Linear Approximation and Asymptotic Expansion of聽Solutions in Many Small Parameters for a Nonlinear Kirchhoff Wave Equation with Mixed Nonhomogeneous Conditions

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• 作者：Le Thi Phuong Ngoc (1)
  Nguyen Thanh Long (2)
  
• 关键词：Faedo鈥揋alerkin method ; Linear recurrent sequence ; Asymptotic expansion of order N+1 ; 35L20 ; 35L70 ; 35Q72
• 刊名：Acta Applicandae Mathematicae
• 出版年：2010
• 出版时间：November 2010
• 年：2010
• 卷：112
• 期：2
• 页码：137-169
• 全文大小：705KB
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• 作者单位：Le Thi Phuong Ngoc (1)
  Nguyen Thanh Long (2)
  
  1. Nhatrang Educational College, 01 Nguyen Chanh Str., Nhatrang City, Vietnam
  2. Department of Mathematics and Computer Science, University of Natural Science, Vietnam National University Ho Chi Minh City, 227 Nguyen Van Cu Str., Dist. 5, Ho Chi Minh City, Vietnam
  
• ISSN：1572-9036

文摘

In this paper, we consider the following nonlinear Kirchhoff wave equation 1 $$\left\{\begin{array}{l}u_{tt}-\frac{\partial }{\partial x}(\mu (u,\Vert u_{x}\Vert ^{2})u_{x})=f(x,t,u,u_{x},u_{t}),\quad 0<x<1,\ 0<t<T,\\[3pt]u_{x}(0,t)=g(t),\qquad u(1,t)=0,\\[3pt]u(x,0)=\widetilde{u}_{0}(x),\qquad u_{t}(x,0)=\widetilde{u}_{1}(x),\end{array}\right.$$ where $\widetilde{u}_{0}$ , $\widetilde{u}_{1}$ , 渭, f, g are given functions and $\Vert u_{x}\Vert ^{2}=\int_{0}^{1}u_{x}^{2}(x,t)dx.$ To the problem聽(1), we associate a linear recursive scheme for which the existence of a local and unique weak solution is proved by applying the Faedo鈥揋alerkin method and the weak compact method. In particular, motivated by the asymptotic expansion of a weak solution in only one, two or three small parameters in the researches before now, an asymptotic expansion of a weak solution in many small parameters appeared on both sides of (1)1 is studied.