Regularity and geometric character of solution of a degenerate parabolic equation

TL;DR

This paper investigates the regularity and geometric properties of solutions to a degenerate parabolic equation, improving previous Hölder estimates and revealing the geometric structure of the free boundary.

Contribution

It improves Hölder regularity estimates for solutions and demonstrates the geometric significance of the free boundary in degenerate parabolic equations.

Findings

Weak solutions are Hölder continuous with specific exponents depending on m.

The surface defined by (u(x,t))^β forms a Riemannian manifold tangent to the boundary.

Explicit propagation speed and dependence on nonlinearity are derived.

Abstract

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.