On study of partial differential equations describing natural phenomena

论描述自然现象的偏微分方程的研究

基本信息

批准号：
15204009
负责人：
HAYASHI Nakao
金额：
$ 13.64万
依托单位：
Osaka University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (A)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2005
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15204009/
关键词：
Nonlinear Schredinger equation Modified KdV equation Modified wave operator Asymptotics of solutions Nonlinear damped wave equation Airy type equations Nonlinear dissipative equations Schredinger type equation 消散形波動方程式臨界冪非線形項分数冪熱方程式

项目摘要

1,P.I.Naumkin and I studied the Burgers equation with pumping and showed a existence in time of solutions and asymptotic behavior of solutions by using a suitable transformation and the structure of nonlinear term.2,E.I.Kaikina and I studied the KdV equations in a half line with 0 boundary value at the origin. Airy function is oscillating rapidly in the left hand side and decaying exponentially in the right hand side. We showed asymptotics of solutions to the KdV equation by making use of this property.3,E.I.Kaikina, P.I.Naumkin and I studied nonlinear complex dissipative equations with sub-critical nonlinearities and showed a solution is stable in the neighborhood of a self similar, solution.4,P.I.Naumkin, Shimomura, Tonegawa and I did a joint work on nonlinear Schredinger equations with cubic nonlinearities. It was known that there exists a modified wave operator under some geometric assumptions on the final data. We succeeded to remove a strong geometric assumption by finding a new way to get a second approximate solution of the problem.5,E.I.Kaikina, P.I.Naumkin and I studied nonlinear damped wave equations with super-critical or critical nonlinearities. In the previous works, it was known that a global existence theorem holds in space dimension is less than 5. We improved this result for any space dimension by using the weighted Sobolev spaces and estimates of solutions linear problem. Furthermore, in the critical case we showed asymptotics of solutions. The result implies the decay order in time of solutions is higher than that of solutions to linear problem. We obtained the results by using the method we found in the study of nonlinear dissipative equations

1，P.I.Naumkin和我使用泵送研究了汉堡方程，并通过使用合适的转换和非线性项的结构在溶液和溶液的渐近行为中表现出存在。通风的功能在左侧迅速振荡，在右侧呈指数衰减。我们通过利用该属性来展示KDV方程解的渐近学。非线性。众所周知，在最终数据上的一些几何假设下，有一个修改后的波浪运算符。我们通过找到一种新的方法来解决问题的新方法，成功地消除了强大的几何假设。5，E.I.Kaikina，P.I.Naumkin和我研究了具有超临界或关键非线性的非线性阻尼波方程。在以前的作品中，众所周知，通过使用加权Sobolev空间和解决方案线性问题的估计值，我们在空间维度中的全局存在定理少于5。我们改善了此结果。此外，在关键情况下，我们显示了解决方案的渐近学。结果意味着解决方案时期的衰减顺序高于线性问题的解决方案。我们使用在非线性耗散方程研究中发现的方法获得了结果