Regularity and geometric character of solution of a degenerate parabolic equation

This work studies the regularity and the geometric significance of solution of the Cauchy problem for a degenerate parabolic equation $u_{t}=\Delta{}u^{m}$. Our main objective is to improve the H$\ddot{o}$lder estimate obtained by pioneers and then, to show the geometric characteristic of free boundary of degenerate parabolic equation. To be exact, the present work will show that: (1) the weak solution $u(x,t)\in{}C^{\alpha,\frac{\alpha}{2}}(\mathbb{R}^{n}\times\mathbb{R}^{+})$, where $\alpha\in(0,1)$ when $m\geq2$ and $\alpha=1$ when $m\in(1,2)$; (2) the surface $\phi=(u(x,t))^{\beta}$ is a complete Riemannian manifold, which is tangent to $\mathbb{R}^{n}$ at the boundary of the positivity set of $u(x,t)$. (3) the function $(u(x,t))^{\beta}$ is a classical solution to another degenerate parabolic equation if $\beta$ is large sufficiently; Moreover, some explicit expressions about the speed of propagation of $u(x,t)$ and the continuous dependence on the nonlinearity of the equation are obtained. Recalling the older H$\ddot{o}$lder estimate ($u(x,t)\in{}C^{\alpha,\frac{\alpha}{2}}(\mathbb{R}^{n}\times\mathbb{R}^{+})$ with $0<\alpha<1$ for all $m>1$), we see our result (1) improves the older result and, based on this conclusion, we can obtain (2), which shows the geometric characteristic of free boundary.