46. Near field asymptotics for the porous medium equation in exterior domains. The critical two-dimensional case

NOEMI WOLANSKI; QUIRÓS, FERNANDO; CORTÁZAR, CARMEN

SIAM JOURNAL ON MATHEMATICAL ANALYSIS

SIAM PUBLICATIONS

Lugar: Philadelphia-USA; Año: 2018 vol. 50 p. 2664 - 2680

0036-1410

We consider the porous medium equation in an exterior  two-dimensional domain that excludes a hole,  with zero Dirichlet data on  its boundary. Gilding and Goncerzewicz proved in 2007 that in the far-field scale,   which is the adequate one to describe the movement of the free boundary,  solutions to this problem with  integrable and compactly supported initial data behave as an instantaneous point-source solution for the equation with a variable mass that decays to 0 in a precise way, determined by the initial data and the hole. In this paper, starting from their result  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.   Spatial dimension two is critical for this problem, and involves logarithmic corrections.