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Regular measurable dynamics for reaction-diffusion equations on narrow domains with rough noise
School of Mathematics and Statistics, Southwest University, Chongqing 400715, China |
This paper is concerned with the regular random dynamics for the reaction-diffusion equation defined on a thin domain and perturbed by rough noise, where the usual Winner process is replaced by a general stochastic process satisfied the basic convergence. A bi-spatial attractor is obtained when the non-initial space is $p$-times Lebesgue space or Sobolev space. The measurability of the solution operator is proved, which leads to the measurability of the attractor in both state spaces. Finally, the upper semi-continuity of attractors under the $p$-norm is established when the narrow domain degenerates onto a lower dimensional set. Both methods of symbolical truncation and spectral decomposition provide all required uniform estimates in both Lebesgue and Sobolev spaces.
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Existence and upper semicontinuity of bi-spatial pullback attractors for smoothing cocycles, Nonlinear Anal., 128 (2015), 303-324.
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Existence and upper semicontinuity of random attractors for stochastic degenerate parabolic equations with multiplicative noises, Appl. Math. Comput., 271 (2015), 777-789.
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A damped hyperbolic equation on thin domains, Trans. Amer. Math. Soc., 329 (1992), 185-219.
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Limiting behavior of non-autonomous stochastic reaction-diffusion equations on thin domains, Discrete Contin. Dyn. Syst., 38 (2018), 187-208.
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[20] |
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Upper semi-continuity and regularity of random attractors on $p$-times integrable spaces and applications, Nonlinear Anal., 109 (2014), 33-44.
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[21] |
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.
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[22] |
Y. Li and B. Guo,
Random attractors for quasi-continuous random dynamical systems and applications to stochastic reaction-diffusion equations, J. Differential Equations, 245 (2008), 1775-1800.
doi: 10.1016/j.jde.2008.06.031. |
[23] |
Y. Li and J. Yin,
A modified proof of pullback attractors in a Sobolev space for stochastic Fitzhugh-Nagumo equations, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 1203-1223.
doi: 10.3934/dcdsb.2016.21.1203. |
[24] |
Y. Li and J. Yin,
Existence, regularity and approximation of global attractors for weakly dissipative $p$-Laplace equations, Discrete Contin. Dyn. Syst. Ser. S, 9 (2016), 1939-1957.
doi: 10.3934/dcdss.2016079. |
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Z. Lian and K. Lu,
Lyapunov exponents and invariant manifolds for random dynamical systems in a Banach space, Memoirs Amer. Math. Soc., 206 (2010), 1-106.
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Z. Lian, P. Liu and K. Lu,
Existence of SRB measures for a class of partially hyperbolic attractors in Banach spaces, Discrete Contin. Dyn. Syst., 37 (2017), 3905-3920.
doi: 10.3934/dcds.2017164. |
[27] |
M. Prizzi,
A remark on reaction-diffusion equations in unbounded domains, Discrete Contin. Dyn. Syst., 9 (2002), 281-286.
doi: 10.3934/dcds.2003.9.281. |
[28] |
M. Prizzi and K. P. Rybakowski,
The effect of domain squeezing upon the dynamics of reaction-diffusion equations, J. Differential Equations, 173 (2001), 271-320.
doi: 10.1006/jdeq.2000.3917. |
[29] |
M. Prizzi and K. P. Rybakowski,
Recent results on thin domain problems Ⅱ, Topol. Methods Nonlinear Anal., 19 (2002), 199-219.
doi: 10.12775/TMNA.2002.010. |
[30] |
G. Raugel and G. R. Sell,
Navier-Stokes equations on thin 3D domains. Ⅰ. Global attractors and global regularity of solutions, J. Amer. Math. Soc., 6 (1993), 503-568.
doi: 10.2307/2152776. |
[31] |
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. |
[32] |
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. |
[33] |
M. Wang and Y. Tang,
Attractors in $H^2$ and $L^{2p-2}$ for reaction-diffusion equations on unbounded domains, Commun. Pure Appl. Anal., 12 (2013), 1111-1121.
doi: 10.3934/cpaa.2013.12.1111. |
[34] |
J. Yin and Y. Li,
Two types of upper semi-continuity of bi-spatial attractors for non-autonomous stochastic $p$-Laplacian equations on $R^n$, Math. Methods Appl. Sci., 40 (2017), 4863-4879.
doi: 10.1002/mma.4353. |
[35] |
J. Yin, Y. Li and H. Cui,
Box-counting dimensions and upper semicontinuities of bi-spatial attractors for stochastic degenerate parabolic equations on an unbounded domain, J. Math. Anal. Appl., 450 (2017), 1180-1207.
doi: 10.1016/j.jmaa.2017.01.064. |
[36] |
J. Yin, Y. Li and H. Zhao,
Random attractors for stochastic semi-linear degenerate parabolic equations with additive noise in $L^q$, Appl. Math. Comput., 225 (2013), 526-540.
