## 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

### Keywords:

Neumann boundary condition; initial boundary layers