American Institute of Mathematical Sciences

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July  2018, 38(7): 3663-3685. doi: 10.3934/dcds.2018158

Regular measurable dynamics for reaction-diffusion equations on narrow domains with rough noise

Fuzhi Li , Yangrong Li , and  Renhai Wang

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

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

Received  November 2017 Published  April 2018

Fund Project: This work was supported by the Natural Science Foundation of China Grant 11571283.

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.

Keywords: Bi-spatial random attractor, stochastic reaction-diffusion equation, rough noise, thin domain, upper semi-continuity, regularity, symbolical truncation.
Mathematics Subject Classification: Primary: 35B40; Secondary: 37L55, 60H15.
Citation: Fuzhi Li, Yangrong Li, Renhai Wang. Regular measurable dynamics for reaction-diffusion equations on narrow domains with rough noise. Discrete & Continuous Dynamical Systems, 2018, 38 (7) : 3663-3685. doi: 10.3934/dcds.2018158
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[20]

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[21]

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[22]

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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.  Google Scholar

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[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

I. Chueshov, Monotone Random Systems Theory and Applications, vol. 1779, Springer Science & Business Media, 2002. doi: 10.1007/b83277.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

J. K. Hale and G. Raugel, Reaction-diffusion equations on thin domains, J. Math. Pures Appl., 71 (1992), 33-95.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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