# American Institute of Mathematical Sciences

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

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

 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.

Citation: Lu Yang, Meihua Yang. Long-time behavior of stochastic reaction-diffusion equation with dynamical boundary condition. Discrete and Continuous Dynamical Systems - B, 2017, 22 (7) : 2627-2650. doi: 10.3934/dcdsb.2017102
##### References:
 [1] R. Adams and J. Fourier, Sobolev Spaces, 2nd ed., Academic Press, 2003. [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. [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. [4] L. Arnold, Random Dynamical Systems, Springer, New York, 1998.  doi: 10.1007/978-3-662-12878-7. [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. [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. [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. [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. [9] I. Chueshov, Monotone Random Systems Theory and Applications Lecture Notes in Mathematics, 1779. Springer-Verlag, Berlin, 2002. doi: 10.1007/b83277. [10] I. Chueshov and B. Schmalfuß, Parabolic stochastic partial differential equations with dynamical boundary conditions, Differential Integral Equations, 17 (2004), 751-780. [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. [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. [13] H. Crauel, Random point attractors versus random set attractors, J. London Math. Soc., Ⅱ. Ser., 63 (2001), 413-427.  doi: 10.1017/S0024610700001915. [14] H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341.  doi: 10.1007/BF02219225. [15] H. Crauel and F. Flandoli, Attractor for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393.  doi: 10.1007/BF01193705. [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. [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. [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. [19] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge Univ. Press, Cambridge, 1992.  doi: 10.1017/CBO9780511666223. [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. [21] A. Debussche, On the finite dimensionality of random attractors, Stochastic Analysis and Applications, 15 (2007), 473-491.  doi: 10.1080/07362999708809490. [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. [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. [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. [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. [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. [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. [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. [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. [30] J. Robinson, Infinite-Dimensional Dynamical systems an Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge University Press, 2001. [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. [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. [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. [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. [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. [36] H. Wu, Convergence to equilibrium for the semilinear parabolic equation with dynamical boundary condition, Adv. Math. Sci. Appl., 17 (2007), 67-88. [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. [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. [39] W. Zhao, $H^1$-random attractors for stochastic reaction diffusion equations with additive noise, Nonl. Anal., 84 (2013), 61-72. [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. [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. [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.

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##### References:
 [1] R. Adams and J. Fourier, Sobolev Spaces, 2nd ed., Academic Press, 2003. [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. [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. [4] L. Arnold, Random Dynamical Systems, Springer, New York, 1998.  doi: 10.1007/978-3-662-12878-7. [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. [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. [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. [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. [9] I. Chueshov, Monotone Random Systems Theory and Applications Lecture Notes in Mathematics, 1779. Springer-Verlag, Berlin, 2002. doi: 10.1007/b83277. [10] I. Chueshov and B. Schmalfuß, Parabolic stochastic partial differential equations with dynamical boundary conditions, Differential Integral Equations, 17 (2004), 751-780. [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. [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. [13] H. Crauel, Random point attractors versus random set attractors, J. London Math. Soc., Ⅱ. Ser., 63 (2001), 413-427.  doi: 10.1017/S0024610700001915. [14] H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341.  doi: 10.1007/BF02219225. [15] H. Crauel and F. Flandoli, Attractor for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393.  doi: 10.1007/BF01193705. [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. [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. [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. [19] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge Univ. Press, Cambridge, 1992.  doi: 10.1017/CBO9780511666223. [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. [21] A. Debussche, On the finite dimensionality of random attractors, Stochastic Analysis and Applications, 15 (2007), 473-491.  doi: 10.1080/07362999708809490. [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. [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. [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. [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. [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. [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. [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. [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. [30] J. Robinson, Infinite-Dimensional Dynamical systems an Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge University Press, 2001. [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. [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. [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. [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. [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. [36] H. Wu, Convergence to equilibrium for the semilinear parabolic equation with dynamical boundary condition, Adv. Math. Sci. Appl., 17 (2007), 67-88. [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. [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. [39] W. Zhao, $H^1$-random attractors for stochastic reaction diffusion equations with additive noise, Nonl. Anal., 84 (2013), 61-72. [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. [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. [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.
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