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Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles
High-order Wong-Zakai approximations for non-autonomous stochastic $ p $-Laplacian equations on $ \mathbb{R}^N $
1. | Chongqing Key Laboratory of Social Economy and Applied Statistics, School of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, China |
2. | Chongqing Key Laboratory of Social Economy and Applied Statistics, School of Science, Chongqing University of Posts and Telecommunications, Chongqing 400065, China |
In this paper, we investigate the approximations of stochastic $ p $-Laplacian equation with additive white noise by a family of piecewise deterministic partial differential equations driven by a stationary stochastic process. We firstly obtain the tempered pullback attractors for the random $ p $-Laplacian equation with a general diffusion. We secondly prove the convergence of solutions and the upper semi-continuity of pullback attractors of the Wong-Zakai approximation equations in a Hilbert space for the additive case. Thirdly, by a truncation technique, the uniform compactness of pullback attractor with respect to the quantity of approximations is derived in the space of $ q $-times integrable functions, where the upper semi-continuity of the attractors of the approximation equations is well established.
References:
[1] |
S. Al-azzawi, J. Liu and X. Liu,
Convergence rate of synchronization of systems with additive noise, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 227-245.
doi: 10.3934/dcdsb.2017012. |
[2] |
L. Arnold, Random Dynamical System, Springer-Verlag, Berlin, 1998.
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H. Crauel and F. Flandoli,
Attracors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393.
doi: 10.1007/BF01193705. |
[5] |
H. Crauel, A. Debussche and F. Flandoli,
Random attractors, J. Dynam. Differ. Equ., 9 (1997), 307-341.
doi: 10.1007/BF02219225. |
[6] |
P. G. Geredeli,
On the existence of regular global attractor for $p$-Laplacian evolution equations, Appl. Math. Optim, 71 (2015), 517-532.
doi: 10.1007/s00245-014-9268-y. |
[7] |
P. G. Geredeli and A. Kh. Khanmamedov,
Long-time dynamics of the parabolic $p$-Laplacian equation, Commun. Pure Appl. Anal., 12 (2013), 735-754.
doi: 10.3934/cpaa.2013.12.735. |
[8] |
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. |
[9] |
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. |
[10] |
I. Gyongy,
On the approximation of stochastic partial differential equations Ⅰ, Stochastics, 25 (1988), 59-85.
doi: 10.1080/17442508808833533. |
[11] |
I. Gyongy,
On the approximation of stochastic partial differential equations Ⅱ, Stochastics, 26 (1989), 129-164.
doi: 10.1080/17442508908833554. |
[12] |
I. Gyongy and A. Shmatkov,
Rate of convergence of Wong-Zakai approximations for stochastic partial differential equations, Appl. Math. Optim., 54 (2006), 315-341.
doi: 10.1007/s00245-006-1001-z. |
[13] |
H. Hu, X. Liu and et al, On smooth approximation for random attractor of stochastic partial differential equations with multiplicative noise, Stoch. Dyn., 18 (2018), 22pp.
doi: 10.1142/S0219493718500405. |
[14] |
A. Kh. Khanmamedov,
Existence of a global attractor for the parabolic equation with nonlinear Laplacian principal part in an unbounded domain, J. Math. Anal. Appl., 316 (2006), 601-615.
doi: 10.1016/j.jmaa.2005.05.003. |
[15] |
A. Krause and B. Wang,
Pullback attractors of non-autonomous stochastic degenerate parabolic equations on unbounded domains, J. Math. Anal. Appl., 417 (2014), 1018-1038.
doi: 10.1016/j.jmaa.2014.03.037. |
[16] |
A. Krause, M. Lewis and B. Wang,
Dynamics of the non-autonomous stochastic $p$-Laplace equation driven by multiplicative noise, Appl. Math. Comput., 246 (2014), 365-376.
doi: 10.1016/j.amc.2014.08.033. |
[17] |
J. Li, H. Cui and Y. Li,
Existence and upper semicontinuity of random attractors of stochastic $p$-Laplacian equations on unbounded domains, Electron J. Differ. Equ., 2014 (2014), 1-27.
|
[18] |
Y. Li, A. Gu and J. Li,
Existences and continuity of bi-spatial random attractors and application to stochasitic semilinear Laplacian equations, J. Differ. Equ., 258 (2015), 504-534.
doi: 10.1016/j.jde.2014.09.021. |
[19] |
Y. Li and J. Yin,
Existence, regularity and approximation of global attractors for weaky dissipative $p$-Laplacian equations, Discrete Contin. Dyn. Syst. Ser. S, 9 (2016), 1939-1957.