doi: 10.1016/j.amc.2013.09.051. |
[37] |
W. Zhao,
$H^1$-random attractors and random equilibria for stochastic reaction-diffusion equations with multiplicative noises, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), 2707-2721.
doi: 10.1016/j.cnsns.2013.03.012. |
[38] |
W. Zhao and Y. Li,
($L^2$, $L^p$)-random attractors for stochastic reaction-diffusion equation on unbounded domains, Nonlinear Anal., 75 (2012), 485-502.
doi: 10.1016/j.na.2011.08.050. |
[39] |
W. Zhao and Y. Li,
Random attractors for stochastic semi-linear degenerate parabolic equations with additive noises, Dyn. Partial Diff. Equ., 11 (2014), 269-298.
doi: 10.4310/DPDE.2014.v11.n3.a4. |
show all references
References:
[1] |
A. Adili and B. Wang,
Random attractors for stochastic FitzHugh-Nagumo systems driven by deterministic non-autonomous forcing, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 643-666.
doi: 10.3934/dcdsb.2013.18.643. |
[2] |
F. Antoci and M. Prizzi,
Reaction-diffusion equations on unbounded thin domains, Topol. Methods Nonlinear Anal., 18 (2001), 283-302.
doi: 10.12775/TMNA.2001.035. |
[3] |
L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, 1998.
doi: 10.1007/978-3-662-12878-7. |
[4] |
J. M. Arrieta and A. N. Carvalho,
Spectral convergence and nonlinear dynamics of reaction-diffusion equations under perturbations of the domain, J. Differential Equations, 199 (2004), 143-178.
doi: 10.1016/j.jde.2003.09.004. |
[5] |
J. M. Arrieta, A. N. Carvalho, R. P. Silva and M. C. Pereira,
Semilinear parabolic problems in thin domains with a highly oscillatory boundary, Nonlinear Anal., 74 (2011), 5111-5132.
doi: 10.1016/j.na.2011.05.006. |
[6] |
P. W. Bates, K. Lu and B. X. 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. |
[7] |
T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuss and J. Valero,
Asymptotic behaviour of a stochastic semilinear dissipative functional equation without uniqueness of solutions, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 439-455.
doi: 10.3934/dcdsb.2010.14.439. |
[8] |
T. Caraballo and J. A. Langa,
Stability and random attractors for a reaction-diffusion equation with multiplicative noise, Disrete Contin. Dyn. Syst., 6 (2000), 875-892.
doi: 10.3934/dcds.2000.6.875. |
[9] |
I. Chueshov, Monotone Random Systems Theory and Applications, vol. 1779, Springer Science & Business Media, 2002.
doi: 10.1007/b83277. |
[10] |
I. Chueshov and S. Kuksin,
Random kick-forced 3D Navier-Stokes equations in a thin domain, Arch. Ration. Mech. Anal, 188 (2008), 117-153.
doi: 10.1007/s00205-007-0068-2. |
[11] |
I. S. Ciuperca,
Reaction-Diffusion equations on thin domains with varying order of thinness, J. Differential Equations, 126 (1996), 244-291.
doi: 10.1006/jdeq.1996.0051. |
[12] |
H. Cui, Y. Li and J. Yin,
Existence and upper semicontinuity of bi-spatial pullback attractors for smoothing cocycles, Nonlinear Anal., 128 (2015), 303-324.
doi: 10.1016/j.na.2015.08.009. |
[13] |
H. Cui and Y. Li,
Existence and upper semicontinuity of random attractors for stochastic degenerate parabolic equations with multiplicative noises, Appl. Math. Comput., 271 (2015), 777-789.
doi: 10.1016/j.amc.2015.09.031. |
[14] |
H. Cui, J. A. Langa and Y. Li,
Measurability of random attractors for quasi strong-to-weak continuous random dynamical systems, J. Dynamics Differential Equations, (2017), 1-26.
doi: 10.1007/s10884-017-9617-z. |
[15] |
J. K. Hale and G. Raugel,
A damped hyperbolic equation on thin domains, Trans. Amer. Math. Soc., 329 (1992), 185-219.
doi: 10.1090/S0002-9947-1992-1040261-1. |
[16] |
J. K. Hale and G. Raugel,
Reaction-diffusion equations on thin domains, J. Math. Pures Appl., 71 (1992), 33-95.
|
[17] |
J. K. Hale and G. Raugel,
A reaction-diffusion equation on a thin L-shaped domain, Proc. Roy. Soc. Edinburgh Sect. A, 125 (1995), 283-327.
doi: 10.1017/S0308210500028043. |
[18] |
D. Li, K. Lu, B. Wang and X. Wang,
Limiting behavior of non-autonomous stochastic reaction-diffusion equations on thin domains, Discrete Contin. Dyn. Syst., 38 (2018), 187-208.
