Stabilization problem for nonlinear wave eq

非线性波方程的镇定问题

基本信息

批准号：
10440053
负责人：
NAKAO Mitsuhiro
金额：
$ 6.72万
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：
1998
资助国家：
日本
起止时间：
1998 至 2000
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-10440053/
关键词：
Nonlinear wave Global existence Energy decay Smoothing effect Exterior domain Nomlinear Wave L^p-estimate Nonlinear Wave Equation Energy decay Dissipation Global Solution Hyperbolic Equation nonlinear wave eguation decay estima

项目摘要

The head investigator. Nakao. has considered the stabilization problem for the nonlinear wave equations in interior and exterior domains and also the behaviours of solutions for nonlinear heat equations.For exterior problems. we first proved the local energy decay for linear wave equations. and on the basis of this we have derived L^p estimates. Further. by use of these estimates we have discussed on the global existence of semilinear wave equations. We note that in our argument no geometrical conditions on the shape of obstacles.For interior problems. we have proved global existence of smooth solutions for the quasilinear wave equations with a very weak dissipative term. In this procedure we have showed a unique continuation property for the wave equation with a variable coefficient. Nakao's inequality was used for the decay estimate. which is an originality of this paper.Concernibg nonlinear heat equations we have treated meancurvature type and m-laplacian type quasilinear equations under various nonlinear perturbations. We have derived sharp estimates of solutions including asymptotics as t → ∞ and smoothing effects near t = O.Investigator Kawashima has mainly treated the equations concerning gas dynamics and shown many interesting results. Investigator Shibata has considered the visco-elastic wave equations and also exterior problems concerning fluid dynamicsand prove many new results by use of spectral analysis. Investigator Kato has discussed on the global solutions of a non-Newtonian flow equation.

首席研究员 Nakao 考虑了内部和外部域中的非线性波动方程的稳定性问题以及非线性热方程的解的行为。对于外部问题，我们首先证明了线性波动方程的局部能量衰减。在此基础上，我们进一步通过使用这些估计来讨论半线性波动方程的全局存在性，我们注意到在我们的论证中没有关于障碍物形状的几何条件。我们已经证明全球具有非常弱耗散项的拟线性波动方程的光滑解的存在性在这个过程中，我们展示了具有可变系数的波动方程的独特连续性，该性质被用于衰减估计。关于非线性热方程，我们在各种非线性扰动下处理了平均曲率型和 m-拉普拉斯型拟线性方程，我们导出了包括渐近方程在内的解的锐估计： t → ∞ 和 t = O 附近的平滑效应。川岛研究员主要研究了有关气体动力学的方程，并给出了许多有趣的结果。柴田研究员考虑了粘弹性波动方程以及有关流体动力学的外部问题，并证明了许多新结果。研究者加藤讨论了非牛顿流动方程的全局解。