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

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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 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.

show all references

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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