Random Exponential Attractor for Stochastic Non-autonomous Suspension Bridge Equation with Linear Multiplicative White Noise

In this paper, we mainly consider the long-time behavior of stochastic non-autonomous suspension bridge equation by linear multiplicative white noise with small coefficient. First step,the well-posedness for the cocycle associated with the considered system is established. Second step, the existence of random attractor for the system is investigated. Third step, the upper semicontinuity of random attractor is also provided when the coefficient of random term approaches zero. Fourth step, we prove the regularity of random attractor in a higher regular space by the“iteration” method. Finally, we give the existence of a random exponential attractor for the considered system, which implies the finiteness of fractal dimension of random attractor.