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Zeitschrift für Analysis und ihre Anwendungen


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Volume 15, Issue 3, 1996, pp. 637–650
DOI: 10.4171/ZAA/720

Published online: 1996-09-30

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

Carsten Ebmeyer[1]

(1) Universität Bonn, Germany

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 $\beta' ≥ 0$ (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 $|\{ 0 < \beta' (u(x,t)) < \epsilon \} | ≤ c \epsilon^{\frac{1}{2}}$.

Keywords: Free boundary problems, finite speed of propagation, porous medium equations

Ebmeyer Carsten: A Non-Degeneracy Property for a Class of Degenerate Parabolic Equations. Z. Anal. Anwend. 15 (1996), 637-650. doi: 10.4171/ZAA/720