# zbMATH — the first resource for mathematics

Random attractors for stochastic reaction-diffusion equations on unbounded domains. (English) Zbl 1155.35112
Summary: The existence of a pullback attractor is established for a stochastic reaction-diffusion equation on all $$n$$-dimensional space. The nonlinearity is dissipative for large values of the state and the stochastic nature of the equation appears as spatially distributed temporal white noise. The reaction-diffusion equation is recast as a random dynamical system and asymptotic compactness for this is demonstrated by using uniform a priori estimates for far-field values of solutions.

##### MSC:
 35R60 PDEs with randomness, stochastic partial differential equations 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 35B40 Asymptotic behavior of solutions to PDEs 35B41 Attractors
Full Text:
##### References:
 [1] Antoci, F.; Prizzi, M., Reaction – diffusion equations on unbounded thin domains, Topol. methods nonlinear anal., 18, 283-302, (2001) · Zbl 1007.35010 [2] Antoci, F.; Prizzi, M., Attractors and global averaging of non-autonomous reaction – diffusion equations in $$\mathbb{R}^n$$, Topol. methods nonlinear anal., 20, 229-259, (2002) · Zbl 1039.35021 [3] Arnold, L., Random dynamical systems, (1998), Springer-Verlag [4] Arrieta, J.M.; Cholewa, J.W.; Dlotko, T.; Rodriguez-Bernal, A., Asymptotic behavior and attractors for reaction diffusion equations in unbounded domains, Nonlinear anal., 56, 515-554, (2004) · Zbl 1058.35102 [5] Babin, A.V.; Vishik, M.I., Attractors of evolution equations, (1992), North-Holland Amsterdam · Zbl 0778.58002 [6] Ball, J.M., Continuity properties and global attractors of generalized semiflows and the navier – stokes equations, J. nonlinear sci., 7, 475-502, (1997) · Zbl 0903.58020 [7] Ball, J.M., Global attractors for damped semilinear wave equations, Discrete contin. dyn. syst., 10, 31-52, (2004) · Zbl 1056.37084 [8] Bates, P.W.; Lisei, H.; Lu, K., Attractors for stochastic lattice dynamical systems, Stoch. dyn., 6, 1-21, (2006) · Zbl 1105.60041 [9] Bates, P.W.; Lu, K.; Wang, B., Attractors for lattice dynamical systems, Internat. J. bifur. chaos appl. sci. engrg., 11, 143-153, (2001) · Zbl 1091.37515 [10] Caraballo, T.; Langa, J.A.; Robinson, J.C., A stochastic pitchfork bifurcation in a reaction – diffusion equation, Proc. R. soc. lond. ser. A, 457, 2041-2061, (2001) · Zbl 0996.60070 [11] Caraballo, T.; Lukaszewicz, G.; Real, J., Pullback attractors for asymptotically compact nonautonomous dynamical systems, Nonlinear anal., 64, 484-498, (2006) · Zbl 1128.37019 [12] Chepyzhov, V.V.; Vishik, M.I., Trajectory attractors for reaction – diffusion systems, Topol. methods nonlinear anal., 7, 49-76, (1996) · Zbl 0894.35010 [13] Crauel, H., Random point attractors versus random set attractors, J. London math. soc., 63, 413-427, (2002) · Zbl 1011.37032 [14] Crauel, H.; Debussche, A.; Flandoli, F., Random attractors, J. dynam. differential equations, 9, 307-341, (1997) · Zbl 0884.58064 [15] Crauel, H.; Flandoli, F., Attractors for random dynamical systems, Probab. theory related fields, 100, 365-393, (1994) · Zbl 0819.58023 [16] Da Prato, G.; Zabczyk, J., Stochastic equations in infinite dimensions, (1992), Cambridge Univ. Press · Zbl 0761.60052 [17] 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 [18] Ghidaglia, J.M., A note on the strong convergence towards attractors for damped forced KdV equations, J. differential equations, 110, 356-359, (1994) · Zbl 0805.35114 [19] Goubet, O.; Rosa, R., Asymptotic smoothing and the global attractor of a weakly damped KdV equation on the real line, J. differential equations, 185, 25-53, (2002) · Zbl 1034.35122 [20] Hale, J.K., Asymptotic behavior of dissipative systems, (1988), Amer. Math. Soc. Providence, RI · Zbl 0642.58013 [21] Ju, N., The $$H^1$$-compact global attractor for the solutions to the navier – stokes equations in two-dimensional unbounded domains, Nonlinearity, 13, 1227-1238, (2000) · Zbl 0979.35116 [22] Moise, I.; Rosa, R., On the regularity of the global attractor of a weakly damped, forced korteweg – de Vries equation, Adv. differential equations, 2, 257-296, (1997) · Zbl 1023.35525 [23] Moise, I.; Rosa, R.; Wang, X., Attractors for non-compact semigroups via energy equations, Nonlinearity, 11, 1369-1393, (1998) · Zbl 0914.35023 [24] Morillas, F.; Valero, J., Attractors for reaction – diffusion equations in $$\mathbb{R}^n$$ with continuous nonlinearity, Asymptot. anal., 44, 111-130, (2005) · Zbl 1083.35022 [25] Prizzi, M., Averaging, Conley index continuation and recurrent dynamics in almost-periodic equations, J. differential equations, 210, 429-451, (2005) · Zbl 1065.37017 [26] Robinson, J.C., Infinite-dimensional dynamical systems, (2001), Cambridge Univ. Press Cambridge, UK · Zbl 1026.37500 [27] Rodriguez-Bernal, A.; Wang, B., Attractors for partly dissipative reaction diffusion systems in $$\mathbb{R}^n$$, J. math. anal. appl., 252, 790-803, (2000) · Zbl 0977.35028 [28] Rosa, R., The global attractor for the 2D navier – stokes flow on some unbounded domains, Nonlinear anal., 32, 71-85, (1998) · Zbl 0901.35070 [29] Sell, G.R.; You, Y., Dynamics of evolutionary equations, (2002), Springer-Verlag New York · Zbl 1254.37002 [30] Stanislavova, M.; Stefanov, A.; Wang, B., Asymptotic smoothing and attractors for the generalized benjamin – bona – mahony equation on $$\mathbb{R}^3$$, J. differential equations, 219, 451-483, (2005) · Zbl 1160.35354 [31] Sun, C.; Zhong, C., Attractors for the semilinear reaction – diffusion equation with distribution derivatives in unbounded domains, Nonlinear anal., 63, 49-65, (2005) · Zbl 1082.35036 [32] Temam, R., Infinite-dimensional dynamical systems in mechanics and physics, (1997), Springer-Verlag New York · Zbl 0871.35001 [33] Wang, B., Attractors for reaction – diffusion equations in unbounded domains, Phys. D, 128, 41-52, (1999) · Zbl 0953.35022 [34] Wang, X., An energy equation for the weakly damped driven nonlinear Schrödinger equations and its application to their attractors, Phys. D, 88, 167-175, (1995) · Zbl 0900.35372
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.