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 $ L^2( {\mathbb{R}}^N) $ space. We establish the Wong-Zakai approximations of solutions in $ L^l( {\mathbb{R}}^N) $ for arbitrary $ l\geq q $ in the sense of upper semi-continuity of their random attractors, where $ q $ is the growth exponent of the nonlinearity. The $ L^l $-pre-compactness of attractors is proved by using the truncation estimate in $ L^q $ and the higher-order bound of solutions.
Citation: Guifen Liu, Wenqiang Zhao. Regularity of Wong-Zakai approximation for non-autonomous stochastic quasi-linear parabolic equation on $ {\mathbb{R}}^N $[J]. Electronic Research Archive, 2021, 29(6): 3655-3686. doi: 10.3934/era.2021056
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 $ L^2( {\mathbb{R}}^N) $ space. We establish the Wong-Zakai approximations of solutions in $ L^l( {\mathbb{R}}^N) $ for arbitrary $ l\geq q $ in the sense of upper semi-continuity of their random attractors, where $ q $ is the growth exponent of the nonlinearity. The $ L^l $-pre-compactness of attractors is proved by using the truncation estimate in $ L^q $ and the higher-order bound of solutions.
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