Dynamics of the nonlinear stochastic heat equation with singular perturbation

The approximation in probability for a singular perturbed nonlinear stochastic heat equation is studied. First the approximation result in the sense of probability is obtained for solutions defined on any finite time interval. Furthermore it is proved that the long time behavior of the stochastic system is described by a global random attractor which is upper semi-continuous with respect to the singular perturbed parameter. This also means the long time effectivity of the approximation with probability one.