×

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
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Aronson, D. C. and Ph. Bénilan: Régularité des solutions de l’équation des milieux poreux dans R’. C.R. Acad. Sd. Paris Sér. A-B 288 (1979), A103 - A105. · Zbl 0397.35034
[2] [3] Aronson, D. C., Caffarelli, L.A. and J. L. Vázquez: Interfaces with a corner point in one-dimensional porous medium flow. Comm. Pure AppI. Math. 38 (1985), 375 - 404. · Zbl 0544.35058 · doi:10.1002/cpa.3160380404
[3] Caffarelli, L. A. and A. Friedman: Continuity of the density of a gas flow in a porous medium. Trans. Amer. Math. Soc. 252 (1979), 99 - 113. · Zbl 0425.35060 · doi:10.2307/1998079
[4] Caffarelli, L. A. and A. Friedman: Regularity of the free boundary of a gas flow in an n-dimensional porous medium. Indiana Univ. Math. J. 29 (1980), 361 - 391. · Zbl 0439.76085 · doi:10.1512/iumj.1980.29.29027
[5] [9] Caffarelli, L. A., Vázquez, J. L. and N. I. Wolanski: Lipschitz continuity of solutions and interfaces of the n-dimensional porous medium equation. Indiana Univ. Math. J. 36 (1987), 373 - 401. · Zbl 0644.35058 · doi:10.1512/iumj.1987.36.36022
[6] Ebmeyer, C.: Konvergenzraten finiter Elemente liAr die Poróse-Medien-Gleichung im R”. Bonner Math. Schriften 287 (1996), 1 - 69. · Zbl 0889.76035
[7] Hongjun, Y.: holder continuity of interfaces for the porous medium equation with absorp- tion. Comm. Part. Duff. Equ. 18 (1993), 965 - 976. · Zbl 0818.35053 · doi:10.1080/03605309308820957
[8] Knerr, B. F.: The porous medium equation in one dimension. Trans. Amer. Math. Soc. 234 (1977), 381 - 415. · Zbl 0365.35030 · doi:10.2307/1997927
[9] 389. · JFM 26.0679.01
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.