In this paper, we study the dynamical behavior of the solution for the stochastic reaction–diffusion equation with the nonlinearity satisfying the polynomial growth of arbitrary order \(p\geq2\) and any space dimension N. Based on the inductive principle, the higher-order integrability of the difference of the solutions near the initial data is established, and then the (norm-to-norm) continuity of solutions with respect to the initial data in \(H_{0}^{1}(U)\) is first obtained. As an application, we show the existence of \((L^{2}(U),L^{p}(U))\) and \((L^{2}(U),H_{0}^{1}(U))\)-pullback random attractors, respectively.

中文翻译：

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带乘性噪声的随机反应扩散方程的长时间行为

在本文中，我们研究了具有满足任意阶\（p \ geq2 \）和任意空间维N的多项式增长的非线性的随机反应扩散方程解的动力学行为。基于归纳原理，建立初始数据附近的解的差分的高阶可积性，然后针对\（H_ {0}中的初始数据，解的（范对范）连续性首先获得^ {1}（U）\）。作为应用程序，我们显示\（（L ^ {2}（U），L ^ {p}（U））\）和\（（L ^ {2}（U），H_ {0} ^ {1}（U））\）-分别拉回随机吸引子。