A non-degeneracy property for a class of degenerate parabolic equations (Q1924307)

Summary: 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 \(\beta'\geq 0\) (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 Hölder continuous. Further, we estimate the Lebesgue measure of the set where \(u>0\) is small and obtain the non-degeneracy property \(|\{0< \beta'(u(x,t))< \varepsilon\}|\leq c\varepsilon^{1/2}\).