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September  2017, 22(7): 2627-2650. doi: 10.3934/dcdsb.2017102

Long-time behavior of stochastic reaction-diffusion equation with dynamical boundary condition

Lu Yang 1, and  Meihua Yang 2,,

1. 

School of Mathematics and Statistics, Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou, Gansu 730000, China
2. 

School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China

* Corresponding author

Received  July 2016 Revised  September 2016 Published  March 2017

Fund Project: The first author was supported by the NSFC grants(11361053,11471148), the Fundamental Research Funds for the Central Universities Grant (lzujbky-2016-98) and the State Scholarship Fund (201506185006) of China Scholarship Council. The second author was supported by the NSFC grant(11571125) and the NCET-12-0204.

In this paper, we study the dynamic behavior of a stochastic reaction-diffusion equation with dynamical boundary condition, where the nonlinear terms $f$ and $h$ satisfy the polynomial growth condition of arbitrary order. Some higher-order integrability of the difference of the solutions near the initial time, and the continuous dependence result with respect to initial data in $H^1(\mathcal{O})× H^{\frac 1 2}(Γ)$ were established. As a direct application, we can obtain the existence of pullback random attractor $A$ in the spaces $L^{p}(\mathcal{O})× L^{p}(Γ)$ and $H^1(\mathcal{O})× H^{\frac 1 2}(Γ)$ immediately.

Keywords: Stochastic reaction-diffusion equation, dynamical boundary condition, higher-order integrability, continuity, pullback random attractor.
Mathematics Subject Classification: Primary:37L05, 35B40;Secondary:35B41.
Citation: Lu Yang, Meihua Yang. Long-time behavior of stochastic reaction-diffusion equation with dynamical boundary condition. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2627-2650. doi: 10.3934/dcdsb.2017102
References:
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M. Anguiano, P. Marín-Rubio and J. Real, Pullback attractors for non-autonomous reactiondiffusion equations with dynamical boundary conditions, J. Math. Anal. Appl., 383 (2011), 608-618.  doi: 10.1016/j.jmaa.2011.05.046.  Google Scholar

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D. Cao, C. Sun and M. Yang, Dynamics for a stochastic reaction-diffusion equation with additive noise, J. Differential Equations, 259 (2015), 838-872.  doi: 10.1016/j.jde.2015.02.020.  Google Scholar

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[11]

I. Chueshov and B. Schmalfuß, Qualitative behavior of a class of stochastic parabolic PDES with dynamical boundary conditions, Discrete Contin. Dyn. Syst, 18 (2007), 315-338.  doi: 10.3934/dcds.2007.18.315.  Google Scholar

[12]

H. Crauel, Global random attractors are uniquely determined by attracting deterministic compact sets, Ann. Mat. Pura Appl., Ⅳ. Ser., 176 (1999), 57-72.  doi: 10.1007/BF02505989.  Google Scholar

[13]

H. Crauel, Random point attractors versus random set attractors, J. London Math. Soc., Ⅱ. Ser., 63 (2001), 413-427.  doi: 10.1017/S0024610700001915.  Google Scholar

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H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341.  doi: 10.1007/BF02219225.  Google Scholar

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H. Crauel and F. Flandoli, Attractor for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393.  doi: 10.1007/BF01193705.  Google Scholar

[16]

H. Crauel and F. Flandoli, Hausdorff dimension of invariant sets for random dynamical systems, J. Dynam. Differential Equations, 10 (1998), 449-474.  doi: 10.1023/A:1022605313961.  Google Scholar

[17]

H. Crauel, P. Kloeden and J. Real, Stochastic partial differential equations with additive noise on time-varying domains, Bol. Soc. Esp. Mat. Apl. SeMA, 51 (2010), 41-48.   Google Scholar

[18]

H. Crauel, P. Kloeden and M. Yang, Random attractors of stochastic reaction-diffusion equations on variable domains, Stoch. Dyn., 11 (2011), 301-314.  doi: 10.1142/S0219493711003292.  Google Scholar

