The Cauchy problem for the nonlinear damped wave equation with slowly decaying data

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• 作者：Masahiro Ikeda ; Takahisa Inui ; Yuta Wakasugi
• 关键词：Nonlinear damped wave equation ; Slowly decaying data ; Global well ; posedness ; Asymptotic behavior ; Blow ; up ; Estimates of lifespan
• 刊名：Nonlinear Differential Equations and Applications NoDEA
• 出版年：2017
• 出版时间：April 2017
• 年：2017
• 卷：24
• 期：2
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Analysis;
• 出版者：Springer International Publishing
• ISSN：1420-9004
• 卷排序：24

文摘

We study the Cauchy problem for the nonlinear damped wave equation and establish the large data local well-posedness and small data global well-posedness with slowly decaying initial data. We also prove that the asymptotic profile of the global solution is given by a solution of the corresponding parabolic problem, which shows that the solution of the damped wave equation has the diffusion phenomena. Moreover, we show blow-up of solution and give the estimate of the lifespan for a subcritical nonlinearity. In particular, we determine the critical exponent for any space dimension.