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A non-degeneracy property for a class of degenerate parabolic equations. (English) Zbl 0858.35069
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 $$|\{0< \beta'(u(x,t))< \varepsilon\}|\leq c\varepsilon^{1/2}$$.

##### MSC:
 35K65 Degenerate parabolic equations 35R35 Free boundary problems for PDEs 76S05 Flows in porous media; filtration; seepage
##### Keywords:
speed of propagation; porous medium equations
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##### References:
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