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The aim of this article is to study the asymptotic behaviour for large timesof solutions to a certain class of stochastic partial differential equations ofparabolic type. In particular, we will prove the backward uniqueness result andthe existence of the spectral limit for abstract SPDEs and then show how theseresults can be applied to some concrete linear and nonlinear SPDEs. Forexample, we will consider linear parabolic SPDEs with gradient noise andstochastic NSEs with multiplicative noise. Our results generalize the resultsproved in \cite{[Ghidaglia-1986]} for deterministic PDEs.

Backward Uniqueness and the existence of the spectral limit for some parabolic SPDEs