Global solvability and convergence to stationary solutions in singular quasilinear stochastic PDEs

We consider singular quasilinear stochastic partial differential equations (SPDEs) studied in \cite{FHSX}, which are defined in paracontrolled sense. The main aim of the present article is to establish the global-in-time solvability for a particular class of SPDEs with origin in particle systems and, under a certain additional condition on the noise, prove the convergence of the solutions to stationary solutions as $t\to\infty$. We apply the method of energy inequality and Poincare inequality. It is essential that the Poincare constant can be taken uniformly in an approximating sequence of the noise. We also use the continuity of the solutions in the enhanced noise, initial values and coefficients of the equation, which we prove in this article for general SPDEs discussed in \cite{FHSX} except that in the enhanced noise. Moreover, we apply the initial layer property of improving regularity of the solutions in a short time.