This paper investigates the asymptotic behavior of solutions for non-autonomous fractional stochastic reaction–diffusion equations with multiplicative noise in $\mathbb{R}$. We prove the existence and uniqueness of the tempered pullback random attractor for the equations in $L^2\left(\mathbb{R}\right)$ and obtain the finite fractal dimension for the pullback random attractor. Two main difficulties here are that the fractional Laplacian operator is non-local and the Sobolev embedding is not compact on unbounded domains. To solve this, we derive the tail-estimates of solutions of the equation and decompose the solutions into a sum of three parts, which one part is finite-dimensional and other two parts are quickly decay in mean sense.

本文研究了具有乘性噪声的非自治分数随机反应扩散方程解的渐近行为$\mathbb{R}$. 我们证明了方程的回火随机吸引子的存在性和唯一性${\mathrm{大号}}^2\left(\mathbb{R}\right)$并获得回拉随机吸引子的有限分形维数。这里的两个主要困难是分数拉普拉斯算子是非局部的，并且 Sobolev 嵌入在无界域上不是紧凑的。为了解决这个问题，我们推导出方程解的尾估计，并将解分解为三部分的总和，其中一部分是有限维的，另外两部分在平均意义上快速衰减。