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

1) p-拉普拉斯算子是对一般拉普拉斯算子的非线性修正。这些算子或它们的扰动算子出现在弹性膜、非线性扩散现象等模型方程中。此外，从数学或技术的角度来看，p∞处解的极限状态是非常有趣的。许多作者研究了p-拉普拉斯算子的特征值问题。由于这个问题可以作为变分问题来处理，因此已经知道了许多结果。然而，它在p∞处的极限问题是已知的，因为它不能用变分问题来描述。利用黏度解的概念，给出了这类问题的极限特征值和相关特征函数的一些结果。2)在生态模型中，当扩散非线性依赖于种群压力时，反应扩散方程具有p-Laplace算子的非线性扩散。Yamada研究了这类方程，得到了一个解的唯一性和全局存在性，以及平稳解集上的声学结果。研究了具有线性扩散的3种合作竞争扩散系统，得到了共存解存在的充分必要条件。3) Murakami研究了若干高维时滞微分方程解的渐近性，得到了从平衡点分岔的周期解的存在性，特别是分岔周期解的显式表达式。4) Kohda给出了抛物型问题的初始值保证解爆破的若干条件。此外，他还展示了爆破溶液在爆破时间附近的行为，即爆破模式。

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.