REGULARITY OF RANDOM ATTRACTORS FOR A STOCHASTIC DEGENERATE PARABOLIC EQUATIONS DRIVEN BY MULTIPLICATIVE NOISE

doi:10.1016/S0252-9602(16)30009-1

数学物理学报(英文版) ›› 2016, Vol. 36 ›› Issue (2): 409-427.doi: 10.1016/S0252-9602(16)30009-1

REGULARITY OF RANDOM ATTRACTORS FOR A STOCHASTIC DEGENERATE PARABOLIC EQUATIONS DRIVEN BY MULTIPLICATIVE NOISE

赵文强

School of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, China

收稿日期:2014-03-09 修回日期:2014-08-07 出版日期:2016-04-25 发布日期:2016-04-25
作者简介:Wenqiang ZHAO,E-mail:gshzhao@sina.com
基金资助:
This work was supported by China NSF (11271388), Scientific and Technological Research Program of Chongqing Municipal Education Commission (KJ1400430), and Basis and Frontier Research Project of Chongqing (cstc2014jcyjA00035).

REGULARITY OF RANDOM ATTRACTORS FOR A STOCHASTIC DEGENERATE PARABOLIC EQUATIONS DRIVEN BY MULTIPLICATIVE NOISE

Wenqiang ZHAO

School of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, China

Received:2014-03-09 Revised:2014-08-07 Online:2016-04-25 Published:2016-04-25
Supported by:
This work was supported by China NSF (11271388), Scientific and Technological Research Program of Chongqing Municipal Education Commission (KJ1400430), and Basis and Frontier Research Project of Chongqing (cstc2014jcyjA00035).

摘要/Abstract

摘要：

We study the regularity of random attractors for a class of degenerate parabolic equations with leading term div(σ(x)∇u) and multiplicative noises. Under some mild conditions on the diffusion variable σ(x) and without any restriction on the upper growth p of nonlinearity, except that p>2, we show the existences of random attractor in D₀^{1, 2}(D_{N, σ})∩ L^?(D_N)(?∈[2, 2p-2]) space, where D_N is an arbitrary (bounded or unbounded) domain in R^N, N≥2. For this purpose, some abstract results based on the omega-limit compactness are established.

关键词: Random dynamical systems, stochastic degenerate parabolic equation, multiplicative noise, random attractors, Wiener process

Abstract:

Key words: Random dynamical systems, stochastic degenerate parabolic equation, multiplicative noise, random attractors, Wiener process

中图分类号:

35B40

赵文强. REGULARITY OF RANDOM ATTRACTORS FOR A STOCHASTIC DEGENERATE PARABOLIC EQUATIONS DRIVEN BY MULTIPLICATIVE NOISE[J]. 数学物理学报(英文版), 2016, 36(2): 409-427.

Wenqiang ZHAO. REGULARITY OF RANDOM ATTRACTORS FOR A STOCHASTIC DEGENERATE PARABOLIC EQUATIONS DRIVEN BY MULTIPLICATIVE NOISE[J]. Acta mathematica scientia,Series B, 2016, 36(2): 409-427.

