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Random Attractors for Stochastic Reaction-Diffusion Equations with Distribution Derivatives on Unbounded Domains
College of Mathematics and Statistics, Northwest Normal University, Lanzhou, China
College of Mathematics and Statistics, Northwest Normal University, Lanzhou, China
College of Mathematics and Statistics, Northwest Normal University, Lanzhou, China
College of Mathematics and Statistics, Northwest Normal University, Lanzhou, China
- 1 College of Mathematics and Statistics, Northwest Normal University, Lanzhou, China
- 2 College of Mathematics and Statistics, Northwest Normal University, Lanzhou, China
- 3 College of Mathematics and Statistics, Northwest Normal University, Lanzhou, China
- 4 College of Mathematics and Statistics, Northwest Normal University, Lanzhou, China
Applied Mathematics·Volume 06 (2015)·Pages 1790–1807·Published 8 September 2015·DOI10.4236/am.2015.610159
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Abstract
In this paper, we prove the existence of random attractors for a stochastic reaction-diffusion equation with distribution derivatives on unbounded domains. The nonlinearity is dissipative for large values of the state and the stochastic nature of the equation appears spatially distributed temporal white noise. The stochastic reaction-diffusion equation is recast as a continuous random dynamical system and asymptotic compactness for this demonstrated by using uniform estimates far-field values of solutions. The results are new and appear to be optimal.
KeywordsStochastic Reaction-Diffusion EquationRandom AttractorsDistribution DerivativesAsymptotic CompactnessUnbounded Domain
- Flandoli, F. and Schmalfuβ, B. (1996) Random Attractors for the 3D Stochastic Navier-Stokes Equation with Multiplicative Noise. Stochastics and Stochastic Reports, 59, 21-45. http://dx.doi.org/10.1080/17442509608834083
- Antoci, F. and Prizzi, M. (2001) Reaction-Diffusion Equations on Unbounded Thin Domains. Topological Methods in Nonlinear Analysis, 18, 283-302.
- Hale, J.K. (1988) Asymptotic Behavior of Dissipative Systems. American Mathematical Society, Providence.
- Robinson, J.C. (2001) Infinite-Dimensional Dynamical Systems. Cambridge University Press, Cambridge. http://dx.doi.org/10.1007/978-94-010-0732-0
- Wang, B. (1999) Attractors for Reaction-Diffusion Equations in Unbounded Domains. Physica D, 128, 41-52. http://dx.doi.org/10.1016/S0167-2789(98)00304-2
- Crauel, H., Debussche, A. and Flandoli, F. (1997) Random Attractors. Journal of Dynamics and Differential Equations, 9, 307-341. http://dx.doi.org/10.1007/BF02219225
- Caraballo, T., Langa, J.A. and Robinson, J.C. (2001) A Stochastic Pitchfork Bifurcation in a Reaction-Diffusion Equation. Proceedings of the Royal Society A, 457, 2041-2061. http://dx.doi.org/10.1098/rspa.2001.0819
- Bates, P.W., Lu, K.N. and Wang, B.X. (2009) Random Attractors for Stochastic Reaction-Diffusion Equations on Unbounded Domains. Journal of Differential Equations, 246, 845-869. http://dx.doi.org/10.1016/j.jde.2008.05.017
- Rosa, R. (1998) The Global Attractor for the 2D Navier-Stokes Flow on Some Unbounded Domains. Nonlinear Analysis, 32, 71-85. http://dx.doi.org/10.1016/S0362-546X(97)00453-7
- Ball, J.M. (2004) Global Attractors for Damped Semilinear Wave Equations. Discrete and Continuous Dynamical Systems, 10, 31-52. http://dx.doi.org/10.3934/dcds.2004.10.31
- Arnold, L. (1998) Random Dynamical Systems. Springer-Verlag, Berlin. http://dx.doi.org/10.1007/978-3-662-12878-7
- Crauel, H., Debussche, A. and Flandoli, F. (1997) Random Attractors. Journal of Dynamics and Differential Equations 9, 307-341. http://dx.doi.org/10.1007/BF02219225
- Crauel, H. and Flandoli, F. (1994) Attractors for Random Dynamical Systems. Probability Theory and Related Fields, 100, 365-393. http://dx.doi.org/10.1007/BF01193705