Well-posedness and long-time dynamics of fractional nonclassical diffusion equations with locally lipschitz noise

The global well-posedness and long-time dynamics are investigated for a class of non -autonomous, stochastic, fractional, nonclassical diffusion equations on R-N with polynomial growth drifts of arbitrary order p > 2 and locally Lipschitz diffusions. By a regularization method, we establish the well-posedness of the equations in L2(omega,F, Hs(RN)) for any s is an element of (0,1] and N is an element of N in four instances: regular additive noise, general additive noise, globally Lipschitz noise and locally Lipschitz noise. Then we prove that the mean random dynamical systems generated by the solution operators have a unique weak pullback mean random attractor in L-2(omega,F,H-s(R-N)). Our results do not depend on any re-strictions on the triple (N,s,p), and even original in L-2(omega,F, H-1(R-N)) when the fractional Laplacian reduces to the standard one.