Publication: The mesa problem: diffusion patterns for ut=∇⋅(um∇u) as m→+∞ .
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1986
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Oxford University Press
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Abstract
In this paper we consider the limit m→+∞ of solutions of the porous-medium equation ut = ∇•(um∇u)(xεRN), with N > 1. We conjecture that, for initial data with a unique maximum, the evolution is characterized by the onset of a ‘mesa’ region, in which the solution is nearly spatially independent, surrounded by a region in which u is nearly equal to its initial value. The transition between these regions occurs near a surface which is identified with the free boundary in a certain Stefan problem which can be studied using variational inequalities. Moreover, singular-perturbation theory can be used to describe the structure of the transition region.