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，过渡层出现。Matano研究了具有空间非齐次系数的扩散方程的奇异极限。Yanagida利用这种奇异极限技巧证明了稳定周期解的存在性。Funaki研究了出现在具有随机偏差的方程中的界面的行为。(2)非线性热方程的爆破解某些非线性热方程的爆破解在某种意义下可以继续存在于爆破时间之外。Matano研究了这种爆破解的动力学。柳田获得了估计爆炸时间的新公式。(3)自由边值问题的奇异性韦斯应用他的新方法得到了一个两相障碍问题奇异集的Hausdorff维数。(4)复动力系统的奇异性与分支Shishikura得到了一种研究分支问题和Julia集刚性的新的系统方法。

(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.