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In this paper, we study the existence of random periodic solutions forsemilinear SPDEs on a bounded domain with a smooth boundary. We identify themas the solutions of coupled forward-backward infinite horizon stochasticintegral equations on $L^2(D)$ in general cases. For this we use Mercer'sTheorem and eigenvalues and eigenfunctions of the second order differentialoperators in the infinite horizon integral equations. We then use the argumentof the relative compactness of Wiener-Sobolev spaces in $C^0([0, T],L^2(\Omega\times D))$ and generalized Schauder's fixed point theorem to provethe existence of a solution of the integral equations. This is the first paperin literature to study random periodic solutions of SPDEs. Our result is alsonew in finding semi-stable stationary solution for non-dissipative SPDEs, whilein literature the classical method is to use the pull-back technique soresearchers were only able to find stable stationary solutions for dissipativesystems.

Random Periodic Solutions of SPDEs via Integral Equations and Wiener-Sobolev Compact Embedding