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SINGULAR LIMITS AND THE "MESA" PROBLEM

J Bouillet, M Korten, V Marquez

unpublished

For x E R, we discuss the "mesa" type limit of the one-phase Stefan problem in enthalpic variables. This limit is the same as for the porous medium equation, and coincides with the asymptotic limit when time tends to infinity of the soluition of the Stefan problem. We discuss a degenerate 'diffusion problem where the diffusivity is concentrated on the fr ee boundary, related to the limit when In-t 0 in the porous medium equation. The solution to this diffusion reaches in finite time a constant
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... te time a constant state, which turns out to be the same as in the first three cases: a function 0 ::; U oo ::; 1 , which coincides with the initial datum in a set which call be identified by a variational inequality. We show an exam Ie where for n > 1 the speed of the free boundary dcscribillg t.he extinction of the zone 1/,1 (x) > 1 tcnds to-00 as t-t I •. We wish to honour the dear memory of onr late teacher, advisor , and friend Julio E. Douillet by contrIbuting to this volume with a paper-hitherto in preprint form-written jointly with him. We kept the paper (see [BKM]) in its original form (incl uding the dedication to Prof. Mischa Cotlar). Since this paper has been written, many results have been obtained on the problem of singular limits; we have added some references ([BI1), [BI2) , [DDIl) , [I) , [BKMJ, and [S I]); and the behaviour of weak solutions to equation (1.3) has been extensively discussed ([AK) , [KIJ , [K2)). However , we think that t.he computations of this paper may still have some interest: as an appl ication of the result,s of section 4, concerning an elliptic-parabolic problem where the diffusion is concentrated on the free boundary, one can find in [BE] a numerical method for the treatment of solutions to diffusions of the type U l = (<x(u))" .. , in which ", is seen as a limit of a sequence of linear combinations of step functions. Dedicat.ed to Professor Mischa Cotlar on his 75th. birthday.

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