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Little seems to be known about the invariant manifolds for stochastic partialdifferential equations (SPDEs) driven by nonlinear multiplicative noise. Herewe contribute to this aspect and analyze the Lu-Schmalfu{\ss} conjecture[Garrido-Atienza, et al., J. Differential Equations, 248(7):1637--1667, 2010]on the existence of stable manifolds for a class of parabolic SPDEs driven bynonlinear mutiplicative fractional noise. We emphasize that stable manifoldsfor SPDEs are infinite-dimensional objects, and the classical Lyapunov-Perronmethod cannot be applied, since the Lyapunov-Perron operator does not give anyinformation about the backward orbit. However, by means of interpolationtheory, we construct a suitable function space in which the discretizedLyapunov-Perron-type operator has a unique fixed point. Based on this wefurther prove the existence and smoothness of local stable manifolds for suchSPDEs.

Existence of smooth stable manifolds for a class of parabolic SPDEs with fractional noise