doi: 10.3934/dcdss.2016079. |
[20] |
J. L. Lions, Quelques Methodes de Resolution des Problemes aux Limites Non Lineaires, Dunod, Paris, 1969. |
[21] |
K. Lu and B. Wang,
Wong-Zakai approximations and long term behavior of stochastic partial differential equations, J. Dyn. Differ. Equ., 31 (2019), 1341-1371.
doi: 10.1007/s10884-017-9626-y. |
[22] |
K. Lu and Q. Wang,
Chaotic behavior in differential equations driven by a Brownian motion, J. Differ. Equ., 251 (2011), 2853-2895.
doi: 10.1016/j.jde.2011.05.032. |
[23] |
T. Nakayama and S. Tappe,
Wong-Zakai approximations with convergence rate for stochastic partial differential equations, Stoch. Anal. Appl., 36 (2018), 832-857.
doi: 10.1080/07362994.2018.1471402. |
[24] |
J. C. Robinson, Infnite-Dimensional Dyanmical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge University Press, 2001.
doi: 10.1007/978-94-010-0732-0.![]() ![]() ![]() |
[25] |
B. Schmalfuß, Backward cocycle and attractors of stochastic differential equations, International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior, Technische Universität, (1992), 185-192. |
[26] |
J. Shen, K. Lu and W. Zhang,
Heteroclinic chaotic behavior driven by a Brownian motion, J. Differ. Equ., 255 (2013), 4185-4225.
doi: 10.1016/j.jde.2013.08.003. |
[27] |
J. Shen and K. Lu,
Wong-Zakai approximations and center manifolds of stochastic differential equations, J. Differ. Equ., 263 (2017), 4929-4977.
doi: 10.1016/j.jde.2017.06.005. |
[28] |
J. Shen, J. Zhao, K. Lu and B. Wang,
The Wong-Zakai approximations of invariant manifolds and foliations for stochastic evolution equations, J. Differ. Equ., 266 (2019), 4568-4623.
doi: 10.1016/j.jde.2018.10.008. |
[29] |
R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, 1997.
doi: 10.1007/978-1-4612-0645-3. |
[30] |
B. Wang, Existence and upper semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms, Stoch. Dyn., 14 (2014), 1450009.
doi: 10.1142/S0219493714500099. |
[31] |
B. Wang,
Suffcient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differ. Equ., 253 (2012), 1544-1583.
doi: 10.1016/j.jde.2012.05.015. |
[32] |
X. Wang, K. Lu and B. Wang,
Wong-Zakai approximations and attractors forstochastic reaction-diffusion equations onunbounded domains, J. Differ. Equ., 264 (2018), 378-424.
doi: 10.1016/j.jde.2017.09.006. |
[33] |
E. Wong and M. Zakai,
On the relation between ordinary and stochastic differentialequations, Int. J. Eng. Sci., 3 (1965), 213-229.
doi: 10.1016/0020-7225(65)90045-5. |
[34] |
E. Wong and M. Zakai,
On the convergence of ordinary integrals to stochastic integrals, Ann. Math. Stat., 36 (1965), 1560-1564.
doi: 10.1214/aoms/1177699916. |
[35] |
M. Yang, C. Sun and C. Zhong, Existence of a global attractor for a $p$-Laplacian equation in $\mathbb{R}^N$, Nonlinear Anal., 66 (2007), 1-13.
doi: 10.1016/j.na.2005.11.004. |
[36] |
M. Yang, C. Sun and C. Zhong,
Global attractor for $p$-Laplacian equation, J. Math. Anal. Appl., 327 (2007), 1130-1142.
doi: 10.1016/j.jmaa.2006.04.085. |
[37] |
J. Yin, Y. Li and H. Cui,
Two types of upper semi-continuity of bi-spatial attractors for non-autonomous stochastic $p$-Laplacian equations on $\mathbb{R}^N$, Math. Meth. Appl. Sci., 40 (2017), 4863-4879.
doi: 10.1002/mma.4353. |
[38] |
W. Zhao and Y. Zhang,
Compactness and attracting of random attractors for non-autonomous stochastic lattice dynamical systems in weighted space $\ell_\rho^p$, Appl. Math. Comput., 291 (2016), 226-243.
doi: 10.1016/j.amc.2016.06.045. |
[39] |
W. Zhao and Y. Zhang, Higher-order Wong-Zakai approximations of stochastic reaction-diffusion equations on $\mathbb{R}^N$, Physica D, 401 (2020), 132147.
doi: 10.1016/j.physd.2019.132147. |
[40] |
W. Zhao,
Random dynamics of stochastic $p$-Laplacian equations on $\mathbb{R}^N$ with an unbounded additive noise, J. Math. Anal. App., 455 (2017), 1178-1203.
