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July  2021, 26(7): 3863-3878. doi: 10.3934/dcdsb.2020266

Weak pullback mean random attractors for non-autonomous $ p $-Laplacian equations

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

Received  April 2019 Revised  May 2020 Published  July 2021 Early access  September 2020

Fund Project: This work is partially supported by NSF of Chongqing Grant No. cstc2018jcyjA0897, the FRF for the Central Universities Grant No. XDJK2020B049 and K.C. Wong Education Foundation and DAAD

In this paper, we obtain the existence and uniqueness of weak pullback mean random attractors for non-autonomous deterministic $ p $-Laplacian equations with random initial data and non-autonomous stochastic $ p $-Laplacian equations with general diffusion terms in Bochner spaces, respectively.

Citation: Anhui Gu. Weak pullback mean random attractors for non-autonomous $ p $-Laplacian equations. Discrete and Continuous Dynamical Systems - B, 2021, 26 (7) : 3863-3878. doi: 10.3934/dcdsb.2020266
References:
[1]

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

[2]

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.

[3]

W.-J. BeynB. GessP. Lescot and M. Röckner, The global random attractor for a class of stochastic porous media equations, Commun. Partial Differ. Equ., 36 (2011), 446-469.  doi: 10.1080/03605302.2010.523919.

[4]

T. CaraballoM. J. Garrido-Atienza and B. Schmalfuss, Existence of exponetially attracting stationary solutions for delay evolution equations, Discrete Contin. Dyn. Syst., 18 (2007), 271-293.  doi: 10.3934/dcds.2007.18.271.

[5]

T. CaraballoM. J. Garrido-AtienzaB. Schmalfuss and J. Valero, Non-autonomous and random attractors for delay random semilinear equations without uniqueness, Discrete Contin. Dyn. Syst., 21 (2008), 415-443.  doi: 10.3934/dcds.2008.21.415.

[6]

T. CaraballoJ. Real and I. D. Chueshov, Pullback attractors for stochastic heat equations in materials with memory, Discrete Contin. Dyn. Syst. Ser. B, 9 (2008), 525-539.  doi: 10.3934/dcdsb.2008.9.525.

[7]

I. Chueshov and M. Scheutzow, On the structure of attractors and invariant measures for a class of monotone random systems, Dynamical Systems, 19 (2004), 127-144.  doi: 10.1080/1468936042000207792.

[8]

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

[9]

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

[10]

J. Duan and B. Schmalfuss, The 3D quasigeostrophic fluid dynamics under random forcing on boundary, Comm. Math. Sci., 1 (2003), 133-151. 

[11]

B. Fehrman and B. Gess, Well-posedness of nonlinear diffusion equations with nonlinear, conservative noise, Arch. Rational Mech. Anal., 233 (2019), 249–322. doi: 10.1007/s00205-019-01357-w.

[12]

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

[13]

H. GaoM. J. Garrido-Atienza and B. Schmalfuss, Random attractors for stochastic evolution equations driven by fractional Brownian motion, SIAM J. Math. Anal., 46 (2014), 2281-2309.  doi: 10.1137/130930662.

[14]

M. J. Garrido-Atienza, K. Lu and B. Schmalfuss, Random dynamical systems for stochastic partial differential equations driven by a fractional Brownian motion, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 473–493. doi: 10.3934/dcdsb.2010.14.473.

[15]

M. J. Garrido-Atienza and B. Schmalfuss, Ergodicity of the infinite dimensional fractional Brownian motion, J. Dyn. Diff. Equat., 23 (2011), 671-681.  doi: 10.1007/s10884-011-9222-5.

[16]

M. J. Garrido-Atienza, K. Lu and B. Schmalfuss, Random dynamical systems for stochastic evolution equations driven by multiplicative fractional Brownian noise with Hurst parametes $H\in (1/3, 1/2]$, SIAM J. Appl. Dyn. Syst., 15 (2016), 625–654. doi: 10.1137/15M1030303.

[17]

B. GessW. Liu and M. Röckner, Random attractors for a class of stochastic partial differential equations driven by general additive noise, J. Differential Equations, 251 (2011), 1225-1253.  doi: 10.1016/j.jde.2011.02.013.

[18]

B. Gess, Random attractors for degenerate stochastic partial differential equations, J. Dyn. Diff. Equat., 25 (2013), 121-157.  doi: 10.1007/s10884-013-9294-5.

