Korean J. Math. Vol. 26 No. 4 (2018) pp.583-599
DOI: https://doi.org/10.11568/kjm.2018.26.4.583

Existence of random attractors for stochastic non-autonomous reaction-diffusion equation with multiplicative noise on Rn

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Published: Dec 30, 2018
Keywords:
Random attractors, Stochastic non-autonomous reaction-diffusion equation, Random dynamical system.
Mathematics Subject Classification:
35B40, 35B41, 35B45.

Main Article Content

Fadlallah Mustafa Mosa
1. College of Mathematics and Statistics Northwest Normal University Lanzhou, Gansu 730070, China 2. Department of Mathematics and physics, Faculty of Education University of Kassala Kassala, Sudan
Qiaozhen Ma
College of Mathematics and Statistics Northwest Normal University Lanzhou, Gansu 730070, China
Mohamed Y. A. Bakhet
1. College of Mathematics and Statistics Northwest Normal University Lanzhou, Gansu 730070, China 2. Department of Mathematics, College of Education Rumbek University of Science and Technology Rumbek, South Sudan

Abstract

In this paper, we are concerned with the existence of random dynamics for stochastic non-autonomous reaction-diffusion equations driven by a Wiener-type multiplicative noise defined on the unbounded domains.


Article Details

Supporting Agencies

This work was partly supported by the NSFC (11561064) and partly supported by NWNU-LKQN-14-6.

References

[1] L. Arnold, Random Dynamical System, Springer-Verlag, New York, 1998. Google Scholar

[2] P. W. Bates, K. Lu and B. Wang, Random Attractor for stochastic reaction-Diffusion Equation on unbounded domains, J. Differential Equations, 246 (2) (2009) 845-869. Google Scholar

[3] T. Caraballo, P. E. Kloeden and B. Schmalfub, Exponentially stable stationary solutions for stochastic evolution equations and their perturbation, Appl. Math. Optim, 50 (2004) 183-207. Google Scholar

[4] M. Yang, C. Sun and C. Zhao, Global attractor for p-Laplacian equation, J. Math. Anal. Appl. 327 (2) (2007) 1130-1142. Google Scholar

[5] T. Caraballo, J. A. Langa and J.C. Robinson, A stochastic pitchfork bifurcation in a reaction-diffusion equation, Proc. R. Soc. Lond. Ser, A 457 (2001) 2041-2061. Google Scholar

[6] Z. Wang and S. Zhou, Random attractor for stochastic reaction-diffusion equation with multiplicative noise on unbounded domains, J. Math. Anal. Appl ,384 (2011) 160-172. Google Scholar

[7] H. Crauel, Random probability Measure on polish Spaces, Taylor and Francis, London,2002. Google Scholar

[8] S.F. Zhou, Y. X. Tian and Z. J. Wang, Fractal dimension of random attractors for stochastic non-autonomous reaction-diffusion equations, Appl. Math. Comput. 276 (2016) 80-95. Google Scholar

[9] J. Li, H. Cui and Y. Li, Existence and upper semicontinuity of random attractors of stochastic p-Laplacian equation on unbounded domains, Electron. J. Differential Equations 87 (2014) 1-27. Google Scholar

[10] W. Zhao, Long-time random dynamics of stochastic parabolic p-Laplacian equation on R^N, Nonlinear Anal.152 (2017),196-219. Google Scholar

[11] H. Crauel, A.Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997) 307-341. Google Scholar

[12] H. Crauel and F. Flandoli, Attractors for random dynamical system, Probab. Theory Related Fields.100 (1994) 365-393. Google Scholar

[13] S. Zhou, Random exponential attractors for stochastic reaction-diffusion equation with multiplicative noise in R^3, J. Differential Equations (2017). Google Scholar

[14] X. Fan, Attractor for a damped stochastic wave equation of sine-Gordon type with sublinear multiplicative noise, Stoch. Anal. Appl.24 (2006) 767-793. Google Scholar

[15] W. Zhao, Random attractors in H^1 for stochastic two dimensional Micropolar fluid flows with spatial-valued noise, Electron. J. Differential Equations 246 (2014) 1-19. Google Scholar

[16] F. Flandoli and B. SchmalfuB, Random attractors for 3D stochastic Navier-Stokes equation with multiplicative noise, Stoch. Stoch. Rep. 59 (1996) 21-45. Google Scholar

[17] W. Zhao and Y. Li, (L^2, L^P) -random attractors for stochastic reaction-diffusion equation on unbounded domains, Nonlinear Anal. 75 (2) (2012) 485-502. Google Scholar

[18] G. Da Prato and J. Zabczyk, stochastic Equation in Infinite Dimensions, Cambridge Univ. Press, New York, 1992. Google Scholar

[19] I. Chueshov, Monotone Random System Theory and Applications, Springer-Verlag.Berlin,2002. Google Scholar

[20] C. Zhao and S. Zhou, sufficient conditions for the existence of global random attractors for stochastic lattice dynamical systems and applications, J. Math. Anal. Appl. 354 (2009) 78-95. Google Scholar

[21] W. Zhao, Regularity of random attractors for a degenerate parabolic equations driven by additive noise, Appl. Math. Comput. 239 (2014) 358-374. Google Scholar

[22] H. Kesten, An iterated logarithm law for local time, Duke Math.J.32 (3) (1965) 447-456. Google Scholar

[23] M. D. Donsker and S. R. Varadhon, On laws of the iterated logarithm for local time, Comm. Pure Appl. Math. 30 (6) (1977) 707-753. Google Scholar

[24] B. Wang, Suffcient and necessary criteria for existence of pullback attractors for non-compact random dynamical system, J. Differential Equations. 253 (2012) 1544-1583. Google Scholar