THE MATHEMATICAL ANALYSIS TO NON-LINEAR PHENOMENA THROUGH NON-LINEAR PARTIAL DIFFERENTIAL EQUATIONS

非线性偏微分方程对非线性现象的数学分析

基本信息

批准号：
09640276
负责人：
ITO Masayuki
金额：
$ 1.98万
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：
1997
资助国家：
日本
起止时间：
1997 至 1998
项目状态：
已结题

项目摘要

1) The p-Laplace operators is well known as the non-linear modification of the usual Laplacian. These operators or their perturbed operators arise in the model equations for the elastic membrane, nonlinear diffusion phenomena and so on. Moreover, the limit state of solutions at p infinity is of great interest from the mathematical or technological view points. The eigenvalue problem of p-Laplacian has been studied by many authors. Since this problem can be dealt with as a variational problem, many results has been known. However, its limit problem at p infinity had been known because it cannot be described in a variatinal problem. We formulate such problem using the notion of the viscosity solution and obtain some results for the limit eigenvalues and the associate eigenfunctions.2) In the ecological model, a reaction-diffusion equation has the nonlinear diffusion with the p-Laplace operator when the diffusion depends on the population pressure nonlinearly. Yamada has studied such equation and obtain the unique and global existence of a solution and sonic results on the set of stationary solutions. He also study the 3 species cooperative competition-diffusion systems with linear diffusion, and obtain the necessary and sufficient condition to the existence of the coexistence solutions.3) Murakami has studied the asymptotic behavior of the solution for several higher dimensional delay differential equations and obtain the existence of periodic solutions which are bifurcated from the equilibrium, in particular, the explicit expressions of the bifurcated periodic solutions.4) Kohda has obtained some conditions on initial value for parabolic problem which guarantee the blow-up of a solution. Moreover, he had shown the behavior of blow-up solution near blow-up time, that is blow-up patterns.

1）P-Laplace操作员被称为通常的Laplacian的非线性修饰。这些操作员或其受干扰的操作员出现在弹性膜，非线性扩散现象等的模型方程中。此外，从数学或技术观点点，P Infinity在P Infinity处的极限状态引起了极大的兴趣。许多作者已经研究了p-Laplacian的特征值问题。由于可以将这个问题作为变异问题处理，因此已知许多结果。但是，它在p Infinity处的极限问题是已知的，因为它无法在变化问题中描述。我们使用粘度溶液的概念提出了此类问题，并获得了极限特征值和关联特征功能的一些结果。2）在生态模型中，当反应扩散方程与P-Laplace操作员非线性扩散时，当扩散取决于种群非线性非线性时。 Yamada研究了这种方程，并在固定溶液集上获得了解决方案和声音结果的独特而全球的存在。他还研究了具有线性扩散的3个物种合作竞争扩散系统，并获得了共存解决方案存在的必要条件。 Kohda已经获得了抛物线问题的初始值的某些条件，这些条件可以保证溶液的爆炸。此外，他在爆炸时间附近显示了爆破解决方案的行为，即爆炸模式。