Study of the Mathematical structure of singularities

(1)扩散方程的奇异极限在某些扩散方程中，当扩散系数趋于零时，会出现过渡层。Matano研究了具有空间非齐次系数的扩散方程的奇异极限。柳田用这种奇异极限技巧证明了稳定周期解的存在性。Funaki研究了具有随机偏差的方程中出现的界面的行为。(2)在非线性热方程的爆破中，非线性热方程的一些爆破解在一定意义上可以在爆破时间之后继续存在。马塔诺研究了这种爆破解的动力学。柳田得到了估计爆破时间的新公式。(3)自由边界问题中的奇点Weiss应用他的新方法得到了两相障碍问题奇点集的Hausdorff维。(4)复杂动力系统中的奇点与分叉。Shishikura获得了一种新的系统方法来研究Julia集的分叉问题和刚性。

(1) Singular limit of diffusion equations In certain kinds of diffusion equations, transition layers appear as the diffusion coefficients tends to zero. Matano has studied the singular limit for diffusion equations with spatially inhomogeneous coefficients. Yanagida has used this singular limit technique to proved the existence of stable periodic solutions. Funaki has investigated the behavior of interfaces that appear in equations with random deviation.(2) Blow-up in nonlinear heat equations Some blow-up solutions of nonlinear heat equations can be continued beyond the blow-up time in a certain sense. Matano has studied the dynamics of such blow-up solutions. Yanagida has obtained new formula for estimating the blow-up time.(3) Singularities in free boundary problems Weiss has applies his new method obtain the Hausdorff dimension of the singularity set of a 2-phase obstacle problem.(4) Singularities and bifurcation in comlex dynamical system Shishikura has obtained a new systematic method to study bifurcation problems and rigidity of Julia sets.