In this paper, we consider the asymptotic behavior of solutions for nonautonomous stochastic parabolic equation with nonstandard growth condition driven by nonlinear multiplicative noise for the first time. First, by making use of variational method, we prove the existence and uniqueness of solutions, and then the mean random dynamical systems generated by stochastic parabolic equations with variable exponents are obtained. Finally, due to the influence of variable indexes (dependent on space variable), we show the existence of weak mean random attractors under suitable assumptions on the variable exponents and the diffusion term.

中文翻译：

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变指数非自治随机抛物线方程的弱均值随机吸引子

在本文中，我们首次考虑了由非线性乘性噪声驱动的具有非标准增长条件的非自治随机抛物型方程解的渐近行为。首先，利用变分法证明解的存在性和唯一性，然后得到由变指数随机抛物型方程生成的平均随机动力系统。最后，由于变量指标的影响（取决于空间变量），我们在变量指数和扩散项的适当假设下证明了弱均值随机吸引子的存在。