[19]

B. Gess, Random attractors for singular stochastic evolution equations, J. Differential Equations, 255 (2013), 524-559.  doi: 10.1016/j.jde.2013.04.023.

[20]

A. Gu, Asymptotic behavior of random lattice dynamical systems and their Wong-Zakai approximations, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 5737–5767. doi: 10.3934/dcdsb.2019104.

[21]

A. GuK. Lu and B. Wang, Asymptotic behavior of random Navier-Stokes equations driven by Wong-Zakai approximations, Discrete Contin. Dyn. Syst. Ser. A, 39 (2019), 185-218.  doi: 10.3934/dcds.2019008.

[22]

A. GuD. LiB. Wang and H. Yang, Regularity of random attractors for fractional stochastic reaction-diffusion equations on ${\mathbb R}^n$, J. Differential Equations, 264 (2018), 7094-7137.  doi: 10.1016/j.jde.2018.02.011.

[23]

A. Gu and B. Wang, Asymptotic behavior of random FitzHugh-Nagumo systems driven by colored noise, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 1689–1720. doi: 10.3934/dcdsb.2018072.

[24]

J. Huang and W. Shen, Pullback attractors for nonautonomous and random parabolic equations on non-smooth domains, Discrete Contin. Dyn. Syst., 24 (2009), 855-882.  doi: 10.3934/dcds.2009.24.855.

[25]

P. E. Kloeden and J. A. Langa, Flattening, squeezing and the existence of random attractors, Proc. Royal Soc. London Serie A. Math. Phys. Eng. Sci., 463 (2007), 163-181.  doi: 10.1098/rspa.2006.1753.

[26]

P. E. Kloeden and T. Lorenz, Mean-square random dynamical systems, J. Differential Equations, 253 (2012), 1422-1438.  doi: 10.1016/j.jde.2012.05.016.

[27]

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

[28]

P. Lindqvist, Notes on the $p$-Laplace Equation, 2006. Available from: http://www.math.ntnu.no/ lqvist/p-laplace.pdf.

[29]

J.-L. Lions, Quelques Methodes de Resolution des Problemes aux Limites Non Lineaires, Dunod, Paris, 1969.

[30]

K. Lu and B. Wang, Wong-Zakai approximations and long term behavior of stochastic partial differential equations, J. Dyn. Diff. Equat., 31 (2019), 1341-1371.  doi: 10.1007/s10884-017-9626-y.

[31]

Y. Lv and W. Wang, Limiting dynamics for stochastic wave equations, J. Differential Equations, 244 (2008), 1-23.  doi: 10.1016/j.jde.2007.10.009.

[32]

J. Málek, J. Nečas, M. Rokyta and M. R$\rm\mathring{u}$zička, Weak and Measure-Valued Solutions to Evolutionary PDEs, Chapman & Hall, London, 1996.

[33]

C. Prévôt and M. Röckner, A Concise Course on Stochastic Partial Differential Equations, Lecture Notes in Mathematics, Springer, Berlin, 2007.

[34]

B. Schmalfuss, Backward cocycles and attractors of stochastic differential equations, International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior, 185-192, Dresden, 1992.

[35]

Z. ShenS. Zhou and W. Shen, One-dimensional random attractor and rotation number of the stochastic damped sine-Gordon equation, J. Differential Equations, 248 (2010), 1432-1457.  doi: 10.1016/j.jde.2009.10.007.

[36]

R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, American Mathematical Society, Providence, 1997.

[37]

B. Wang, Random attractors for the stochastic Benjamin-Bona-Mahony equation on unbounded domains, J. Differential Equations, 246 (2009), 2506-2537.  doi: 10.1016/j.jde.2008.10.012.

[38]

B. Wang, Asymptotic behavior of stochastic wave equations with critical exponents on ${\mathbb R}^3$, Trans. Amer. Math. Soc., 363 (2011), 3639-3663.  doi: 10.1090/S0002-9947-2011-05247-5.

[39]

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.

[40]

B. Wang, Existence and upper semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms, Stoch. Dyn., 14 (2014), 1450009, 1–31. doi: 10.1142/S0219493714500099.

[41]

B. Wang, Weak pullback attractors for mean random dynamical systems in Bochner spaces, J. Dyn. Diff. Equat., 31 (2019), 2177-2204.  doi: 10.1007/s10884-018-9696-5.

[42]

B. Wang, Weak pullback attractors for stochastic Navier-Stokes equations with nonlinear diffusion terms, Proc. Amer. Math. Soc., 147 (2019), 1627-1638.  doi: 10.1090/proc/14356.

