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Lipschitz continuity of solutions and interfaces of the N-dimensional porous medium equation. (English) Zbl 0644.35058
The authors consider the Cauchy problem for the flow of a gas in a porous medium: \(u_ t=\Delta u^ m\), \(x\in R^ N\), \(t>0\); \(u(x,0)=u_ 0(x)\), \(m>1.\)
If the initial datum has compact support there exists an interface \(\Gamma\) that separates the points where there is gas, \(\Omega\), from the void region where \(u=0.\)
In § 2 the authors prove that as a consequence of the Lipschitz continuity of \(v=(m/m-1)u^{m-1}\), the interface of every solution is a Lipschitz continuous surface for all large times.
In § 3, considering a special class of initial data, the authors are able to bound bellow the speed of the interface and to describe \(\Gamma\) in the form \(t=S(x)\) where S is a Lipschitz continuous function of x.
Reviewer: A.Carabineanu

MSC:
35K55 Nonlinear parabolic equations
35K15 Initial value problems for second-order parabolic equations
76S05 Flows in porous media; filtration; seepage
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