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The relation between the porous medium and the eikonal equations in several space dimensions. (English) Zbl 0697.35012

This paper deals with the Cauchy problem for the porous medium equation \(u_ t=\Delta (u^ m)\) as \(m\to 1\). The change of variable \((m- 1)v=mu^{m-1}\) yields the perturbed eikonal equation \(v_ t=(m- 1)v\Delta v+| Dv|^ 2.\) The initial data is nonnegative to that u and v are nonnegative. Thus the perturbation is of parabolic type with degeneracy as long as v vanishes.
The first theorem states the convergence of the porous medium solution v to the viscosity solution of the eikonal equation. The convergence is locally uniform. The second one deals with compactly supported data: in an appropriate sense, the interface \(S_ m\) of \(v_ m\) (recall that the porous medium equation has the finite speed propagation property for the support) converges to the one of v.
Reviewer: D.Serre

MSC:

35B25 Singular perturbations in context of PDEs
76S05 Flows in porous media; filtration; seepage
35K65 Degenerate parabolic equations
35L80 Degenerate hyperbolic equations
35R35 Free boundary problems for PDEs
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