Neumann problem for the nonlinear Klein-Gordon equation

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• 作者：Ivan Naumkin ^{ivan.naumkin@upmc.fr}
• 关键词：35M13 ; 35B40 ; 35A01 ; 35A02
• 刊名：Nonlinear Analysis: Theory, Methods & Applications
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：149
• 期：Complete
• 页码：81-110
• 全文大小：827 K
• 卷排序：149

文摘

The Neumann initial–boundary value problem for the nonlinear Klein–Gordon equation equation(0.1){vtt+v−vxx=μv3, (t,x)∈R+×R+, v(0,x)=v0(x), vt(0,x)=v1(x), x∈R+,(∂xv)(t,0)=h(t), t∈R+, for real μμ, v0(x)v0(x),v1(x)v1(x) and h(t)h(t), is considered. We prove the global well-posedness for the initial–boundary value problem (0.1) and we present a sharp time decay estimate of the solution in the uniform norm. Also we study the asymptotic behavior of the solution to (0.1). We show that the cubic nonlinearity in the Neumann initial–boundary value problem (0.1) is scattering-critical.