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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 |
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.
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. R |
[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. R |
[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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