Asymptotic behaviour of singular solution of the fast diffusion equation in the punctured euclidean space

Kin Ming Hui; Jinwan Park

doi:10.3934/dcds.2021085

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November 2021, 41(11): 5473-5508. doi: 10.3934/dcds.2021085

Asymptotic behaviour of singular solution of the fast diffusion equation in the punctured euclidean space

Kin Ming Hui ^1, and Jinwan Park ^2,,

1.	Institute of Mathematics, Academia Sinica, Taipei, Taiwan, R. O. C
2.	Research Institute of Mathematics, Seoul National University, Seoul 08826, Republic of Korea

^* Corresponding author: Jinwan Park

Received July 2020 Revised April 2021 Published November 2021 Early access May 2021

Fund Project: The corresponding author is supported by NRF grant 2020R1A6A3A01099425

For

$ n\ge 3 $

$ 0<m<\frac{n-2}{n} $

$ \beta<0 $

and

$ \alpha = \frac{2\beta}{1-m} $

, we prove the existence, uniqueness and asymptotics near the origin of the singular eternal self-similar solutions of the fast diffusion equation in

$ (\mathbb{R}^n\setminus\{0\})\times \mathbb{R} $

of the form

$ U_{\lambda}(x,t) = e^{-\alpha t}f_{\lambda}(e^{-\beta t}x), x\in \mathbb{R}^n\setminus\{0\}, t\in\mathbb{R}, $

where

$ f_{\lambda} $

is a radially symmetric function satisfying

$ \frac{n-1}{m}\Delta f^m+\alpha f+\beta x\cdot\nabla f = 0 \text{ in }\mathbb{R}^n\setminus\{0\}, $

with

$ \underset{\substack{r\to 0}}{\lim}\frac{r^2f(r)^{1-m}}{\log r^{-1}} = \frac{2(n-1)(n-2-nm)}{|\beta|(1-m)} $

and

$ \underset{\substack{r\to\infty}}{\lim}r^{\frac{n-2}{m}}f(r) = \lambda^{\frac{2}{1-m}-\frac{n-2}{m}} $

, for some constant

$ \lambda>0 $

As a consequence we prove the existence and uniqueness of solutions of Cauchy problem for the fast diffusion equation

$ u_t = \frac{n-1}{m}\Delta u^m $

$ (\mathbb{R}^n\setminus\{0\})\times (0,\infty) $

with initial value

$ u_0 $

satisfying

$ f_{\lambda_1}(x)\le u_0(x)\le f_{\lambda_2}(x) $

$ \forall x\in\mathbb{R}^n\setminus\{0\} $

, such that the solution

$ u $

satisfies

$ U_{\lambda_1}(x,t)\le u(x,t)\le U_{\lambda_2}(x,t) $

$ \forall x\in \mathbb{R}^n\setminus\{0\}, t\ge 0 $

, for some constants

$ \lambda_1>\lambda_2>0 $

We also prove the asymptotic large time behaviour of such singular solution

$ u $

when

$ n = 3,4 $

and

$ \frac{n-2}{n+2}\le m<\frac{n-2}{n} $

holds. Asymptotic large time behaviour of such singular solution

$ u $

is also obtained when

$ 3\le n<8 $

$ 1-\sqrt{2/n}\le m<\min\left(\frac{2(n-2)}{3n},\frac{n-2}{n+2}\right) $

, and

$ u(x,t) $

is radially symmetric in

$ x\in\mathbb{R}^n\setminus\{0\} $

for any

$ t>0 $

under appropriate conditions on the initial value

$ u_0 $

Keywords: asymptotic behaviour of solutions, blow-up, fast diffusion equation, singular solution, L¹-contraction, radially symmetric self-similar solution.

Mathematics Subject Classification: Primary: 35B40, 35B44, 35K55, 35K67.

Citation: Kin Ming Hui, Jinwan Park. Asymptotic behaviour of singular solution of the fast diffusion equation in the punctured euclidean space. Discrete and Continuous Dynamical Systems, 2021, 41 (11) : 5473-5508. doi: 10.3934/dcds.2021085