A Non-Degeneracy Property for a Class of Degenerate Parabolic Equations

Abstract

We deal with the initial and boundary value problem for the degenerate parabolic equation $u_t=\Delta \beta (u)$ in the cylinder $\Omega \times (0,T)$, where $\Omega \subset {\mathbb{R}}^n$ is bounded, $\beta (0)=\beta ’(0)=0$, and ${\beta }^{\prime }\ge 0$ (e.g., $\beta (u)=u|u{|}^{m–1}(m>1))$. We study the appearance of the free boundary, and prove under certain hypothesis on $\beta$ that the free boundary has a finite speed of propagation, and is Holder continuous. Further, we estimate the Lebesgue measure of the set where $u>0$ is small and obtain the non-degeneracy property $|\{0<{\beta }^{\prime }(u(x,t))<ϵ\}|\le c{ϵ}^{\frac{1}{2}}$.