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Weak pullback mean random attractors for non-autonomous $ p $-Laplacian equations
School of Mathematics and Statistics, Southwest University, Chongqing 400715, China |
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.
References:
[1] |
L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin 1998.
doi: 10.1007/978-3-662-12878-7. |
[2] |
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. |
[3] |
W.-J. Beyn, B. Gess, P. 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. Caraballo, M. 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. Caraballo, M. J. Garrido-Atienza, B. 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. Caraballo, J. 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. Crauel, A. 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. Gao, M. 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. Gess, W. 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. Gu, K. 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. Gu, D. Li, B. 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. Li, A. 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. Google Scholar |
[35] |
Z. Shen, S. 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. 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. |
[3] |
W.-J. Beyn, B. Gess, P. 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. Caraballo, M. 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. Caraballo, M. J. Garrido-Atienza, B. 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. Caraballo, J. 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. Crauel, A. 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. Gao, M. 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. Gess, W. 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. Gu, K. 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. Gu, D. Li, B. 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. Li, A. 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. Google Scholar |
[35] |
Z. Shen, S. 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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