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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.

本研究的主要目的是解决拟线性波动方程的外部问题。对于这个问题，我们成功地证明了局部耗散作用下光滑解的全局存在性。我们通过两种方式取得了成果；一种是基于局部能量衰减和线性方程解的 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，我们没有对边界的形状提出任何几何条件。关于非线性耗散方程的能量衰减的另一个问题，我们引入了一个新概念“半线性”，并成功地导出了非常微妙的能量衰减估计，并将它们应用于具有非线性源项的方程的全局解的存在性。作为相关问题，我们考虑了存在性和具有一些非线性局部耗散的有界域中非线性波动方程的周期解的稳定性。此外，我们还考虑了外域的基尔霍夫型非线性波动方程。在非线性耗散下，我们已经证明了全局解的各种结果。对于具有诺伊曼型边界耗散的外域波动方程，我们导出了新的能量衰减估计。川岛研究员导出了许多关于玻尔兹曼方程和双曲守恒方程的有趣结果。 Shibata 研究员通过谱分析的方法得出了许多关于压缩纳维-斯托克斯方程的外部问题的有趣结果。研究人员小川证明了对非线性波动方程、非线性薛定格方程和某些谐波演化方程的解的行为和规律性的精确估计。