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

被引：0
作者：
Kazumasa Fujiwara
Masahiro Ikeda
Yuta Wakasugi
机构：
[1] Nagoya University,Graduate School of Mathematics
[2] Keio University,Department of Mathematics, Faculty of Science and Technology, Japan
[3] Center for Advanced Intelligence Project,Laboratory of Mathematics, Graduate School of Advanced Science and Engineering
[4] Hiroshima University,undefined
来源：
Nonlinear Differential Equations and Applications NoDEA | 2021年 / 28卷
关键词：
Fractional Laplacian; Dissipative term; Second order evolution equation; Power nonlinearity; Asymptotic behavior; Global existence; Primary 35L71; Seconary 35A01;
D O I：
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摘要：
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).
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共 66 条
[1]  
Charão RC(2013)Sharp decay rates for wave equations with a fractional damping via new method in the Fourier space J. Math. Anal. Appl. 408 247-255
[2]  
da Luz CR(2015)Estimates for damped fractional wave equations and applications Electron. J. Differ. Equ. 2015 1-14
[3]  
Ikehata R(2003)An optimal estimate for the difference of solutions of two abstract evolution equations J. Differ. Equ. 193 385-395
[4]  
Chen J(1991)Dispersion of small amplitude solutions of the generalized Korteweg–de Vries equation J. Funct. Anal. 100 87-109
[5]  
Fan D(2015)Asymptotic behavior for abstract evolution differential equations of second order J. Differ. Equ. 259 5017-5039
[6]  
Zhang C(2019)An application of J. Math. Anal. Appl. 476 426-463
[7]  
Chill R(2019) estimates for oscillating integrals to parabolic like semi-linear structurally damped -evolution models Discrete Contin. Dyn. Syst. 39 5431-5463
[8]  
Haraux A(2010) estimates for oscillating integrals and their applications to semi-linear models with -evolution like structural damping Chin. Ann. Math. Ser. B 31 237-246
[9]  
Christ FM(1966)High-order energy decay for structural damped systems in the electromagnetical field J. Fac. Sci. Univ. Tokyo Sect. I 109-124
[10]  
Weinstein MI(1951)On the blowing up of solutions of the Cauchy problem for Q. J. Mech. Appl. Math. 4 129-156
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