Regularity of Wong-Zakai approximation for non-autonomous stochastic quasi-linear parabolic equation on <inline-formula><tex-math id="M1">$ {\mathbb{R}}^N $</tex-math></inline-formula>

In this paper, we investigate a non-autonomous stochastic quasi-linear parabolic equation driven by multiplicative white noise by a Wong-Zakai approximation technique. The convergence of the solutions of quasi-linear parabolic equations driven by a family of processes with stationary increment to that of stochastic differential equation with white noise is obtained in the topology of \begin{document}$ L^2( {\mathbb{R}}^N) $\end{document} space. We establish the Wong-Zakai approximations of solutions in \begin{document}$ L^l( {\mathbb{R}}^N) $\end{document} for arbitrary \begin{document}$ l\geq q $\end{document} in the sense of upper semi-continuity of their random attractors, where \begin{document}$ q $\end{document} is the growth exponent of the nonlinearity. The \begin{document}$ L^l $\end{document} -pre-compactness of attractors is proved by using the truncation estimate in \begin{document}$ L^q $\end{document} and the higher-order bound of solutions.