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A. Debussche, On the finite dimensionality of random attractors, Stochastic Analysis and Applications, 15 (2007), 473-491.  doi: 10.1080/07362999708809490.  Google Scholar

[22]

Z. Fan and C. Zhong, Attractors for parabolic equations with dynamic boundary conditions, Nonlinear Anal., 68 (2008), 1723-1732.  doi: 10.1016/j.na.2007.01.005.  Google Scholar

[23]

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

[24]

C. Gal and M. Warma, Well-posedness and the global attractor of some quasi-linear parabolic equations with nonlinear dynamic boundary conditions, Differential Integral Equations, 23 (2010), 327-358.   Google Scholar

[25]

B. Gess, W. Liu and M. Röcknera, 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.  Google Scholar

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[27]

J. Langa and J. Robinson, Fractal dimension of a random invariant set, J. Math. Pures Appl., 85 (2006), 269-294.  doi: 10.1016/j.matpur.2005.08.001.  Google Scholar

[28]

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

[29]

J. Li, Y. Li and B. Wang, Random attractors of reaction-diffusion equations with multiplicative noise in $L^p$, Appl. Math. Comput., 215 (2010), 3399-3407.  doi: 10.1016/j.amc.2009.10.033.  Google Scholar

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M. Scheutzow, Comparison of various concepts of a random attractor: A case study, Archiv der Mathematik, 78 (2002), 233-240.  doi: 10.1007/s00013-002-8241-1.  Google Scholar

[32]

B. Schmalfuß, Backward cocycles and attractors of stochastic differential equations, in: V. Reitmann, T. Riedrich, N. Koksch (Eds. ), International Seminar on Applied Mathematics Nonlinear Dynamics: Attractor Approximation and Global Behaviour, 1992,185-192. Google Scholar

[33]

C. Sun and W. Tan, Non-autonomous reaction-diffusion model with dynamic boundary conditions, J. Math.Anal.Appl., 443 (2016), 1007-1032.  doi: 10.1016/j.jmaa.2016.05.054.  Google Scholar

[34]

B. Wang, Random attractors for the stochastic FitzHugh-Nagumo system on unbounded domains, Nonl. Anal., 71 (2009), 2811-2828.  doi: 10.1016/j.na.2009.01.131.  Google Scholar

[35]

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

[36]

H. Wu, Convergence to equilibrium for the semilinear parabolic equation with dynamical boundary condition, Adv. Math. Sci. Appl., 17 (2007), 67-88.   Google Scholar

[37]

L. Yang and M. Yang, Long-time behavior of reaction-diffusion equations with dynamical boundary condition, Nonlinear Anal., 74 (2011), 3876-3883.  doi: 10.1016/j.na.2011.02.022.  Google Scholar

[38]

C. Zhao and J. Duan, Random attractor for the Ladyzhenskaya model with additive noise, J. Math. Anal. Appl., 362 (2010), 241-251.  doi: 10.1016/j.jmaa.2009.08.050.  Google Scholar

[39]

W. Zhao, $H^1$-random attractors for stochastic reaction diffusion equations with additive noise, Nonl. Anal., 84 (2013), 61-72.   Google Scholar

[40]

W. Zhao, $H^1$-random attractors and random equilibria for stochastic reaction-diffusion equations with multiplicative noises, Commu. Nonl. Sci. Num. Simu., 18 (2013), 2707-2721.   Google Scholar

[41]

W. Zhao and Y. Li, $(L^2, L^p)$-random attractors for stochastic reaction-diffusion equation on unbounded domains, Nonl. Anal., 75 (2012), 485-502.  doi: 10.1016/j.na.2011.08.050.  Google Scholar

[42]

C. Zhong, M. Yang and C. Sun, The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations, J. Differential Equations, 223 (2006), 367-399.  doi: 10.1016/j.jde.2005.06.008.  Google Scholar

show all references

References:
[1] R. Adams and J. Fourier, Sobolev Spaces, 2nd ed., Academic Press, 2003.   Google Scholar
[2]