[43]

B. Wang and B. Guo, Asymptotic behavior of non-autonomous stochastic parabolic equations with nonlinear Laplacian principal part, Electron. J. Differ. Eq., (2013), No. 191, 25 pp.

[44]

S. Zhou, Random exponential attractor for stochastic reaction-diffusion equation with multiplicative noise in ${\mathbb R}^3$, J. Differential Equations, 263 (2017), 6347-6383.  doi: 10.1016/j.jde.2017.07.013.

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. 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.

[3]

W.-J. BeynB. GessP. Lescot and M. Röckner, The global random attractor for a class of stochastic porous media equations, Commun. Partial Differ. Equ., 36 (2011), 446-469.  doi: 10.1080/03605302.2010.523919.

[4]

T. CaraballoM. J. Garrido-Atienza and B. Schmalfuss, Existence of exponetially attracting stationary solutions for delay evolution equations, Discrete Contin. Dyn. Syst., 18 (2007), 271-293.  doi: 10.3934/dcds.2007.18.271.

[5]

T. CaraballoM. J. Garrido-AtienzaB. Schmalfuss and J. Valero, Non-autonomous and random attractors for delay random semilinear equations without uniqueness, Discrete Contin. Dyn. Syst., 21 (2008), 415-443.  doi: 10.3934/dcds.2008.21.415.

[6]

T. CaraballoJ. Real and I. D. Chueshov, Pullback attractors for stochastic heat equations in materials with memory, Discrete Contin. Dyn. Syst. Ser. B, 9 (2008), 525-539.  doi: 10.3934/dcdsb.2008.9.525.

[7]

I. Chueshov and M. Scheutzow, On the structure of attractors and invariant measures for a class of monotone random systems, Dynamical Systems, 19 (2004), 127-144.  doi: 10.1080/1468936042000207792.

[8]

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

[9]

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

[10]

J. Duan and B. Schmalfuss, The 3D quasigeostrophic fluid dynamics under random forcing on boundary, Comm. Math. Sci., 1 (2003), 133-151. 

[11]

B. Fehrman and B. Gess, Well-posedness of nonlinear diffusion equations with nonlinear, conservative noise, Arch. Rational Mech. Anal., 233 (2019), 249–322. doi: 10.1007/s00205-019-01357-w.

[12]

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

[13]

H. GaoM. J. Garrido-Atienza and B. Schmalfuss, Random attractors for stochastic evolution equations driven by fractional Brownian motion, SIAM J. Math. Anal., 46 (2014), 2281-2309.  doi: 10.1137/130930662.

[14]

M. J. Garrido-Atienza, K. Lu and B. Schmalfuss, Random dynamical systems for stochastic partial differential equations driven by a fractional Brownian motion, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 473–493. doi: 10.3934/dcdsb.2010.14.473.

[15]

M. J. Garrido-Atienza and B. Schmalfuss, Ergodicity of the infinite dimensional fractional Brownian motion, J. Dyn. Diff. Equat., 23 (2011), 671-681.  doi: 10.1007/s10884-011-9222-5.

[16]

M. J. Garrido-Atienza, K. Lu and B. Schmalfuss, Random dynamical systems for stochastic evolution equations driven by multiplicative fractional Brownian noise with Hurst parametes $H\in (1/3, 1/2]$, SIAM J. Appl. Dyn. Syst., 15 (2016), 625–654. doi: 10.1137/15M1030303.

[17]

B. GessW. Liu and M. Röckner, Random attractors for a class of stochastic partial differential equations driven by general additive noise, J. Differential Equations, 251 (2011), 1225-1253.  doi: 10.1016/j.jde.2011.02.013.

[18]

B. Gess, Random attractors for degenerate stochastic partial differential equations, J. Dyn. Diff. Equat., 25 (2013), 121-157.  doi: 10.1007/s10884-013-9294-5.

[19]

B. Gess, Random attractors for singular stochastic evolution equations, J. Differential Equations, 255 (2013), 524-559.  doi: 10.1016/j.jde.2013.04.023.

[20]

A. Gu, Asymptotic behavior of random lattice dynamical systems and their Wong-Zakai approximations, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 5737–5767. doi: 10.3934/dcdsb.2019104.

[21]

A. GuK. Lu and B. Wang, Asymptotic behavior of random Navier-Stokes equations driven by Wong-Zakai approximations, Discrete Contin. Dyn. Syst. Ser. A, 39 (2019), 185-218.  doi: 10.3934/dcds.2019008.

