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Zbl 0790.35053
Carrillo, José; Díaz, Jesus Ildefonso; Gilardi, Gianni
The propagation of the free boundary of the solution of the dam problem and related problems.
(English)
[J] Appl. Anal. 49, No. 3-4, 255-276 (1993). ISSN 0003-6811; ISSN 1563-504X/e

The following equations are considered (1) $\partial\sb t\chi- \text{div} (\nabla u+\chi{\bold e})=0$ and $\chi\in H(u)$ in an open set $\Omega\subset\bbfR\sp n$, with a proper initial condition and suitable boundary conditions of mixed type. In (1) $H$ is the maximal monotone Heaviside graph and {\bf e} is a given vector in $\bbfR\sp n$. The aim of the paper is to study the growth of the support $S(t)$ of the solution $u(\cdot,t)$: in fact $S(t)$ can grow with finite or infinite speed depending on the boundary data, the initial datum $\chi(\cdot, 0)$, and the shape of $\Omega$. Starting with a local study of the support for small $t$, sharp estimates from above and below are found and sufficient conditions for finite or infinite speed of propagation are given. Moreover global results are derived and a monotonicity result is proved for the mushy region'', where $\chi$ takes values in $]0,1[$.
[G.Gilardi (Pavia)]
MSC 2000:
*35K65 Parabolic equations of degenerate type
35B30 Dependence of solutions of PDE on initial and boundary data
76S05 Flows in porous media

Keywords: dam problem; free boundary; porous media; degenerate parabolic problems; speed of propagation; mushy region

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