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Random attractors for stochastic semi-linear degenerate parabolic equations. (English) Zbl 1222.35042

Summary: The existence of a random attractor is established for a class of stochastic semi-linear degenerate parabolic equations with the leading term of the form \(\text{div}(\sigma (x)\nabla u)\) and additive spatially distributed temporal noise. The nonlinearity is dissipative for large values of the state without restriction on the growth order of the polynomial, while the spatial domain is either bounded or unbounded.

MSC:

35B41 Attractors
35L55 Higher-order hyperbolic systems
35K58 Semilinear parabolic equations
35R60 PDEs with randomness, stochastic partial differential equations
35K65 Degenerate parabolic equations
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[1] A.N. Carvalho, J.A. Langa, J.C. Robinson, Attractors of Infinite Dimensional Nonautonomous Dynamical Systems, Springer, Berlin, 2011 (in press).; A.N. Carvalho, J.A. Langa, J.C. Robinson, Attractors of Infinite Dimensional Nonautonomous Dynamical Systems, Springer, Berlin, 2011 (in press). · Zbl 1263.37002
[2] Cheban, D.; Kloeden, P. E.; Schmalfuß, B., The relationship between pullback, forwards and global attractors of nonautonomous dynamical systems, Nonlinear Dyn. Syst. Theory, 2, 9-28 (2002) · Zbl 1054.34087
[3] P.E. Kloeden, M. Rasmussen, Nonautonomous Dynamical Systems, Amer. Math. Soc., Providence, 2011 (in press).; P.E. Kloeden, M. Rasmussen, Nonautonomous Dynamical Systems, Amer. Math. Soc., Providence, 2011 (in press). · Zbl 1244.37001
[4] Sell, G. R.; You, Y., Dynamics of Evolutionary Equations (2002), Springer: Springer New York · Zbl 1254.37002
[5] 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., 363, 444-453 (2010) · Zbl 1181.35138
[6] Karachalios, N. I.; Zographopoulos, N. B., Convergence towards attractors for a degenerate Ginzburg-Landau equation, Z. Angew. Math. Phys., 56, 11-30 (2005) · Zbl 1181.35133
[7] 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, 25, 361-393 (2006) · Zbl 1090.35035
[8] Bates, P. W.; Lisei, H.; Lu, K., Attractors for stochastic lattice dynamical system, Stoch. Dyn., 6, 1-21 (2006) · Zbl 1105.60041
[9] Bates, P. W.; Lu, Kening; Wang, Bixiang, Random attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 246, 845-869 (2009) · Zbl 1155.35112
[10] Crauel, H.; Debussche, A.; Flandoli, F., Random attractors, J. Dynam. Differential Equations, 9, 307-341 (1997) · Zbl 0884.58064
[11] Flandoli, F.; Schmalfuß, B., Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative noise, Stoch. Stoch. Rep., 59, 21-45 (1996) · Zbl 0870.60057
[12] Kloeden, P. E.; Langa, J. A., Flattening, squeezing and the existence of random attractors, Proc. R. Soc. Lond. Ser. A, 463, 163-181 (2007) · Zbl 1133.37323
[13] Schmalfuß, B., Backward cocycles and attractors of stochastic differential equations, (Reitmann, V.; Riedrich, T.; Koksch, N., Nonlinear Dynamics: Attractor Approximation and Global Behaviour. Nonlinear Dynamics: Attractor Approximation and Global Behaviour, International Seminar on Applied Mathematics (1992)), 185-192
[14] Yang, Meihua; Duan, Jinqiao; Kloeden, P. E., Asymptotic behavior of solutions for random wave equations with nonlinear damping and white noise, Nonlinear Anal. RWA, 12, 464-478 (2011) · Zbl 1202.35124
[15] Barbu, V.; Da Prato, G.; Röckner, M., Stochastic nonlinear diffusion equations with singular diffusivity, SIAM J. Math. Anal., 41, 1106-1120 (2009) · Zbl 1203.60079
[16] W.-J. Beyn, B. Gess, P. Lescot, M. Röckner, The global random attractor for a class of stochastic porous media equations, BiBoS-Preprint 10-02-338.; W.-J. Beyn, B. Gess, P. Lescot, M. Röckner, The global random attractor for a class of stochastic porous media equations, BiBoS-Preprint 10-02-338.
[17] B. Gess, Wei Liu, M. Röckner, Random attractors for a class of stochastic partial differential equations driven by general additive noise, Preprint.; B. Gess, Wei Liu, M. Röckner, Random attractors for a class of stochastic partial differential equations driven by general additive noise, Preprint. · Zbl 1228.35062
[18] Anh, C. T.; Bao, T. Q., Pullback attractors for a non-autonomous semi-linear degenerate parabolic equation, Glasg. Math. J., 52, 537-554 (2010) · Zbl 1213.35120
[19] Anh, C. T.; Hung, P. Q., Global existence and long-time behavior of solutions to a class of degenerate parabolic equations, Ann. Polon. Math., 93, 217-230 (2008) · Zbl 1145.35341
[20] Caldiroli, P.; Musina, R., On a variational degenerate elliptic problem, NoDEA Nonlinear Differential Equations Appl., 7, 187-199 (2000) · Zbl 0960.35039
[21] Abdelaoui, B.; Peral, I., On the quasilinear elliptic equations related to some Caffarelli-Kohn-Nirenberg inequalities, Commun. Pure Appl. Anal., 2, 539-566 (2003) · Zbl 1148.35324
[22] Cazenave, T.; Haraux, T., Introduction to Semilinear Evolution Equations (1998), Oxford University Press: Oxford University Press Oxford · Zbl 0926.35049
[23] Zeidler, E., Nonlinear Functional Analysis and its Applications (1990), Springer: Springer New York · Zbl 0684.47028
[24] Arnold, L., Random Dynamical Systems (1998), Springer: Springer New York · Zbl 0906.34001
[25] Crauel, H.; Flandoli, F., Attractor for random dynamical systems, Probab. Theory Related Fields, 10, 365-393 (1994) · Zbl 0819.58023
[26] Da Prato, G.; Zabczyk, J., Stochastic Equations in Infinite Dimensions (1992), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0761.60052
[27] Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations (1983), Springer: Springer New York · Zbl 0516.47023
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