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arXiv · 1610.04772

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

Carmen Cortázar·Fernando Quirós·Noemí Wolanski
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Abstract

We consider the porous medium equation in an exterior two-dimensional domain which excludes a hole, with zero Dirichlet data on its boundary. Gilding and Goncerzewicz proved in [Gilding-Goncerzewicz-2007] that in the far field scale, $x=\xi t^{\frac1{2m}}/(\log t)^{\frac{m-1}{2m}}$, $\xi\ne 0$, solutions to this problem with an 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. However, their result does not say much about the behavior when $|x|=o\big(t^{\frac1{2m}}/(\log t)^{\frac{m-1}{2m}}\big)$, in the so called near field scale, except that the solution is $o\big((t\log t)^{-\frac1m}\big)$ there. In particular, it does not give a sharp decay rate, neither a nontrivial asymptotic profile, on compact sets. In this paper we characterize the large time behavior in such scale, thus completing the results of [Gilding-Goncerzewicz-2007].

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Carmen Cortázar, Fernando Quirós, Noemí Wolanski. 2016-10-15. Near field asymptotics for the porous medium equation in exterior domains. The critical two-dimensional case. https://arxiv.org/abs/1610.04772

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