Stochastic modified Boussinesq approximate equation driven by fractional Brownian motion

The current paper is devoted to the dynamics of a stochastic modified Boussinesq approximate equation driven by fractional Brownian motion with H ∈ ( 1 2 , 1 ) . Based on the different diffusion operators △ 2 and −△ in the stochastic system, we combine two types of operators Φ 1 = I and a Hilbert-Schmidt operator Φ 2 to guarantee the convergence of the corresponding Wiener-type stochastic integrals. Then the existence and regularity of the stochastic convolution for the corresponding additive linear stochastic equation can be shown. By the Banach modified fixed point theorem in the selected intersection space, the existence and uniqueness of the global mild solution are obtained. Finally, the existence of a random attractor for the random dynamical system generated by the mild solution for the modified Boussinesq approximation equation is also established. MSC: 35B40, 35Q35, 76D05.