一类退化抛物方程解的几何性质

The Geometric Characteristic of a Degenerate Parabolic Equation

本文讨论非线性退化抛物方程u_t=△φ(u)的Cauchy问题弱解u(x,t)的正则性与几何性质.本文证明:若正数β足够大,则曲面ψ=ψ(x,t)=[φ(u)]~β是随时间t的连续变化而漂浮于空间R^(n+1)中的n维完备黎曼流形,它与实欧氏空R^n相切于低维流形(?)H_n(t),而H_u(t)={x∈R^n:u(x,t)>0);函数ψ(x,t)在经典的意义下满足另一退化抛物方程.

This work studies the regularity and the geometric significance of solution of the Cauchy problem for a degenerate parabolic equation u_t = △φ(u).Our objective is to show that:(1) the function ψ(x,t) = {φ(u)]~β is a classical solution to another degenerate parabolic equation if β is large sufficiently;(2) the surface ψ = ψ(x,t) is a complete Riemannian manifold,which is tangent to R^n at the boundary of the positivity set of u(x,t).