| Abstract: |
| Let $\mathcal{H}\subset\mathbb{R}^N$ be a non-empty bounded open set. We consider the porous medium equation in the complement of $\mathcal{H}$, with zero Dirichlet data on its boundary and nonnegative compactly supported integrable initial data. When $N=1$, Kamin and V\`azquez, in 1991, studied the large time behavior of solutions to this problem in the far-field scale, which is the adequate one to describe the movement of the free boundary. Gilding and Goncerzewicz, in 2007, performed an analogous study in dimension $N=2$. Starting from their results in the far field, we study the large time behavior in the near field, in scales that evolve more slowly than the free boundary. In this way we get, in particular, the final profile and decay rate on compact sets. In contrast with the case of high dimensions, $N\ge3$, in low dimensions the decay rate of solutions in the near field is not the same as in the far field. |
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