Interface motion and blow-up phenomena in nonlinear partial differential equations

本研究项目的目的是通过开发基于无限维动力系统理论和随机方法的渐近方法，并在必要时进行数值模拟，对与界面运动和爆破现象相关的各种非线性问题进行理论研究。我们得到了以下结果。(1)本文给出了Allen-Cahn型非线性扩散方程奇异极限的一个最优估计，这是以前所不知道的。对于FitzHugh-Nagumo系统和Lotka-Volterra竞争系统，我们也得到了类似的结果. (2)我们阐明了非线性扩散方程爆破解的整体动力学性质。这是通过将现有的全局吸引子理论扩展到发生爆破的情况而实现的。(3)在具有幂非线性项的非线性热方程中，解的爆破分为第一类和第二类，第二类比第一类更难分析。 ...更多信息 前者通过使用基于辫群理论的拓扑方法，我们成功地确定了所有可能的II型爆破率。(4)利用变分方法研究了二维格点周期空间上Allen-Cahn方程的平稳性问题。我们证明了多层定态解存在的一个充分必要条件是单层解的集合在某处有一个缺口，换句话说，这个集合不应该是一个连续体（叶理）。(5)我们考虑了边界为锯齿形的二维无限条中的周期行波。当边界起伏越来越细时，我们确定了这种行波的均匀化极限。(6)从微观模型出发，通过流体力学极限推导出各种偏微分方程。这些方程包括Stefan问题和一些随机微分方程。(7)建立了各种椭圆型方程（如燃烧模型）中自由边界的正则性。(8)建立了曲率相关运动方程中V形行波的稳定性。少

The aim of this research project is to make a theoretical study of various nonlinear problems related to interfacial motions and blow-up phenomena, by developing asymptotic methods based on the theory of infinite dimensional dynamical systems and stochastic methods, and also performing numerical simulations if necessary. We have obtained the following results.(1) We have given an optimal estimate concerning the singular limit of Allen-Cahn type nonlinear diffusion equations, that has not been known previously. We also obtained similar results for FitzHugh-Nagumo systems and Lotka-Volterra competition systems.(2) We have clarified the global dynamics of blow-up solutions in nonlinear diffusion equations. This was made possible by extending the existing theory on global attractors to the case where blow-up occurs.(3) In nonlinear heat equations with power nonlinearity, the blow-up of solutions is classified into type I and type II, the latter being much more difficult to analyze than the … More former. By using a topological method based on the braid group theory, we have succeeded in determining all possible type II blow-up rates.(4) We studied the stationary problem for the Allen-Cahn equation on the 2-dimensional space having lattice periodicity by using variational methods. We have shown that a necessary and sufficient condition for the existence of multi-layered stationary solution is that the set of single layered solutions has a gap somewhere ; in other words, this set should not be a continuum (foliation).(5) We considered periodic traveling waves in a two-dimensional infinite strip whose boundaries are saw-tooth shaped. We determined the homogenization limit of such traveling waves as one lets the boundary undulation finer and finer.(6) Various partial differential equations are derived from microscopic models via hydrodynamic limit. Those equations include the Stefan problem and some stochastic differential equations.(7) Regularity of free boundaries arising in various elliptic equations such as the combustion model has bee established.(8) Stability of V-shaped traveling wave in the equation of curvature-dependent motion has been established. Less