Random attractors for 3D Benjamin–Bona–Mahony equations derived by a Laplace-multiplier noise

This paper contributes the dynamics for stochastic Benjamin–Bona–Mahony (BBM) equations on an unbounded 3D-channel with a multiplicative noise. An interesting feature is that the noise has a Laplace-operator multiplier, which seems not to appear in any literature for the study of stochastic PDE. After translating the stochastic BBM equation into a random equation and deducing a random dynamical system, we obtain both existence and semi-continuity of random attractors for this random system in the Sobolev space. The convergence of the system can be verified without the lower bound assumption of the nonlinear derivative. The tail-estimate is achieved by using a square of the usual cutoff function and by a careful analysis of the solution’s biquadrate. A spectrum method is also applied to prove the collective limit-set compactness.