# Article

 Title: The long-time behaviour of the solutions to semilinear stochastic partial differential equations on the whole space (English) Author: Manthey, Ralf Language: English Journal: Mathematica Bohemica ISSN: 0862-7959 (print) ISSN: 2464-7136 (online) Volume: 126 Issue: 1 Year: 2001 Pages: 15-39 Summary lang: English . Category: math . Summary: The Cauchy problem for a stochastic partial differential equation with a spatial correlated Gaussian noise is considered. The "drift" is continuous, one-sided linearily bounded and of at most polynomial growth while the "diffusion" is globally Lipschitz continuous. In the paper statements on existence and uniqueness of solutions, their pathwise spatial growth and on their ultimate boundedness as well as on asymptotical exponential stability in mean square in a certain Hilbert space of weighted functions are proved. (English) Keyword: nuclear and cylindrical noise Keyword: existence and uniqueness of the solution Keyword: spatial growth Keyword: ultimate boundedness Keyword: asymptotic mean square stability Keyword: Cauchy problem MSC: 35B40 MSC: 35R60 MSC: 60H15 idZBL: Zbl 0982.60055 idMR: MR1825855 DOI: 10.21136/MB.2001.133912 . Date available: 2009-09-24T21:46:52Z Last updated: 2020-07-29 Stable URL: http://hdl.handle.net/10338.dmlcz/133912 . Reference: [1] G. Da Prato, J. Zabczyk: Stochastic equations in infinite dimensions.Cambridge University Press, 1992. MR 1207136 Reference: [2] G. Da Prato, J. Zabczyk: Convergence to equilibrium for classical and quantum spin systems.Probab. Theory Relat. Fields 103 (1995), 529–553. MR 1360204, 10.1007/BF01246338 Reference: [3] K. Iwata: An infinite dimensional stochastic differential equation with state space $\mathbb{C}(\mathbb{R})$.Probab. Theory Relat. Fields 74 (1987), 141–159. MR 0863723, 10.1007/BF01845644 Reference: [4] R. Manthey: On the Cauchy problem for reaction-diffusion equations with white noise.Math. Nachr. 136 (1988), 209–228. Zbl 0658.60089, MR 0952473, 10.1002/mana.19881360114 Reference: [5] R. Manthey: On semilinear stochastic partial differential equations on the real line.Stochastics Stochastics Rep. 57 (1996), 213–234. Zbl 0887.60070, MR 1425366, 10.1080/17442509608834061 Reference: [6] R. Manthey, K. Mittmann: A growth estimate for continuous random fields.Math. Bohem. 121 (1996), 397–413. MR 1428142 Reference: [7] R. Manthey, K. Mittmann: The initial value problem for stochastic reaction-diffusion equations with continuous reaction.Stochastic Anal. Appl. 15 (1997), 555–583. MR 1464406, 10.1080/07362999708809495 Reference: [8] R. Manthey, C. Stiewe: Existence and uniqueness of a solution to Volterra’s population equation with diffusion and noise.Stochastics 41 (1992), 135–161. MR 1275580 Reference: [9] R. Manthey, T. Zausinger: Stochastic evolution equations in ${\mathbb{L}}^{2\nu }_{\rho }$.Stochastics Stochastics Rep. 66 (1999), 37–85. MR 1687799, 10.1080/17442509908834186 Reference: [10] R. Marcus: Stochastic diffusion on an unbounded domain.Pacific J. Math. 84 (1979), 143–153. Zbl 0423.60056, MR 0559632, 10.2140/pjm.1979.84.143 Reference: [11] R. I. Ovsepian, A. Pełczyński: On the existence of a fundamental total biorthogonal sequence in every separable Banach space and related constructions of uniformly bounded orthonormal systems in ${\mathbb{L}}^2$.Studia Mathematica 54 (1975), 149–159. MR 0394137, 10.4064/sm-54-2-149-159 Reference: [12] T. Shiga: Two contrasting properties of solutions for one-dimensional stochastic partial differential equations.Can. J. Math. 46 (1994), 415–437. Zbl 0801.60050, MR 1271224, 10.4153/CJM-1994-022-8 .

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