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The mesa-limit of the porous-medium equation and the Hele-Shaw problem. (English) Zbl 1011.35080

Summary: We are interested in the limit, as \(m\to\infty\), of the solution \(u_m\) of the porous-medium equation \(u_t=\Delta u^m\) in a bounded domain \(\Omega\) with Neumann boundary condition, \({\partial u^m\over \partial n}=g\) on \(\partial\Omega\), and initial datum \(u(0)=u_0\geq 0\). It is well known by now that this kind of limits turns out to be singular. In the case \(g\equiv 0\), it was proved that there exists an initial boundary layer \(\underline u_0\), the so-called mesa, and \(u_m(t)\to\underline u_0\), in \(L^1 (\Omega)\), for any \(t>0\), as \(m\to\infty\). In this work, we generalize this result to the case of arbitrary \(g\in L^2(\partial\Omega)\), we prove that the initial boundary layer is still \(\underline u_0\) and in general (even in the regular case) the limit function is not a solution of a Hele-Shaw problem. There exists a time interval \(I\) where the limit of \(u_m\), as \(m\to\infty\), is the unique solution of a Hele-Shaw problem and elsewhere, \(u_m\) converges to the constant function \({1\over|\Omega|} (\int_\Omega u_0+t \int_{\partial \Omega}g)\).

MSC:

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35K65 Degenerate parabolic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35B40 Asymptotic behavior of solutions to PDEs
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