Invariant measures for the stochastic strongly damped wave equation

In this paper, a stochastic strongly damped wave equation with polynomial drift and diffusion terms is studied. First we prove there exists a unique solution for this equation by using a truncation method. Then, we establish the tightness of a family of probability distributions of solutions and obtain the existence of invariant measures by introducing an appropriate Lyapunov function and utilizing a decomposition approach. Finally, the regularity of the invariant measure is investigated, that is, the invariant measure is supported by a regular space.