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

• Published in: Nonlinear differential equations and applications Vol. 28; no. 6
• Main Authors: Fujiwara, Kazumasa, Ikeda, Masahiro, Wakasugi, Yuta
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01-12-2021; Springer Nature B.V
• Subjects: Analysis; Cauchy problems; Damping; Estimates; Evolution; Function space; Mathematics; Mathematics and Statistics; Nonlinearity; Sobolev space; Second order evolution equation; Seconary 35A01; Asymptotic behavior; Global existence; Dissipative term; Power nonlinearity; Fractional Laplacian; Primary 35L71
• ISSN: 1021-9722; 1420-9004
• DOI: 10.1007/s00030-021-00723-6

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).