Qualitative behavior of a class of stochastic parabolic PDEs with dynamical boundary conditions

Previous Article
The exponential stability of neutral stochastic delay partial differential equations
DCDS Home
This Issue
Next Article
Nonlinear Iwasawa decomposition of control flows

February & March 2007, 18(2&3): 315-338. doi: 10.3934/dcds.2007.18.315

Qualitative behavior of a class of stochastic parabolic PDEs with dynamical boundary conditions

Igor Chueshov ^1, and Björn Schmalfuss ^2,

1.	Department of Mechanics and Mathematics, Kharkov National University, Kharkov, 61077, Ukraine
2.	Mathematical Institute, University of Paderborn, Paderborn, 33098, Germany

Received February 2006 Revised September 2006 Published March 2007

Abstract
Related Papers

We consider non-linear parabolic stochastic partial differential equations with dynamical boundary conditions and with a noise which acts in the domain but also on the boundary and is presented by the temporal generalized derivative of an infinite dimensional Wiener process. We prove that solutions to this stochastic partial differential equation generate a random dynamical system. Under additional conditions we show that this system is monotone. Our main result states the existence of a compact global (pullback) attractor.

Keywords: stochastic PDE, monotonicity., random dynamical systems, dynamical boundary conditions, pullback attractor.

Mathematics Subject Classification: primary: 37L55; secondary: 37H10, 37L30, 60H1.

Citation: Igor Chueshov, Björn Schmalfuss. Qualitative behavior of a class of stochastic parabolic PDEs with dynamical boundary conditions. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 315-338. doi: 10.3934/dcds.2007.18.315

[1]	Peter Brune, Björn Schmalfuss. Inertial manifolds for stochastic pde with dynamical boundary conditions. Communications on Pure & Applied Analysis, 2011, 10 (3) : 831-846. doi: 10.3934/cpaa.2011.10.831
[2]	Wenqiang Zhao. Pullback attractors for bi-spatial continuous random dynamical systems and application to stochastic fractional power dissipative equation on an unbounded domain. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3395-3438. doi: 10.3934/dcdsb.2018326
[3]	Yejuan Wang, Chengkui Zhong, Shengfan Zhou. Pullback attractors of nonautonomous dynamical systems. Discrete & Continuous Dynamical Systems - A, 2006, 16 (3) : 587-614. doi: 10.3934/dcds.2006.16.587
[4]	Wen Tan. The regularity of pullback attractor for a non-autonomous p-Laplacian equation with dynamical boundary condition. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 529-546. doi: 10.3934/dcdsb.2018194
[5]	Tomás Caraballo, Stefanie Sonner. Random pullback exponential attractors: General existence results for random dynamical systems in Banach spaces. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6383-6403. doi: 10.3934/dcds.2017277
[6]	Lianfa He, Hongwen Zheng, Yujun Zhu. Shadowing in random dynamical systems. Discrete & Continuous Dynamical Systems - A, 2005, 12 (2) : 355-362. doi: 10.3934/dcds.2005.12.355
[7]	Philippe Marie, Jérôme Rousseau. Recurrence for random dynamical systems. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 1-16. doi: 10.3934/dcds.2011.30.1
[8]	Felix X.-F. Ye, Hong Qian. Stochastic dynamics Ⅱ: Finite random dynamical systems, linear representation, and entropy production. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-26. doi: 10.3934/dcdsb.2019122
[9]	Guanggan Chen, Jian Zhang. Asymptotic behavior for a stochastic wave equation with dynamical boundary conditions. Discrete & Continuous Dynamical Systems - B, 2012, 17 (5) : 1441-1453. doi: 10.3934/dcdsb.2012.17.1441
[10]	Yejuan Wang. On the upper semicontinuity of pullback attractors for multi-valued noncompact random dynamical systems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3669-3708. doi: 10.3934/dcdsb.2016116
[11]	Lu Yang, Meihua Yang, Peter E. Kloeden. Pullback attractors for non-autonomous quasi-linear parabolic equations with dynamical boundary conditions. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2635-2651. doi: 10.3934/dcdsb.2012.17.2635
[12]	Yujun Zhu. Preimage entropy for random dynamical systems. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 829-851. doi: 10.3934/dcds.2007.18.829
[13]	Ji Li, Kening Lu, Peter W. Bates. Invariant foliations for random dynamical systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3639-3666. doi: 10.3934/dcds.2014.34.3639
[14]	Weigu Li, Kening Lu. Takens theorem for random dynamical systems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3191-3207. doi: 10.3934/dcdsb.2016093
[15]	Bixiang Wang. Stochastic bifurcation of pathwise random almost periodic and almost automorphic solutions for random dynamical systems. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3745-3769. doi: 10.3934/dcds.2015.35.3745
[16]	María J. Garrido–Atienza, Kening Lu, Björn Schmalfuss. Random dynamical systems for stochastic partial differential equations driven by a fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 473-493. doi: 10.3934/dcdsb.2010.14.473
[17]	Björn Schmalfuss. Attractors for nonautonomous and random dynamical systems perturbed by impulses. Discrete & Continuous Dynamical Systems - A, 2003, 9 (3) : 727-744. doi: 10.3934/dcds.2003.9.727
[18]	Ivan Werner. Equilibrium states and invariant measures for random dynamical systems. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1285-1326. doi: 10.3934/dcds.2015.35.1285
[19]	Weigu Li, Kening Lu. A Siegel theorem for dynamical systems under random perturbations. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 635-642. doi: 10.3934/dcdsb.2008.9.635
[20]	Yuri Kifer. Computations in dynamical systems via random perturbations. Discrete & Continuous Dynamical Systems - A, 1997, 3 (4) : 457-476. doi: 10.3934/dcds.1997.3.457

2017 Impact Factor: 1.179

American Institute of Mathematical Sciences