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The long-time behaviour of the solutions to semilinear stochastic partial differential equations on the whole space
Mathematica Bohemica, Tome 126 (2001) no. 1, pp. 15-39
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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.
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.
DOI : 10.21136/MB.2001.133912
Classification : 35B40, 35R60, 60H15
Keywords: nuclear and cylindrical noise; existence and uniqueness of the solution; spatial growth; ultimate boundedness; asymptotic mean square stability; Cauchy problem 
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Manthey, Ralf. The long-time behaviour of the solutions to semilinear stochastic partial differential equations on the whole space. Mathematica Bohemica, Tome 126 (2001) no. 1, pp. 15-39. doi: 10.21136/MB.2001.133912

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