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，关键是分析奇点的出现机制，如爆炸层、集中层和过渡层。在这项研究中，我们的目标是开发新的方法来分析稳定模式的形成，高维空间的模式动力学，以及标量反应扩散方程的长时间行为。首先，研究了阴影系统中平稳解的稳定性，证明了在非线性项具有斜梯度结构的假设下，空间结构的凹凸性起着重要的作用。其次，研究了fujita型方程解的性质，明确了解的收敛率与初始数据衰减率的关系。第三，研究了权值不定的椭圆型问题的主特征值最小化问题。我们证明了最小化器必须是bang-bang型的，并得到了一维空间下的最小化器。我们还考虑了圆柱域的情况，并发现一定会发生某种对称破缺。

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.