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The propagation of the free boundary of the solution of the dam problem and related problems. (English) Zbl 0790.35053

The following equations are considered (1) \(\partial_ t\chi- \text{div} (\nabla u+\chi{\mathbf e})=0\) and \(\chi\in H(u)\) in an open set \(\Omega\subset\mathbb{R}^ n\), with a proper initial condition and suitable boundary conditions of mixed type. In (1) \(H\) is the maximal monotone Heaviside graph and e is a given vector in \(\mathbb{R}^ n\). 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 \(]0,1[\).
Reviewer: G.Gilardi (Pavia)

MSC:

35K65 Degenerate parabolic equations
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
76S05 Flows in porous media; filtration; seepage
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