In the present paper, we prove time decay estimates of solutions in weighted Sobolev spaces to the second order evolution equation with fractional Laplacian and damping for data in Besov spaces. Our estimates generalize the estimates obtained in the previous studies (Karch in Stud Math 143:175–197, 2000; Ikeda et al. in Nonlinear Differ. Equ. Appl. 24:10, 2017). The second aim of this article is to apply these estimates to prove small data global well-posedness for the Cauchy problem of the equation with power nonlinearities. Especially, the estimates obtained in this paper enable us to treat more general conditions on the nonlinearities and the spatial dimension than the results in the papers (Chen et al. in Electron. J. Differ. Equ. 2015:1–14, 2015; Ikeda et al. in Nonlinear Differ. Equ. Appl. 24:10, 2017).

在本文中，我们证明了加权 Sobolev 空间中解的时间衰减估计到带有分数拉普拉斯算子和 Besov 空间中数据阻尼的二阶演化方程。我们的估计概括了先前研究中获得的估计（Karch in Stud Math 143:175–197, 2000；Ikeda et al. in Nonlinear Differ. Equ. Appl. 24:10, 2017）。本文的第二个目的是应用这些估计来证明具有幂非线性的方程的柯西问题的小数据全局适定性。特别是，与论文中的结果相比，本文获得的估计使我们能够处理更多关于非线性和空间维度的一般条件（Chen 等人在 Electron. J. Differ. Equ. 2015:1-14, 2015; Ikeda 等人在 Nonlinear Differ. Equ. Appl. 24:10, 2017 中）。