Global solution structure and the stability of nonlocal nonlinear second order boundary value problems with definite integrals

首先，Lou-Ni-Yotsutani [DCDS 2004]研究了数学生物学中出现的交叉扩散方程的极限方程。我们证明了它有不同种类的奇异解，并揭示了所有解的结构。该问题是一个非局部非线性椭圆型边界问题，没有已知的求解方法。我们发现了一种将现代偏微分方程方法与经典分析和代数相结合的新方法。我们的方法适用于许多有趣的问题。Oseen的螺旋流问题就是其中之一，我们在Ikeda-Kondo-Okamoto-Yotsutani [CPAA 2003]中得到了完整的分岔图。Matsumoto-Murai-Yotsutani [Pisa， 2005]给出了确定给定长度下能量最小曲线问题的完整答案。[2] [CPAA 2005, j .数学与物理]。[2005]揭示了具有周期边界条件的一维金兹堡-朗道方程的完全全局分岔分支。

First, Lou-Ni-Yotsutani [DCDS 2004] investigated a limiting equation to a cross-diffusion equation that appears in mathematical biology. We showed that it has different kinds of singular solutions and revealed the structure of all solutions.This problem is a nonlocal nonlinear elliptic boundary problem, for which no method was known to solve it. We discovered a new method, which are the combination of the modern method of PDE and classical analysis and algebra.There are a lot of interesting problems for which our method is applicable.A problem of the Oseen's spiral flow is one of them, for which we obtained the complete bifurcation diagram in Ikeda-Kondo-Okamoto-Yotsutani [CPAA 2003].Matsumoto-Murai-Yotsutani [Pisa, 2005] gave the complete answer for a problem to determine curves with the least energy under the given length.Second, Kosugi-Morita-Yotsutani [CPAA 2005, J.Math.Phy. 2005] have revealed the complete Global bifurcation branches one dimensional Ginzburg-Landau equations with periodic boundary conditions.