Strong averaging principle for nonautonomous multi-scale SPDEs with fully local monotone and almost periodic coefficients

In this paper, we consider a class of nonautonomous multi-scale stochastic partial differential equations with fully local monotone coefficients. By introducing the evolution system of measures for time-inhomogeneous Markov semigroups, we study the averaging principle for such kind of system. Specifically, we first prove the slow component in the multi-scale stochastic system converges strongly to the solution of an averaged equation, whose coefficients retain the dependence of the scaling parameter. Furthermore, if the coefficients satisfy uniformly almost periodic conditions, we establish that the slow component converges strongly to the solution of another averaged equation, whose coefficients are independent of the scaling parameter. The main contribution of this paper extends the basic nonautonomous framework investigated by Cheng and Liu in [11] to a fully coupled framework, as well as the autonomous framework explored by Liu et al. in [27] to the more general nonautonomous framework. Additionally, we improve the locally monotone coefficients discussed in [11,27] to the fully local monotone coefficients, thus our results can be applied to a wide range of cases in nonlinear nonautonomous stochastic partial differential equations, such as multi-scale stochastic Cahn-Hilliard-heat equation and multi-scale stochastic liquid-crystal-porous-media equation.