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Exterior problem for nonlinear wave equations

非线性波动方程的外问题

基本信息

批准号：
13440049
负责人：
NAKAO Mitsuhiro
金额：
$ 7.3万
依托单位：
KYUSHU UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
2001
资助国家：
日本
起止时间：
2001 至 2004
项目状态：
已结题

项目摘要

The main purpose of this research is concerned with the exterior problem for the quasi-linear wave equations. For this problem we have been successful in proving the global existence of smooth solutions under the effect of localized dissipation. We have achieved the results through two ways; one is based on the local energy decay and L^p estimates of solutions for linear equation, and the other one is the method to utilize total energy decay for the llinearized equation. Both ways are intended to make the effects of dissipation as weaker as possible, but, we have made no geometrical conditions on the shape of the boundary.Concerning another problem on the energy decay for the equation with nonlinear dissipations we introduced anew concept ‘Half linear' and has been successful in deriving very delicate decay estimates of energy and applied them to the existence of global solutions for the equations with a nonlinear source term.As related problems we have considered the existence and stability of periodic solutions for the nonlinear wave equations in bounded domains with some nonlinear localized dissipations. Further, we have considered the Kirchhoff type nonlinear wave equations in exterior domains. Under a nonlinear dissipations we have proved various results on global solutions. For the wave equation in exterior domains with a Neumann type boundary dissipation we have derived a new energy decay estimate.Investigator Kawashima has derived many interesting results concerning Boltzman equations and hyperbolic conservation equations. Investigator Shibata has derived by the method of spectral analysis, many interesting results concerning the exterior problem for the compressive Navier-Stokes equations. Investigator Ogawa has proved precise estimates of solutions concerning behaviors and regularities of solutions for the nonlinear wave equations, nonlinear Shroadinger equations and some harmonic evolution equation.

本文主要研究拟线性波动方程的外问题。对于这个问题，我们已经成功地证明了在局部耗散的影响下光滑解的整体存在性。我们通过两种方法得到了这些结果：一种是基于线性方程解的局部能量衰减和L^p估计，另一种是利用线性化方程的总能量衰减的方法.这两种方法都是为了使耗散的影响尽可能弱，但是，关于另一个非线性耗散方程的能量衰减问题，我们引进了“半线性”的新概念，并成功地导出了能量衰减的精确估计，并应用于非线性耗散方程整体解的存在性问题。作为相关问题，我们考虑了具有非线性局部耗散的有界区域上的非线性波动方程周期解的存在性和稳定性。在此基础上，我们考虑了外部区域上的Kirchhoff型非线性波动方程。在非线性耗散下，我们证明了关于整体解的各种结果。对于具有Neumann型边界耗散的外区域波动方程，我们得到了一个新的能量衰减估计，研究者Kawashima在Boltzman方程和双曲守恒方程中得到了许多有趣的结果。研究者柴田用谱分析方法导出了压缩Navier-Stokes方程外问题的许多有趣结果。Ogawa研究员证明了非线性波动方程、非线性Shroadinger方程和某些调和发展方程的解的性态和收敛性的精确估计。