M. Anguiano, P. Marín-Rubio and J. Real, Pullback attractors for non-autonomous reactiondiffusion equations with dynamical boundary conditions, J. Math. Anal. Appl., 383 (2011), 608-618.  doi: 10.1016/j.jmaa.2011.05.046.  Google Scholar

[3]

M. Anguiano, P. Marín-Rubio and J. Real, Regularity results and exponential growth for pullback attractors of a non-autonomous reaction-diffusion model with dynamical boundary conditions, Nonlinear Analysis: Real World Applications, 20 (2014), 112-125.  doi: 10.1016/j.nonrwa.2014.05.003.  Google Scholar

[4] L. Arnold, Random Dynamical Systems, Springer, New York, 1998.  doi: 10.1007/978-3-662-12878-7.  Google Scholar
[5]

T. Bao, Regularity of pullback random attractors for stochastic Fitzhugh-Nagumo system on unbounded domains, Discrete Contin. Dyn. Syst., 35 (2015), 441-466.  doi: 10.3934/dcds.2015.35.441.  Google Scholar

[6]

P. 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.  Google Scholar

[7]

D. Cao, C. Sun and M. Yang, Dynamics for a stochastic reaction-diffusion equation with additive noise, J. Differential Equations, 259 (2015), 838-872.  doi: 10.1016/j.jde.2015.02.020.  Google Scholar

[8]

T. Caraballo, H. Crauel, J. Langa and J. Robinson, The effect of noise on the Chafee-Infante equation: A nonlinear case study, Proceedings Amer. Math. Soc., 135 (2007), 373-382.   Google Scholar

[9]

I. Chueshov, Monotone Random Systems Theory and Applications Lecture Notes in Mathematics, 1779. Springer-Verlag, Berlin, 2002. doi: 10.1007/b83277.  Google Scholar

[10]

I. Chueshov and B. Schmalfuß, Parabolic stochastic partial differential equations with dynamical boundary conditions, Differential Integral Equations, 17 (2004), 751-780.   Google Scholar

[11]

I. Chueshov and B. Schmalfuß, Qualitative behavior of a class of stochastic parabolic PDES with dynamical boundary conditions, Discrete Contin. Dyn. Syst, 18 (2007), 315-338.  doi: 10.3934/dcds.2007.18.315.  Google Scholar

[12]

H. Crauel, Global random attractors are uniquely determined by attracting deterministic compact sets, Ann. Mat. Pura Appl., Ⅳ. Ser., 176 (1999), 57-72.  doi: 10.1007/BF02505989.  Google Scholar

[13]

H. Crauel, Random point attractors versus random set attractors, J. London Math. Soc., Ⅱ. Ser., 63 (2001), 413-427.  doi: 10.1017/S0024610700001915.  Google Scholar

[14]

H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341.  doi: 10.1007/BF02219225.  Google Scholar

[15]

H. Crauel and F. Flandoli, Attractor for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393.  doi: 10.1007/BF01193705.  Google Scholar

[16]

H. Crauel and F. Flandoli, Hausdorff dimension of invariant sets for random dynamical systems, J. Dynam. Differential Equations, 10 (1998), 449-474.  doi: 10.1023/A:1022605313961.  Google Scholar

[17]

H. Crauel, P. Kloeden and J. Real, Stochastic partial differential equations with additive noise on time-varying domains, Bol. Soc. Esp. Mat. Apl. SeMA, 51 (2010), 41-48.   Google Scholar

[18]

H. Crauel, P. Kloeden and M. Yang, Random attractors of stochastic reaction-diffusion equations on variable domains, Stoch. Dyn., 11 (2011), 301-314.  doi: 10.1142/S0219493711003292.  Google Scholar

[19] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge Univ. Press, Cambridge, 1992.  doi: 10.1017/CBO9780511666223.  Google Scholar
[20]

A. Debussche, Hausdorff dimension of a random invariant set, J. Math. Pures Appl., 77 (1998), 967-988.  doi: 10.1016/S0021-7824(99)80001-4.  Google Scholar

[21]

A. Debussche, On the finite dimensionality of random attractors, Stochastic Analysis and Applications, 15 (2007), 473-491.  doi: 10.1080/07362999708809490.  Google Scholar

