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Regularity of the free boundary for the porous medium equation. (English) Zbl 0910.35145
This 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.

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
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References:
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