Well-posedness of stochastic partial differential equations with Lyapunov condition

In this paper we show the existence and uniqueness of strong solutions for a large class of SPDE where the coefficients satisfy the local monotonicity and Lyapunov condition (one-sided linear growth condition). Moreover, some new invariance result and stronger regularity estimate are also established for the solutions. As examples, the main result is applied to stochastic tamed 3D Navier–Stokes equations, stochastic generalized curve shortening flow, singular stochastic p-Laplace equations, stochastic fast diffusion equations, stochastic Burgers type equations and stochastic reaction–diffusion equations.