doi: 10.1016/j.jmaa.2017.06.025. |
[41] |
W. Zhao,
Long-time random dynamics of stochastic parabolic $p$-Laplacian equations on $\mathbb{R}^N$, Nonliner Anal., 152 (2017), 196-219.
doi: 10.1016/j.na.2017.01.004. |
show all references
References:
[1] |
S. Al-azzawi, J. Liu and X. Liu,
Convergence rate of synchronization of systems with additive noise, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 227-245.
doi: 10.3934/dcdsb.2017012. |
[2] |
L. Arnold, Random Dynamical System, Springer-Verlag, Berlin, 1998.
doi: 10.1007/978-3-662-12878-7. |
[3] |
I. Chueshov, Monotone Random Systems Theory and Applications, Springer-Verlag, Berlin, 2002.
doi: 10.1007/b83277. |
[4] |
H. Crauel and F. Flandoli,
Attracors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393.
doi: 10.1007/BF01193705. |
[5] |
H. Crauel, A. Debussche and F. Flandoli,
Random attractors, J. Dynam. Differ. Equ., 9 (1997), 307-341.
doi: 10.1007/BF02219225. |
[6] |
P. G. Geredeli,
On the existence of regular global attractor for $p$-Laplacian evolution equations, Appl. Math. Optim, 71 (2015), 517-532.
doi: 10.1007/s00245-014-9268-y. |
[7] |
P. G. Geredeli and A. Kh. Khanmamedov,
Long-time dynamics of the parabolic $p$-Laplacian equation, Commun. Pure Appl. Anal., 12 (2013), 735-754.
doi: 10.3934/cpaa.2013.12.735. |
[8] |
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. |
[9] |
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. |
[10] |
I. Gyongy,
On the approximation of stochastic partial differential equations Ⅰ, Stochastics, 25 (1988), 59-85.
doi: 10.1080/17442508808833533. |
[11] |
I. Gyongy,
On the approximation of stochastic partial differential equations Ⅱ, Stochastics, 26 (1989), 129-164.
doi: 10.1080/17442508908833554. |
[12] |
I. Gyongy and A. Shmatkov,
Rate of convergence of Wong-Zakai approximations for stochastic partial differential equations, Appl. Math. Optim., 54 (2006), 315-341.
doi: 10.1007/s00245-006-1001-z. |
[13] |
H. Hu, X. Liu and et al, On smooth approximation for random attractor of stochastic partial differential equations with multiplicative noise, Stoch. Dyn., 18 (2018), 22pp.
doi: 10.1142/S0219493718500405. |
[14] |
A. Kh. Khanmamedov,
Existence of a global attractor for the parabolic equation with nonlinear Laplacian principal part in an unbounded domain, J. Math. Anal. Appl., 316 (2006), 601-615.
doi: 10.1016/j.jmaa.2005.05.003. |
[15] |
A. Krause and B. Wang,
Pullback attractors of non-autonomous stochastic degenerate parabolic equations on unbounded domains, J. Math. Anal. Appl., 417 (2014), 1018-1038.
doi: 10.1016/j.jmaa.2014.03.037. |
[16] |
A. Krause, M. Lewis and B. Wang,
Dynamics of the non-autonomous stochastic $p$-Laplace equation driven by multiplicative noise, Appl. Math. Comput., 246 (2014), 365-376.
doi: 10.1016/j.amc.2014.08.033. |
[17] |
J. Li, H. Cui and Y. Li,
Existence and upper semicontinuity of random attractors of stochastic $p$-Laplacian equations on unbounded domains, Electron J. Differ. Equ., 2014 (2014), 1-27.
|
[18] |
Y. Li, A. Gu and J. Li,
Existences and continuity of bi-spatial random attractors and application to stochasitic semilinear Laplacian equations, J. Differ. Equ., 258 (2015), 504-534.
doi: 10.1016/j.jde.2014.09.021. |
[19] |
Y. Li and J. Yin,
Existence, regularity and approximation of global attractors for weaky dissipative $p$-Laplacian equations, Discrete Contin. Dyn. Syst. Ser. S, 9 (2016), 1939-1957.
doi: 10.3934/dcdss.2016079. |
[20] |
J. L. Lions, Quelques Methodes de Resolution des Problemes aux Limites Non Lineaires, Dunod, Paris, 1969. |
[21] |
K. Lu and B. Wang,
Wong-Zakai approximations and long term behavior of stochastic partial differential equations, J. Dyn. Differ. Equ., 31 (2019), 1341-1371.
