# zbMATH — the first resource for mathematics

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
Full Text: