# Averaging principles for non-autonomous two-time-scale stochastic reaction-diffusion equations with polynomial growth

## TL;DR

This paper establishes an averaging principle for complex stochastic reaction-diffusion equations with non-Lipschitz coefficients and polynomial growth, expanding the applicability of averaging methods in stochastic PDEs.

## Contribution

It develops an averaging principle for non-autonomous stochastic reaction-diffusion equations with polynomial growth and non-Lipschitz drift, including existence, uniqueness, and measure analysis.

## Key findings

- Proved existence and uniqueness of mild solutions.
- Established the existence of time-dependent evolution measures.
- Verified the validity of the averaging principle.

## Abstract

In this paper, we develop the averaging principle for a class of two-time-scale stochastic reaction-diffusion equations driven by Wiener processes and Poisson random measures. We assume that all coefficients of the equation have polynomial growth, and the drift term of the equation is non-Lipschitz. Hence, the classical formulation of the averaging principle under the Lipschitz condition is no longer available. To prove the validity of the averaging principle, the existence and uniqueness of the mild solution are proved firstly. Then, the existence of time-dependent evolution family of measures associated with the fast equation is studied, by which the averaged coefficient is obtained. Finally, the validity of the averaging principle is verified.