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>

Abstract

&lt;p style='text-indent:20px;'&gt;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 &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ L^2( {\mathbb{R}}^N) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; space. We establish the Wong-Zakai approximations of solutions in &lt;inline-formula&gt;&lt;tex-math id="M3"&gt;\begin{document}$ L^l( {\mathbb{R}}^N) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; for arbitrary &lt;inline-formula&gt;&lt;tex-math id="M4"&gt;\begin{document}$ l\geq q $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; in the sense of upper semi-continuity of their random attractors, where &lt;inline-formula&gt;&lt;tex-math id="M5"&gt;\begin{document}$ q $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; is the growth exponent of the nonlinearity. The &lt;inline-formula&gt;&lt;tex-math id="M6"&gt;\begin{document}$ L^l $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-pre-compactness of attractors is proved by using the truncation estimate in &lt;inline-formula&gt;&lt;tex-math id="M7"&gt;\begin{document}$ L^q $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; and the higher-order bound of solutions.&lt;/p&gt;