Super fast vanishing solutions of the fast diffusion equation

Shu-Yu Hsu

doi:10.3934/dcds.2020232

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2020, Volume 40, Issue 9: 5383-5414. Doi: 10.3934/dcds.2020232

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Super fast vanishing solutions of the fast diffusion equation

Shu-Yu Hsu

Department of Mathematics, National Chung Cheng University, 168 University Road, Min-Hsiung, Chia-Yi 621, Taiwan

Received: October 31, 2019

Revised: March 31, 2020

Published: June 2020

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Abstract

We will extend a recent result of B. Choi, P. Daskalopoulos and J. King [5]. For any $ n\ge 3 $, $ 0<m<\frac{n-2}{n+2} $ and $ \gamma>0 $, we will construct subsolutions and supersolutions of the fast diffusion equation $ u_t = \frac{n-1}{m}\Delta u^m $ in $ \mathbb{R}^n\times (t_0, T) $, $ t_0<T $, which decay at the rate $ (T-t)^{\frac{1+\gamma}{1-m}} $ as $ t\nearrow T $. As a consequence we obtain the existence of unique solution of the Cauchy problem $ u_t = \frac{n-1}{m}\Delta u^m $ in $ \mathbb{R}^n\times (t_0, T) $, $ u(x, t_0) = u_0(x) $ in $ \mathbb{R}^n $, which decay at the rate $ (T-t)^{\frac{1+\gamma}{1-m}} $ as $ t\nearrow T $ when $ u_0 $ satisfies appropriate decay condition.

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Mathematics Subject Classification: Primary: 35K55; Secondary: 35A01, 35B44.

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