JournalszaaVol. 15, No. 3pp. 637–650

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

  • Carsten Ebmeyer

    Universität Bonn, Germany
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Abstract

We deal with the initial and boundary value problem for the degenerate parabolic equation ut=Δβ(u)u_t = \Delta \beta (u) in the cylinder Ω×(0,T)\Omega \times (0,T), where ΩRn\Omega \subset \mathbb R^n is bounded, β(0)=β(0)=0\beta (0) = \beta’(0) = 0, and β0\beta' ≥ 0 (e.g., β(u)=uum1 (m>1))\beta (u) = u |u|^{m–1} \ (m > 1)). We study the appearance of the free boundary, and prove under certain hypothesis on β\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>0u > 0 is small and obtain the non-degeneracy property {0<β(u(x,t))<ϵ}cϵ12|\{ 0 < \beta' (u(x,t)) < \epsilon \} | ≤ c \epsilon^{\frac{1}{2}}.

Cite this article

Carsten Ebmeyer, A Non-Degeneracy Property for a Class of Degenerate Parabolic Equations. Z. Anal. Anwend. 15 (1996), no. 3, pp. 637–650

DOI 10.4171/ZAA/720