Asympotic behaviors of spatial critical points and zeros of solutions of parabolic equations

（1）水平表面的变化与时间上的差异解决方案，我们考虑了欧几里德空间中边界Lipschitz域的热平衡问题的初始解决方案，并在Euclidean空间中的分离超表面（1937年）和Segre （1938年）的帮助下，我们对同一表面在时间上是变化的解决方案进行分类。更进一步，我们可以用非线性扩散方程来解决这一点，就像porous medium equation一样，我们可以得到类似的分类理论。（2）具有单一维度极化介质equation解决方案的明显变化的界面的不对称行为，我们认为Cauchy和一个维度演化p-Laplacian equation中具有p>1的非零、边界和非负初始数据具有紧凑型支持的初始肮脏问题。这是在一个有限的时间，在一点上的一组空间关键点的解决方案，说x = ... More x（t） for time t。在这项研究中，我们展示了在有限的时间x（t）之后是C-D11-D1。Furthermore，我们可以与通用的porous medium equations进行交易，与签名更改一起获得C-D11-D1界面的规范性。此外，在一个维度进化p-Laplacian equation的初始-Dirichlet问题中，我们展示了存在着一个积极的常量β=β（ρ）这样的x（t）t β D1-β D1倾向于一些积极的常量为t → ∞。（3）热流的固定关键点和域的对称性，我们考虑了平面上边界和简单连接域的初始-肮脏问题。通过在复杂分析中使用Riemann映射理论的一种新方法，我们在角度2π/3下给出了域变化的特征，使热流的静态临界点的使用。（以前，只有球和中心对称域的特征才被发现。）Furthermore，我们认为热流动在sphere S-D1 N-D1和在超自然空间H-D1 N-D1，并提出了一些结果相关的结果，如在Magnanini和Sakaguchi的Euclidean空间中曾提供的那样（1997、1999）。非常精确。我们获得了地理球和中心对称域的特征，以使热流的静态关键点的使用。Less（低）

(1) Level surfaces invariant with time of solutions of diffusion equationsWe consider solutions of the initial-Neumann problem for the heat equation on bounded Lipschitz domains in Euclidean space, and with the help of the classification theorem of isoparametric hypersurfaces in Euclidean space of Levi-Civita (1937) and Segre (1938), we classify the solutions whose isothermal surfaces are invariant with time. Furthermore, we can deal with nonlinear diffusion equations such as the porous medium equation, and we get similar classification theorems.(2) Asymptotic behaviors of the interfaces with sign changes of solutions of the one-dimensional porous medium equationWe consider the Cauchy and the initial-Dirichlet problems for the one-dimensional evolution p-Laplacian equation with p>1 for nonzero, bounded, and nonnegative initial data having compact support. It was shown that after a finite time the set of spatial critical points of the solution u in {u > 0} consists of one point, say x = … More x(t) for time t. In this research, we show that after a finite time x(t) is CィイD11ィエD1 in t. Furthermore, we can deal with generalized porous medium equations with sign changes, and we get CィイD11ィエD1 regularity of the interfaces with sign changes. Also, in the initial-Dirichlet problem for the one-dimensional evolution p-Laplacian equation, we show that there exists a positive constant β=β(ρ) such that x(t)tィイD1-βィエD1 tends to some positive constant as t → ∞.(3) Stationary critical points of the heat flow and the symmetries of the domainsWe consider the initial-Dirichlet problem for the heat equation on bounded and simply connected domains in the plane. By a new method with the help of the Riemann Mapping theorem in complex analysis, we give a characterization of domains invariant under the rotation of angle 2π/3 by making use of the stationary critical points of the heat flow. (Previously, only the characterizations of balls and centrosymmetric domains were obtained.) Furthermore, we consider stationary critical points of the heat flow in sphere SィイD1NィエD1 and in hyperbolic space HィイD1NィエD1, and prove several results corresponding to those in Euclidean space which have been proved in Magnanini and Sakaguchi (1997, 1999). Precisely. We get the characterizations of geodesic balls and centrosymmetric domains by making use of the stationary critical points of the heat flow. Less