. She, Backward controliability of pullback trajectory attractors with applications to multi-valued Jeffreys-Oldroyd equations, Evol. Equ. Control Theory, 7 (2018), 617-637.  doi: 10.3934/eect.2018030.

[24]

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.

[25]

F. Wang and Y. Li, Random attractors for multi-valued multi-stochastic delayed $p$-Laplace lattice equations, J. Difference Equat. Appl., 27 (2021), 1232-1258.  doi: 10.1080/10236198.2021.1976771.

[26]

R. Wang and Y. Li, Asymptotic autonomy of random attractors for BBM equations with Laplace-multiplier noise, J. Appl. Anal. Comput., 10 (2020), 1199-1222.  doi: 10.11948/20180145.

[27]

R. Wang and B. Wang, Random dynamics of $p$-Laplacian lattice systems driven by infinite-dimensional nonlinear noise, Stoch. Proc. Appl., 130 (2020), 7431-7462.  doi: 10.1016/j.spa.2020.08.002.

[28]

S. Wang and Y. Li, Longtime robustness of pullback random attractors for stochastic magneto-hydrodynamics equations, Physica D, 382 (2018), 46-57.  doi: 10.1016/j.physd.2018.07.003.

[29]

X. WangK. Lu and B. Wang, Exponential stability of non-autonomous stochastic delay lattice systems with multiplicative noise, J. Dynam. Differential Equations, 28 (2016), 1309-1335.  doi: 10.1007/s10884-015-9448-8.

[30]

S. Yang and Y. Li, Forward controllability of a random attractor for the non-autonomous stochastic sine-Gordon equation on an unbounded domain, Evol. Equ. Control Theory, 9 (2020), 581-604.  doi: 10.3934/eect.2020025.

[31]

S. Zhou, Random exponential attractor for cocycle and application to non-autonomous stochastic lattice systems with multiplicative white noise, J. Differential Equations, 263 (2017), 2247-2279.  doi: 10.1016/j.jde.2017.03.044.

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