Existence of Neumann and singular solutions of the fast diffusion equation

Kin Ming Hui; Sunghoon Kim

doi:10.3934/dcds.2015.35.4859

Article Contents

2015, Volume 35, Issue 10: 4859-4887. Doi: 10.3934/dcds.2015.35.4859

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Existence of Neumann and singular solutions of the fast diffusion equation

Kin Ming Hui^1, and
Sunghoon Kim^2,

1.
Institute of Mathematics, Academia Sinica, Taipei, 10617
2.
Department of Mathematics, School of Natural Sciences, The Catholic University of Korea, 43 Jibong-ro, Wonmi-gu, Bucheon-si, Gyeonggi-do, 420-743

Received: May 31, 2014

Revised: January 31, 2015

Published: September 2015

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Abstract

Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^n$, $n\ge 3$, $0 < m \le \frac{n-2}{n}$, $a_1,a_2,\dots, a_{i_0}\in\Omega$, $\delta_0 = \min_{1 \le i \le i_0} \mbox{dist} (a_i,∂\Omega)$ and let $\Omega_{\delta}=\Omega\setminus\cup_{i=1}^{i_0}B_{\delta}(a_i)$ and $\hat{\Omega}=\Omega\setminus\{a_1\,\dots,a_{i_0}\}$. For any $0<\delta<\delta_0$ we will prove the existence and uniqueness of positive solution of the Neumann problem for the equation $u_t=\Delta u^m$ in $\Omega_{\delta}\times (0,T)$ for some $T>0$. We will prove the existence of singular solutions of this equation in $\hat{\Omega}\times (0,T)$ for some $T>0$ that blow-up at the points $a_1,\dots, a_{i_0}$.

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Mathematics Subject Classification: 35A01, 35K67.

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