Martingale solutions and invariant measures for the stochastic strongly damped wave equation with critical nonlinearity

This paper investigates the stochastic strongly damped wave equation with critical nonlinearity driven by a spatially homogeneous Winer process on the whole space. We first establish the existence of martingale solutions for the equation. Moreover, we prove that the martingale solution of the equation with deterministic initial data is pathwise unique and obtain a pathwise unique strong solution in the sense of probability. Finally, we construct an invariant measure for the equation with critical nonlinearity by using a weak Feller approach and construct an invariant measure for the equation with subcritical nonlinearity by using a decomposition method. We mention that the invariant measure constructed by the decomposition method is more regular than that constructed by the weak Feller approach.