doi: 10.1007/s10884-017-9626-y. |
[22] |
K. Lu and Q. Wang,
Chaotic behavior in differential equations driven by a Brownian motion, J. Differ. Equ., 251 (2011), 2853-2895.
doi: 10.1016/j.jde.2011.05.032. |
[23] |
T. Nakayama and S. Tappe,
Wong-Zakai approximations with convergence rate for stochastic partial differential equations, Stoch. Anal. Appl., 36 (2018), 832-857.
doi: 10.1080/07362994.2018.1471402. |
[24] |
J. C. Robinson, Infnite-Dimensional Dyanmical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge University Press, 2001.
doi: 10.1007/978-94-010-0732-0.![]() ![]() ![]() |
[25] |
B. Schmalfuß, Backward cocycle and attractors of stochastic differential equations, International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior, Technische Universität, (1992), 185-192. |
[26] |
J. Shen, K. Lu and W. Zhang,
Heteroclinic chaotic behavior driven by a Brownian motion, J. Differ. Equ., 255 (2013), 4185-4225.
doi: 10.1016/j.jde.2013.08.003. |
[27] |
J. Shen and K. Lu,
Wong-Zakai approximations and center manifolds of stochastic differential equations, J. Differ. Equ., 263 (2017), 4929-4977.
doi: 10.1016/j.jde.2017.06.005. |
[28] |
J. Shen, J. Zhao, K. Lu and B. Wang,
The Wong-Zakai approximations of invariant manifolds and foliations for stochastic evolution equations, J. Differ. Equ., 266 (2019), 4568-4623.
doi: 10.1016/j.jde.2018.10.008. |
[29] |
R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, 1997.
doi: 10.1007/978-1-4612-0645-3. |
[30] |
B. Wang, Existence and upper semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms, Stoch. Dyn., 14 (2014), 1450009.
doi: 10.1142/S0219493714500099. |
[31] |
B. Wang,
Suffcient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differ. Equ., 253 (2012), 1544-1583.
doi: 10.1016/j.jde.2012.05.015. |
[32] |
X. Wang, K. Lu and B. Wang,
Wong-Zakai approximations and attractors forstochastic reaction-diffusion equations onunbounded domains, J. Differ. Equ., 264 (2018), 378-424.
doi: 10.1016/j.jde.2017.09.006. |
[33] |
E. Wong and M. Zakai,
On the relation between ordinary and stochastic differentialequations, Int. J. Eng. Sci., 3 (1965), 213-229.
doi: 10.1016/0020-7225(65)90045-5. |
[34] |
E. Wong and M. Zakai,
On the convergence of ordinary integrals to stochastic integrals, Ann. Math. Stat., 36 (1965), 1560-1564.
doi: 10.1214/aoms/1177699916. |
[35] |
M. Yang, C. Sun and C. Zhong, Existence of a global attractor for a $p$-Laplacian equation in $\mathbb{R}^N$, Nonlinear Anal., 66 (2007), 1-13.
doi: 10.1016/j.na.2005.11.004. |
[36] |
M. Yang, C. Sun and C. Zhong,
Global attractor for $p$-Laplacian equation, J. Math. Anal. Appl., 327 (2007), 1130-1142.
doi: 10.1016/j.jmaa.2006.04.085. |
[37] |
J. Yin, Y. Li and H. Cui,
Two types of upper semi-continuity of bi-spatial attractors for non-autonomous stochastic $p$-Laplacian equations on $\mathbb{R}^N$, Math. Meth. Appl. Sci., 40 (2017), 4863-4879.
doi: 10.1002/mma.4353. |
[38] |
W. Zhao and Y. Zhang,
Compactness and attracting of random attractors for non-autonomous stochastic lattice dynamical systems in weighted space $\ell_\rho^p$, Appl. Math. Comput., 291 (2016), 226-243.
doi: 10.1016/j.amc.2016.06.045. |
[39] |
W. Zhao and Y. Zhang, Higher-order Wong-Zakai approximations of stochastic reaction-diffusion equations on $\mathbb{R}^N$, Physica D, 401 (2020), 132147.
doi: 10.1016/j.physd.2019.132147. |
[40] |
W. Zhao,
Random dynamics of stochastic $p$-Laplacian equations on $\mathbb{R}^N$ with an unbounded additive noise, J. Math. Anal. App., 455 (2017), 1178-1203.
doi: 10.1016/j.jmaa.2017.06.025. |
[41] |
W. Zhao,
Long-time random dynamics of stochastic parabolic $p$-Laplacian equations on $\mathbb{R}^N$, Nonliner Anal., 152 (2017), 196-219.
doi: 10.1016/j.na.2017.01.004. |
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