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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.

首席调查员。中尾。研究了非线性波动方程内外域的镇定问题和非线性热方程解的性质。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，我们得到了L^p估计。进一步。利用这些估计，我们讨论了半线性波动方程的整体存在性。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。解决内部问题。证明了一类极弱耗散项拟线性波动方程光滑解的整体存在性。在这个过程中，我们证明了变系数波动方程的一个独特的延拓性质。中尾不等式用于衰减估计。这是本文的一个独创性之处。关于非线性热方程，我们讨论了各种非线性扰动下的曲率型和m-拉普拉斯型拟线性方程。我们得到了解的尖锐估计，包括t→∞时的渐近性和t = o附近的平滑效应。研究者Kawashima主要处理了有关气体动力学的方程，并给出了许多有趣的结果。研究者Shibata考虑了粘弹性波动方程和流体动力学的外部问题，并利用谱分析证明了许多新的结果。加藤研究员讨论了一个非牛顿流动方程的整体解。