Regularity of the free boundary for the porous medium equation.

*(English)*Zbl 0910.35145This paper deals with the initial value problem for the porous medium equation \(u_t=\Delta u^m\) with compactly supported initial data, written for pressure \(f=mu^{m-1}\). The authors prove that, under rather general assumptions on the initial data, the free boundary is a smooth surface locally in time. First, a model linear degenerate equation is studied on the half-space \(x\geq 0\). The basic idea is to establish Schauder type coercive estimates for solutions of the model equation. Then the result is extended to a certain class of quasilinear degenerate evolution equations. Then the proof of regularity of the free boundary is given. Using a global change of coordinates, the authors transform the free boundary problem to a fixed boundary problem for a degenerate quasilinear equation, which can be solved in appropriately defined Hölder spaces.

Reviewer: P.B.Dubovskiĭ (Moskva)

##### MSC:

35R35 | Free boundary problems for PDEs |

35K60 | Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations |

35B65 | Smoothness and regularity of solutions to PDEs |

##### Keywords:

porous medium equation; free boundary; diffusion; regularity; degenerate equations; Schauder type coercive estimates; quasilinear degenerate evolution equations
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\textit{P. Daskalopoulos} and \textit{R. Hamilton}, J. Am. Math. Soc. 11, No. 4, 899--965 (1998; Zbl 0910.35145)

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