MATHEMATICA BOHEMICA, Vol. 126, No. 1, pp. 15-39 (2001)

The long-time behaviour of the solutions to semilinear stochastic partial differential equations on the whole space

Ralf Manthey

Ralf Manthey, Mathematisches Institut, Universität Erlangen-Nürnberg, Bismarckstr. 1 $1/2$, D-91054 Erlangen, Germany

Abstract: 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.

Keywords: Cauchy problem, nuclear and cylindrical noise, existence and uniqueness of the solution, spatial growth, ultimate boundedness, asymptotic mean square stability

Classification (MSC2000): 60H15, 35R60

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