[22]

A. GuD. LiB. Wang and H. Yang, Regularity of random attractors for fractional stochastic reaction-diffusion equations on ${\mathbb R}^n$, J. Differential Equations, 264 (2018), 7094-7137.  doi: 10.1016/j.jde.2018.02.011.

[23]

A. Gu and B. Wang, Asymptotic behavior of random FitzHugh-Nagumo systems driven by colored noise, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 1689–1720. doi: 10.3934/dcdsb.2018072.

[24]

J. Huang and W. Shen, Pullback attractors for nonautonomous and random parabolic equations on non-smooth domains, Discrete Contin. Dyn. Syst., 24 (2009), 855-882.  doi: 10.3934/dcds.2009.24.855.

[25]

P. E. Kloeden and J. A. Langa, Flattening, squeezing and the existence of random attractors, Proc. Royal Soc. London Serie A. Math. Phys. Eng. Sci., 463 (2007), 163-181.  doi: 10.1098/rspa.2006.1753.

[26]

P. E. Kloeden and T. Lorenz, Mean-square random dynamical systems, J. Differential Equations, 253 (2012), 1422-1438.  doi: 10.1016/j.jde.2012.05.016.

[27]

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

[28]

P. Lindqvist, Notes on the $p$-Laplace Equation, 2006. Available from: http://www.math.ntnu.no/ lqvist/p-laplace.pdf.

[29]

J.-L. Lions, Quelques Methodes de Resolution des Problemes aux Limites Non Lineaires, Dunod, Paris, 1969.

[30]

K. Lu and B. Wang, Wong-Zakai approximations and long term behavior of stochastic partial differential equations, J. Dyn. Diff. Equat., 31 (2019), 1341-1371.  doi: 10.1007/s10884-017-9626-y.

[31]

Y. Lv and W. Wang, Limiting dynamics for stochastic wave equations, J. Differential Equations, 244 (2008), 1-23.  doi: 10.1016/j.jde.2007.10.009.

[32]

J. Málek, J. Nečas, M. Rokyta and M. R$\rm\mathring{u}$zička, Weak and Measure-Valued Solutions to Evolutionary PDEs, Chapman & Hall, London, 1996.

[33]

C. Prévôt and M. Röckner, A Concise Course on Stochastic Partial Differential Equations, Lecture Notes in Mathematics, Springer, Berlin, 2007.

[34]

B. Schmalfuss, Backward cocycles and attractors of stochastic differential equations, International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior, 185-192, Dresden, 1992.

[35]

Z. ShenS. Zhou and W. Shen, One-dimensional random attractor and rotation number of the stochastic damped sine-Gordon equation, J. Differential Equations, 248 (2010), 1432-1457.  doi: 10.1016/j.jde.2009.10.007.

[36]

R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, American Mathematical Society, Providence, 1997.

[37]

B. Wang, Random attractors for the stochastic Benjamin-Bona-Mahony equation on unbounded domains, J. Differential Equations, 246 (2009), 2506-2537.  doi: 10.1016/j.jde.2008.10.012.

[38]

B. Wang, Asymptotic behavior of stochastic wave equations with critical exponents on ${\mathbb R}^3$, Trans. Amer. Math. Soc., 363 (2011), 3639-3663.  doi: 10.1090/S0002-9947-2011-05247-5.

[39]

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.

[40]

B. Wang, Existence and upper semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms, Stoch. Dyn., 14 (2014), 1450009, 1–31. doi: 10.1142/S0219493714500099.

[41]

B. Wang, Weak pullback attractors for mean random dynamical systems in Bochner spaces, J. Dyn. Diff. Equat., 31 (2019), 2177-2204.  doi: 10.1007/s10884-018-9696-5.

[42]

B. Wang, Weak pullback attractors for stochastic Navier-Stokes equations with nonlinear diffusion terms, Proc. Amer. Math. Soc., 147 (2019), 1627-1638.  doi: 10.1090/proc/14356.

[43]

B. Wang and B. Guo, Asymptotic behavior of non-autonomous stochastic parabolic equations with nonlinear Laplacian principal part, Electron. J. Differ. Eq., (2013), No. 191, 25 pp.

[44]

S. Zhou, Random exponential attractor for stochastic reaction-diffusion equation with multiplicative noise in ${\mathbb R}^3$, J. Differential Equations, 263 (2017), 6347-6383.  doi: 10.1016/j.jde.2017.07.013.

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