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The stochastic 3D Navier–Stokes equation with damping driven by a multiplicative noise is considered in this paper. The stability of weak solutions to the stochastic 3D Navier–Stokes equations with damping is proved for any β > 3 with any α > 0 and α ≥ 1 2 as β = 3. The weak solutions converge exponentially in the mean square and almost surely exponentially to the stationary solutions are proved for any β > 3 with any α > 0 and α ≥ 1 2 as β = 3. The stabilization of these equations is obtained for any β > 3 with any α > 0 and α ≥ 1 2 as β = 3. [ABSTRACT FROM AUTHOR]

In this paper, bifurcation of the generalized Swift–Hohenberg equation is considered. We first study the bifurcation of the generalized Swift–Hohenberg equation in one spatial dimension with three kinds of boundary conditions. With the help of Liapunov–Schmidt reduction, the original equation is transformed to the reduced system, and then the bifurcation analysis is carried out. Secondly, bifurcation of the generalized Swift–Hohenberg equation in two spatial dimensions with periodic boundary conditions is also considered, using the perturbation method, asymptotic expressions of the nontrivial solutions bifurcated from the trivial solution are obtained. Moreover, the stability of the bifurcated solutions is discussed. [ABSTRACT FROM AUTHOR]