THE STUDY OF NON-LINEAR PHENOMENA BY THE ASYMPTOTIC ANALYSIS

非线性现象的渐近分析研究

基本信息

批准号：
11640124
负责人：
ITO Masayuki
金额：
$ 2.11万
依托单位：
The University of Tokushima
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2001
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640124/
关键词：
quasilinear degenerate elliptic equation eigenvalue problem bifurcation comparison principle difference equation invariant curve degenerate elliptic equation p-Laplacian ∞-Laplacian limit eigenvalue problem delay differential equat

项目摘要

The purpose of this project is to describe nonlinear phenomena mathematically by using asymptotic analysis. And we have the following results.1) Narukawa and Fukagai proposed a mathematical model related to the nonlinear elasticity. This is described by a degenerate quasilinear elliptic equation whose principal part has different orders at 0 and at infinity. They have showed a global bifurcation diagram of positive solutions for a nonlinear eigenvalue problem of such quasilinear equations and, in particular, the coexistence of multiple positive solutions. These results obtained by regularity estimate of weak solutions and modifying the argument given by Ambrosetti, Brezis and Cerami in the semilinear case.2) Ohnuma has investigated a class of singular degenerate parabolic equations including the p-Laplace diffusion equation and the equation of the mean curvature flow, and proved the comparison principle for these equations. He also discovered a strong maximum principle of quasilinear degenerate elliptic equations.3) Murakami showed a necessary and sufficient condition of the asymptotic stability of a fixed point for a higher order linear difference equation. He has also investigated some nonlinear difference equations, derived the formula to compute the stability conditions of the invariant curve caused by the Neimark-Sacker bifurcation and, moreover, given the explicit expression of the invariant curve.4) Kohda has obtained blow-up criteria for a solution of an initial value problem of a semilinear parabolic equation. This is described by using a super-solusion and a sub-solution of the stationary problem. Moreover, he had given some condition which guarantees the blow-up of the solution.

该项目的目的是通过使用渐近分析来数学上描述非线性现象。 1）Narukawa和Fukagai提出了与非线性弹性有关的数学模型。这是通过退化的quasilarear椭圆方程来描述的，其主要部分在0和无穷大处的订单不同。他们显示了用于这种准线性方程的非线性特征值问题的阳性溶液的全球分叉图，尤其是多种阳性溶液的共存。这些结果是通过对弱解决方案的规律性估计获得的，并修改了Ambrosetti，Brezis和cerami在半线性案例中给出的论点。2）OHNUMA研究了一类单数偏离抛物线方程，包括P-Laplace扩散方程，以及平均曲率流的方程，并证明了对比较的平均值。他还发现了准线性退化椭圆方程的强大最大原理。3）穆拉卡米显示出固定点的渐近稳定性的必要和足够条件，用于高阶线性差方程。他还研究了一些非线性差异方程式，得出了计算Neimark-Sack-Sacker分叉引起的不变曲线稳定性条件的公式，而且鉴于不变曲线的明确表达。4）Kohda已获得了针对半初始值的爆破标准，即在半初始价值问题上获得了一个半呈半度性的Parineareareareareareareareareareareation Carabolation。这是通过使用超溶解和固定问题的子解决方案来描述的。此外，他已经提供了一些条件，可以保证解决方案的爆炸。