Abstract
We construct inertial manifolds for a class of random dynamical systems generated by retarded semilinear parabolic equations subjected to additive white noise. These inertial manifolds are finite-dimensional invariant surfaces, which attract exponentially all trajectories. We study the corresponding inertial forms, i.e., the restriction of the stochastic equation to the inertial manifold. These inertial forms are finite-dimensional Ito equations and they completely describe the long-time dynamics of the system under consideration. The existence of inertial manifolds and the properties of inertial forms allow us to show that under mild additional conditions the system has a global (random) attractor in the sense of the theory of random dynamical systems.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
Arnold, L. (1998). Random Dynamical Systems, Springer, Berlin-Heidelberg-New York.
Arnold, L., and Scheutzow, M. (1995). Perfect cocycles through stochastic differential equations. Prob. Theory Rel. Fields 101, 65-88.
Bensoussan, A, and Flandoli, F. (1995). Stochastic inertial manifolds. Stochast. Stoch. Rep. 53, 13-39.
Boutet de Monvel, L., Chueshov, I. D., and Rezounenko, A. V. (1998). Inertial manifolds for retarded semilinear parabolic equations. Nonlin. Anal. 34, 907-925.
Chow, S.-N., and Lu, K. (1988). Invariant manifolds for flows in Banach spaces. J. Diff. Eqs. 74, 285-317.
Chueshov, I. D. (1992). Introduction to the Theory of Inertial Manifolds (Lecture Notes), Kharkov University Press, Kharkov (in Russian).
Chueshov, I. D. (1995). Approximate inertial manifolds of exponential order for semilinear parabolic equations subjected to additive white noise. J. Dynam. Diff. Eqs. 7, 549-566.
Chueshov, I. D, and Girya, T. V. (1994). Inertial manifolds for stochastic dissipative dynamical systems. Doklady Acad. Sci. Ukraine 7, 42-45.
Chueshov, I. D, and Girya, T. V. (1995). Inertial manifolds and forms for semilinear parabolic equations subjected to additive white noise. Lett. Math. Phys. 34, 69-76.
Constantin, P., Foias, C., Nicolaenko, B., and Temam, R. (1989). Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations, Springer, Berlin-Heidelberg-New York.
Crauel, H., and Flandoli, F. (1994). Attractors for random dynamical systems. Prob. Theory Rel. Fields 100, 365-393.
Da Prato, G., and Debussche, A. (1996). Construction of stochastic inertial manifolds using backward integration. Stochast. Stoch. Rep. 59, 305-324.
Da Prato, G., and Zabczyk, J. (1992). Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge.
Da Prato, G., and Zabczyk, J. (1996). Ergodicity for Infinite Dimensional Systems, Cambridge University Press, Cambridge.
Foias, C., Sell, G. R., and Temam, R. (1988). Inertial manifolds for nonlinear evolutionary equations. J. Diff. Eqs. 73, 309-353.
Girya, T. V., and Chueshov, I. D. (1995). Inertial manifolds and stationary measures for stochastically perturbed dissipative dynamical systems. Sbornik: Math. 186, 29-45.
Kager, G., and Scheutzow, M. (1997). Generation of one-sided random dynamical systems by stochastic differential equations. Electronic J. Prob. 2, Paper 8.
Kunita, H. (1990). Stochastic Flows and Stochastic Differential Equations, Cambridge University Press, Cambridge.
Kuo, H.-H. (1975). Gaussian Measure sin Banach Spaces, Springer, Berlin-Heidelberg-New York.
Scheutzow, M. (1996). On the perfection of crude cocycles. Random Comp. Dynam. 4, 235-255.
Sharpe, M. (1988). General Theory of Markov Processes, Academic Press, Boston.
Temam, R. (1988). Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer, Berlin-Heidelberg-New York.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Chueshov, I.D., Scheutzow, M. Inertial Manifolds and Forms for Stochastically Perturbed Retarded Semilinear Parabolic Equations. Journal of Dynamics and Differential Equations 13, 355–380 (2001). https://doi.org/10.1023/A:1016684108862
Issue Date:
DOI: https://doi.org/10.1023/A:1016684108862