参考文献

[1] Dautray R, Lions J L. Mathematical Analysis and Numerical Methods for Science and Technology. Vol. I:Physical origins and classical methods. Berlin:Springer-Verlag, 1985
[2] Eidus D, Kamin S. The filtration equation, in a class of functions decreasing at infinity. Proc Amer Math Soc, 1994, 120(3):825-830
[3] Gurtin M E, Macamy R C. On the Diffusion of biological populations. Math Biosci, 1977, 33:35-49
[4] Murray J D. Mathematical Biology, II:Spatial Models and Biomedical Applications. New York:Springer-Verlag, 2003
[5] Karachalios N I, Zographopoulos N B. On the dynamics of a degenerate parabolic equation:Global bifurcation of stationary states and convergence. Calc Var Partial Differential Equations, 2006, 25(3):361-393
[6] Caldiroli P, Musina R. On a variational degenerate elliptic problem. Nonlinear Differ Equ Appl, 2000, 7(2):187-199
[7] Anh C T, Bao T Q. Pullback attractors for a non-autonomous semi-linear degenerate parabolic equation. Glasgow Math J, 2010, 52(3):537-554
[8] Anh C T, Hung P Q. Global existence and long-time behavior of solutions to a class of degenerate parabolic equations. Ann Polon Math, 2008, 93(3):217-230
[9] Anh C T, Chuong N M, Ke T D. Global attractors for the m-semiflow degenerated by a quasilinear degenerate parabolic equation. J Math Anal Appl, 2010, 363(2):444-453
[10] Niu W S. Global attractors for degenerate semilinear parabolic equations. Nonl Anal, 2013, 77:158-170
[11] Feireisl E, Laurencot P, Simondon F. Global attractors for degenerate parabolic equations on unbounded domains. J Differential Equations, 1996, 129(2):239-261
[12] Yang M H, Kloeden P E. Random attractors for stochastic semi-linear degenerate parabolic equations. Nonlinear Analysis:Real World Applications, 2011, 12(5):2811-2821
[13] Anh C T, Bao T Q, Thanh N V. Regularity of random attractors for stochastic semilinear degenerate parabolic equations. Electronic Journal of Differential Equations, 2012, 207:1-22
[14] Yin J Y, Li Y R, Zhao H J. Random attractors for stochastic semi-linear degenerate parabolic equations with additive noise in L^q. Appl Math Compu, 2013, 225(1):526-540
[15] ZhaoWQ. Regularity of random attractors for a stochastic degenerate parabolic equation driven by additive noises. Appl Math Compu, 2014, 239C(15):358-374
[16] Li Y R, Guo B L. Random attractors for quasi-continuous random dynamical systems and applications to stochastic reaction-diffusion equations. J Differential Equations, 2008, 245(7):1775-1800
[17] Guo B L, Guo C X, Pu X K. Random attractors for a stochastic hydrodynamical equation in Heisenberg paramagnet. Acta Mathematica Scientia, 2011, 31B(2):529-540
[18] Zhao W Q. H¹-random attractors for stochastic reaction diffusion equations with additive noise. Nonlinear Anal, 2013, 84:61-72
[19] Zhao W Q. H¹-random attractors and random equilibria for stochastic reaction diffusion equations with multiplicative noises. Comm Nonlinear Sci Numer Simulat, 2013, 18(10):2707-2721
[20] Arnold L. Random Dynamical System. Berlin:Springer-Verlag, 1998
[21] Chueshov I. Monotone Random Systems Theory and Applications. Berlin:Springer-Verlag, 2002
[22] Crauel H, Debussche A, Flandoli F. Random attractors. J Dynam Diff Equa, 1997, 9(2):307-341
[23] Crauel H, Flandoli F. Attractors for random dynamical systems. Probab Theory Related Fields, 1994, 100(3):365-393
[24] Schmalfuß B. Backward cocycle and attractors of stochastic differential equations//Reitmann V, Riedrich T, Koksch N. International Seminar on Applied Mathematics-Nonlinear Dynamics:Attractor Approximation and Global Behavior. Dresden:Technische Universität, 1992:185-192
[25] Bates PW, Lu K N,Wang B X. Random attractors for stochastic reaction-diffusion equations on unbounded domains. J Differential Equations, 2009, 246(2):845-869
[26] Flandoli F, Schmalfuß B. Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative noise. Stoch Stoch Rep, 1996, 59(1/2):21-45
[27] Zhao W Q, Li Y R. (L², L^p)-random attractors for stochastic reaction-diffusion equation on unbounded domains. Nonlinear Anal, 2012, 75(2):485-502
[28] Temam R. Infinite-Dimensional Dynamical Systems in Mechanics and Physics. New York:Springer-Verlag, 1997
[29] Robinson J C. Infinite-Dimensional Dyanmical Systems:An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors. Cambridge University Press, 2001
[30] Zhong C K, Yang M H, Sun C Y. The existence of global attractors for the norm-to-weak continuous semigroup and its application to the nonlinear reaction-diffusion equations. J Differential Equations, 2006, 223(2):367-399
[31] Li J, Li Y R, Wang B. Random attractors of reaction-diffusion equations with multiplicative noise in L^p. Appl Math Compu, 2010, 215(9):3399-3407

REGULARITY OF RANDOM ATTRACTORS FOR A STOCHASTIC DEGENERATE PARABOLIC EQUATIONS DRIVEN BY MULTIPLICATIVE NOISE

REGULARITY OF RANDOM ATTRACTORS FOR A STOCHASTIC DEGENERATE PARABOLIC EQUATIONS DRIVEN BY MULTIPLICATIVE NOISE

PDF (PC)

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 3

Metrics

本文评价

推荐阅读 10

[1]	赵才地, 李用声, 周盛凡. RANDOM ATTRACTOR FOR A TWO-DIMENSIONAL INCOMPRESSIBLE NON-NEWTONIAN FLUID\\ WITH MULTIPLICATIVE NOISE[J]. 数学物理学报(英文版), 2011, 31(2): 567-575.
[2]	黎育红; Zdzislaw Brzezniak; 周建中. CONCEPTUAL ANALYSIS AND RANDOM ATTRACTOR FOR DISSIPATIVE RANDOM DYNAMICAL SYSTEMS[J]. 数学物理学报(英文版), 2008, 28(2): 253-268.
[3]	张立新. A LIMINF RESULT FOR HANSON-RUSSO TYPE INCREMENTS OF FRACTIONAL BROWNIAN MOTION[J]. 数学物理学报(英文版), 1997, 17(2): 190-197.