Dynamics on Nonlinear Diffusive Systems and Analysis of Singularities

为了理解在非线性扩散系统中观察到的各种非线性现象，一个关键是分析奇异性出现的机制，如爆破，浓度和过渡层。本研究的目的是发展新的方法来分析标量反应扩散方程的稳定斑图的形成、斑图在高维空间中的动力学行为和长时间行为.首先，我们研究了影子系统中定态解的稳定性，并在非线性项具有斜梯度结构的假设下，证明了空间结构的凸性起着重要的作用.其次，我们研究了非线性项具有斜梯度结构的情况下，空间结构的凸性对反应扩散方程的稳定性的影响.研究了Fujita型方程解的性态，明确了解的收敛速度与初值衰减速度之间的关系。第三，研究了不定权椭圆型方程主特征值的极小化问题。证明了极小元必是bang-bang型的，并在一维空间中得到了极小元。我们还考虑了圆柱形区域的情形，发现一定会出现一种破缺现象。

For the understanding of various nonlinear phenomena observed in nonlinear diffusive systems, a key is to analyze the mechanism for the appearance of singularities such as blow-up, concentration, and transition layers. In this research, we aim to develop new methods for the analysis of stable pattern formation, pattern dynamics in higher dimensional space, and long-time behavior of scalar reaction-diffusion equations.First, we studied the stability of stationary solutions in shadow systems, and showed that the convexity of spatial structure plays an important role under the assumption that nonlinear term has skew-gradient structure.Second, we studied the behavior of solutions of Fujita-type equations, and made clear the relation between the convergence rate of solutions and decay rate of initial data.Third, we studied a minimization problem for the principal eigenvalue of elliptic problem with indefinite weight. We showed the minimizer must be of bang-bang type, and obtained a minimizer in the case of one-dimensional space. We also considered the case of cylindrical domain, and found that a sort of symmetry-breaking must occur.