On the Cauchy problem for a class of semilinear second order evolution equations with fractional Laplacian and damping
On the Cauchy problem for a class of semilinear second order evolution equations with fractional Laplacian and damping
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DOI：
10.1007/s00030-021-00723-6
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发表时间：
2021-09
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影响因子：
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通讯作者：
K. Fujiwara;M. Ikeda;Yuta Wakasugi
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文献类型：
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作者：
K. Fujiwara;M. Ikeda;Yuta Wakasugi
本文证明了具有分数阶Laplacian算子和阻尼项的二阶发展方程的解在加权Sobolev空间中的时间衰减估计。我们的估计概括了在以前的研究中获得的估计（Karch in Stud Math 143：175-197，2000; Ikeda et al. in Nonlinear Differ.当量ag 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项的方程的柯西问题的小数据整体适定性。特别地，本文中获得的估计使我们能够处理比论文中的结果更一般的非线性和空间维度的条件（Chen等人，Electron. J. Differ.当量ag 2015：1-14，2015; Ikeda等人，在Nonlinear Differ.当量ag Appl.24：10，2017）。
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).