[22]

Z. Fan and C. Zhong, Attractors for parabolic equations with dynamic boundary conditions, Nonlinear Anal., 68 (2008), 1723-1732.  doi: 10.1016/j.na.2007.01.005.  Google Scholar

[23]

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

[24]

C. Gal and M. Warma, Well-posedness and the global attractor of some quasi-linear parabolic equations with nonlinear dynamic boundary conditions, Differential Integral Equations, 23 (2010), 327-358.   Google Scholar

[25]

B. Gess, W. Liu and M. Röcknera, 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.  Google Scholar

[26]

P. Kloeden and J. Langa, Flattening, squeezing and the existence of random attractors, Proc. Roy. Soc. London A, 463 (2007), 163-181.  doi: 10.1098/rspa.2006.1753.  Google Scholar

[27]

J. Langa and J. Robinson, Fractal dimension of a random invariant set, J. Math. Pures Appl., 85 (2006), 269-294.  doi: 10.1016/j.matpur.2005.08.001.  Google Scholar

[28]

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

[29]

J. Li, Y. Li and B. Wang, Random attractors of reaction-diffusion equations with multiplicative noise in $L^p$, Appl. Math. Comput., 215 (2010), 3399-3407.  doi: 10.1016/j.amc.2009.10.033.  Google Scholar

[30] J. Robinson, Infinite-Dimensional Dynamical systems an Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge University Press, 2001.   Google Scholar
[31]

M. Scheutzow, Comparison of various concepts of a random attractor: A case study, Archiv der Mathematik, 78 (2002), 233-240.  doi: 10.1007/s00013-002-8241-1.  Google Scholar

[32]

B. Schmalfuß, Backward cocycles and attractors of stochastic differential equations, in: V. Reitmann, T. Riedrich, N. Koksch (Eds. ), International Seminar on Applied Mathematics Nonlinear Dynamics: Attractor Approximation and Global Behaviour, 1992,185-192. Google Scholar

[33]

C. Sun and W. Tan, Non-autonomous reaction-diffusion model with dynamic boundary conditions, J. Math.Anal.Appl., 443 (2016), 1007-1032.  doi: 10.1016/j.jmaa.2016.05.054.  Google Scholar

[34]

B. Wang, Random attractors for the stochastic FitzHugh-Nagumo system on unbounded domains, Nonl. Anal., 71 (2009), 2811-2828.  doi: 10.1016/j.na.2009.01.131.  Google Scholar

[35]

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

[36]

H. Wu, Convergence to equilibrium for the semilinear parabolic equation with dynamical boundary condition, Adv. Math. Sci. Appl., 17 (2007), 67-88.   Google Scholar

[37]

L. Yang and M. Yang, Long-time behavior of reaction-diffusion equations with dynamical boundary condition, Nonlinear Anal., 74 (2011), 3876-3883.  doi: 10.1016/j.na.2011.02.022.  Google Scholar

[38]

C. Zhao and J. Duan, Random attractor for the Ladyzhenskaya model with additive noise, J. Math. Anal. Appl., 362 (2010), 241-251.  doi: 10.1016/j.jmaa.2009.08.050.  Google Scholar

[39]

W. Zhao, $H^1$-random attractors for stochastic reaction diffusion equations with additive noise, Nonl. Anal., 84 (2013), 61-72.   Google Scholar

[40]

W. Zhao, $H^1$-random attractors and random equilibria for stochastic reaction-diffusion equations with multiplicative noises, Commu. Nonl. Sci. Num. Simu., 18 (2013), 2707-2721.   Google Scholar

[41]

W. Zhao and Y. Li, $(L^2, L^p)$-random attractors for stochastic reaction-diffusion equation on unbounded domains, Nonl. Anal., 75 (2012), 485-502.  doi: 10.1016/j.na.2011.08.050.  Google Scholar

[42]

C. Zhong, M. Yang and C. Sun, The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations, J. Differential Equations, 223 (2006), 367-399.  doi: 10.1016/j.jde.2005.06.008.  Google Scholar

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