This article is concerned with the random dynamics of a wide class of non-autonomous, non-local, fractional, stochastic p-Laplacian equations driven by multiplicative white noise on the entire space $${\mathbb {R}}^N$$ . We first establish the well-posedness of the equations when the time-dependent non-linear drift terms have polynomial growth of arbitrary orders $$p,q\ge 2$$ . We then prove that the equation has a unique bi-spatial pullback random attractor that is measurable, compact in $$L^2({\mathbb {R}}^N)\cap L^p({\mathbb {R}}^N)\cap L^q({\mathbb {R}}^N)$$ and attracts all random subsets of $$L^2({\mathbb {R}}^N)$$ under the topology of $$L^2({\mathbb {R}}^N)\cap L^p({\mathbb {R}}^N)\cap L^q({\mathbb {R}}^N)$$ . In addition, we establish the upper semi-continuity of these attractors in $$L^2({\mathbb {R}}^N)\cap L^p({\mathbb {R}}^N)\cap L^q({\mathbb {R}}^N)$$ when the density of noise shrinks to zero. The idea of uniform tail estimates and the method of asymptotic a priori estimates are applied to prove the pullback asymptotic compactness of the solutions in $$L^2({\mathbb {R}}^N)\cap L^p({\mathbb {R}}^N)\cap L^q({\mathbb {R}}^N)$$ to overcome the non-compactness of Sobolev embeddings on $${\mathbb {R}}^N$$ as well as the almost sure nondifferentiability of the sample paths of the Wiener process.

中文翻译：

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$${\mathbb {R}}^N$$ 上非自治分数随机 p-拉普拉斯方程的随机动力学

这篇文章关注的是在整个空间 $${\mathbb {R}}^N$$ 上由乘法白噪声驱动的一大类非自治的、非局部的、分数的、随机的 p-Laplacian 方程的随机动力学. 当瞬态非线性漂移项具有任意阶数 $$p,q\ge 2$$ 的多项式增长时，我们首先建立方程的适定性。然后我们证明该方程有一个独特的双空间回拉随机吸引子，它是可测量的，紧凑的 $$L^2({\mathbb {R}}^N)\cap L^p({\mathbb {R}} ^N)\cap L^q({\mathbb {R}}^N)$$ 并在 $ 的拓扑结构下吸引 $$L^2({\mathbb {R}}^N)$$ 的所有随机子集$L^2({\mathbb {R}}^N)\cap L^p({\mathbb {R}}^N)\cap L^q({\mathbb {R}}^N)$$ 。此外，我们在 $$L^2({\mathbb {R}}^N)\cap L^p({\mathbb {R}}^N)\cap L^q({ \mathbb {R}}^N)$$ 当噪声密度缩小到零时。应用统一尾估计思想和渐近先验估计方法证明$$L^2({\mathbb {R}}^N)\cap L^p({\ mathbb {R}}^N)\cap L^q({\mathbb {R}}^N)$$ 来克服 Sobolev 嵌入在 $${\mathbb {R}}^N$$ 上的非紧凑性以及维纳过程的样本路径几乎可以肯定的不可微性。