doi: 10.1016/j.jde.2016.10.024. |
[19] |
D. Li, B. Wang and X. Wang,
Limiting behavior of non-autonomous stochastic reaction-diffusion equations on thin domains, J. Differential Equations, 262 (2017), 1575-1602.
doi: 10.1016/j.jde.2016.10.024. |
[20] |
Y. Li, H. Cui and J. Li,
Upper semi-continuity and regularity of random attractors on $p$-times integrable spaces and applications, Nonlinear Anal., 109 (2014), 33-44.
doi: 10.1016/j.na.2014.06.013. |
[21] |
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. |
[22] |
Y. Li and B. Guo,
Random attractors for quasi-continuous random dynamical systems and applications to stochastic reaction-diffusion equations, J. Differential Equations, 245 (2008), 1775-1800.
doi: 10.1016/j.jde.2008.06.031. |
[23] |
Y. Li and J. Yin,
A modified proof of pullback attractors in a Sobolev space for stochastic Fitzhugh-Nagumo equations, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 1203-1223.
doi: 10.3934/dcdsb.2016.21.1203. |
[24] |
Y. Li and J. Yin,
Existence, regularity and approximation of global attractors for weakly dissipative $p$-Laplace equations, Discrete Contin. Dyn. Syst. Ser. S, 9 (2016), 1939-1957.
doi: 10.3934/dcdss.2016079. |
[25] |
Z. Lian and K. Lu,
Lyapunov exponents and invariant manifolds for random dynamical systems in a Banach space, Memoirs Amer. Math. Soc., 206 (2010), 1-106.
doi: 10.1090/S0065-9266-10-00574-0. |
[26] |
Z. Lian, P. Liu and K. Lu,
Existence of SRB measures for a class of partially hyperbolic attractors in Banach spaces, Discrete Contin. Dyn. Syst., 37 (2017), 3905-3920.
doi: 10.3934/dcds.2017164. |
[27] |
M. Prizzi,
A remark on reaction-diffusion equations in unbounded domains, Discrete Contin. Dyn. Syst., 9 (2002), 281-286.
doi: 10.3934/dcds.2003.9.281. |
[28] |
M. Prizzi and K. P. Rybakowski,
The effect of domain squeezing upon the dynamics of reaction-diffusion equations, J. Differential Equations, 173 (2001), 271-320.
doi: 10.1006/jdeq.2000.3917. |
[29] |
M. Prizzi and K. P. Rybakowski,
Recent results on thin domain problems Ⅱ, Topol. Methods Nonlinear Anal., 19 (2002), 199-219.
doi: 10.12775/TMNA.2002.010. |
[30] |
G. Raugel and G. R. Sell,
Navier-Stokes equations on thin 3D domains. Ⅰ. Global attractors and global regularity of solutions, J. Amer. Math. Soc., 6 (1993), 503-568.
doi: 10.2307/2152776. |
[31] |
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. |
[32] |
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. |
[33] |
M. Wang and Y. Tang,
Attractors in $H^2$ and $L^{2p-2}$ for reaction-diffusion equations on unbounded domains, Commun. Pure Appl. Anal., 12 (2013), 1111-1121.
doi: 10.3934/cpaa.2013.12.1111. |
[34] |
J. Yin and Y. Li,
Two types of upper semi-continuity of bi-spatial attractors for non-autonomous stochastic $p$-Laplacian equations on $R^n$, Math. Methods Appl. Sci., 40 (2017), 4863-4879.
doi: 10.1002/mma.4353. |
[35] |
J. Yin, Y. Li and H. Cui,
Box-counting dimensions and upper semicontinuities of bi-spatial attractors for stochastic degenerate parabolic equations on an unbounded domain, J. Math. Anal. Appl., 450 (2017), 1180-1207.
doi: 10.1016/j.jmaa.2017.01.064. |
[36] |
J. Yin, Y. Li and H. Zhao,
Random attractors for stochastic semi-linear degenerate parabolic equations with additive noise in $L^q$, Appl. Math. Comput., 225 (2013), 526-540.
doi: 10.1016/j.amc.2013.09.051. |
[37] |
W. Zhao,
$H^1$-random attractors and random equilibria for stochastic reaction-diffusion equations with multiplicative noises, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), 2707-2721.
doi: 10.1016/j.cnsns.2013.03.012. |
[38] |
W. Zhao and Y. Li,
($L^2$, $L^p$)-random attractors for stochastic reaction-diffusion equation on unbounded domains, Nonlinear Anal., 75 (2012), 485-502.
doi: 10.1016/j.na.2011.08.050. |
[39] |
W. Zhao and Y. Li,
Random attractors for stochastic semi-linear degenerate parabolic equations with additive noises, Dyn. Partial Diff. Equ., 11 (2014), 269-298.
doi: 10.4310/DPDE.2014.v11.n3.a4. |
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