The Cauchy problem of the semilinear second order evolution equation with fractional Laplacian and damping

2020/03/20 by Kazumasa Fujiwara, Masahiro Ikeda, Fujiwara, Kazumasa +3
Engineering · Mathematics · #35A01 #35L71 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2003.09239

openalex publication_date 2020/03/20 · openalex created_date 2020/03/27 · openalex updated_date 2026/07/28

Abstract

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. 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 previous studies.

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