# American Institute of Mathematical Sciences

December  2021, 26(12): 6425-6462. doi: 10.3934/dcdsb.2021026

## Periodic, almost periodic and almost automorphic solutions for SPDEs with monotone coefficients

 School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China

* Corresponding author: Zhenxin Liu

Received  November 2020 Published  December 2021 Early access  January 2021

Fund Project: The authors sincerely thank the referee for his/her careful reading of the paper. This work is partially supported by NSFC Grants 11522104, 11871132, 11925102, and Xinghai Jieqing and DUT19TD14 funds from Dalian University of Technology

In this paper, we use the variational approach to investigate recurrent properties of solutions for stochastic partial differential equations, which is in contrast to the previous semigroup framework. Consider stochastic differential equations with monotone coefficients. Firstly, we establish the continuous dependence on initial values and coefficients for solutions, which is interesting in its own right. Secondly, we prove the existence of recurrent solutions, which include periodic, almost periodic and almost automorphic solutions. Then we show that these recurrent solutions are globally asymptotically stable in square-mean sense. Finally, for illustration of our results we give two applications, i.e. stochastic reaction diffusion equations and stochastic porous media equations.

Citation: Mengyu Cheng, Zhenxin Liu. Periodic, almost periodic and almost automorphic solutions for SPDEs with monotone coefficients. Discrete & Continuous Dynamical Systems - B, 2021, 26 (12) : 6425-6462. doi: 10.3934/dcdsb.2021026
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