On the Cauchy problem for a class of semilinear second order evolution equations with fractional Laplacian and damping

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).