穿孔欧氏空间中快扩散方程奇异解的渐近性态
偏微分方程分析
2021-01-11 v2
摘要
对于n ≥ 3 n\ge 3 n ≥ 3 0 < m < n − 2 n 0<m<\frac{n-2}{n} 0 < m < n n − 2 β < 0 \beta<0 β < 0 α = 2 β 1 − m \alpha=\frac{2\beta}{1-m} α = 1 − m 2 β ( R n ∖ { 0 } ) × R (\mathbb{R}^n\setminus\{0\})\times \mathbb{R} ( R n ∖ { 0 }) × R U λ ( x , t ) = e − α t f λ ( e − β t x ) , x ∈ R n ∖ { 0 } , t ∈ R , U_{\lambda}(x,t)=e^{-\alpha t}f_{\lambda}(e^{-\beta t}x), x\in \mathbb{R}^n\setminus\{0\}, t\in\mathbb{R}, U λ ( x , t ) = e − α t f λ ( e − β t x ) , x ∈ R n ∖ { 0 } , t ∈ R , f λ f_{\lambda} f λ n − 1 m Δ f m + α f + β x ⋅ ∇ f = 0 于 R n ∖ { 0 } , \frac{n-1}{m}\Delta f^m+\alpha f+\beta x\cdot\nabla f=0 \text{ 于 }\mathbb{R}^n\setminus\{0\}, m n − 1 Δ f m + α f + β x ⋅ ∇ f = 0 于 R n ∖ { 0 } , lim r → 0 r 2 f ( r ) 1 − m log r − 1 = 2 ( n − 1 ) ( n − 2 − n m ) ∣ β ∣ ( 1 − m ) \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)} r → 0 lim l o g r − 1 r 2 f ( r ) 1 − m = ∣ β ∣ ( 1 − m ) 2 ( n − 1 ) ( n − 2 − nm ) lim r → ∞ r n − 2 m f ( r ) = λ 2 1 − m − n − 2 m \underset{\substack{r\to\infty}}{\lim}r^{\frac{n-2}{m}}f(r)=\lambda^{\frac{2}{1-m}-\frac{n-2}{m}} r → ∞ lim r m n − 2 f ( r ) = λ 1 − m 2 − m n − 2 λ > 0 \lambda>0 λ > 0 u t = n − 1 m Δ u m u_t=\frac{n-1}{m}\Delta u^m u t = m n − 1 Δ u m ( R n ∖ { 0 } ) × ( 0 , ∞ ) (\mathbb{R}^n\setminus\{0\})\times (0,\infty) ( R n ∖ { 0 }) × ( 0 , ∞ ) u 0 u_0 u 0 f λ 1 ( x ) ≤ u 0 ( x ) ≤ f λ 2 ( x ) f_{\lambda_1}(x)\le u_0(x)\le f_{\lambda_2}(x) f λ 1 ( x ) ≤ u 0 ( x ) ≤ f λ 2 ( x ) ∀ x ∈ R n ∖ { 0 } \forall x\in\mathbb{R}^n\setminus\{0\} ∀ x ∈ R n ∖ { 0 } U λ 1 ( x , t ) ≤ u ( x , t ) ≤ U λ 2 ( x , t ) U_{\lambda_1}(x,t)\le u(x,t)\le U_{\lambda_2}(x,t) U λ 1 ( x , t ) ≤ u ( x , t ) ≤ U λ 2 ( x , t ) ∀ x ∈ R n ∖ { 0 } , t ≥ 0 \forall x\in \mathbb{R}^n\setminus\{0\}, t\ge 0 ∀ x ∈ R n ∖ { 0 } , t ≥ 0 λ 1 > λ 2 > 0 \lambda_1>\lambda_2>0 λ 1 > λ 2 > 0 n = 3 , 4 n=3,4 n = 3 , 4 n − 2 n + 2 ≤ m < n − 2 n \frac{n-2}{n+2}\le m<\frac{n-2}{n} n + 2 n − 2 ≤ m < n n − 2 u u u t → ∞ t\to\infty t → ∞ 3 ≤ n < 8 3\le n<8 3 ≤ n < 8 1 − 2 / n ≤ m < min ( 2 ( n − 2 ) 3 n , n − 2 n + 2 ) 1-\sqrt{2/n}\le m<\min\left(\frac{2(n-2)}{3n},\frac{n-2}{n+2}\right) 1 − 2/ n ≤ m < min ( 3 n 2 ( n − 2 ) , n + 2 n − 2 ) u ( x , t ) u(x,t) u ( x , t ) t > 0 t>0 t > 0 x ∈ R n ∖ { 0 } x\in\mathbb{R}^n\setminus\{0\} x ∈ R n ∖ { 0 } u 0 u_0 u 0 u u u t → ∞ t\to\infty t → ∞
引用
@article{arxiv.2007.06830,
title = {Asymptotic behaviour of singular solution of the fast diffusion equation in the punctured Euclidean space},
author = {Kin Ming Hui and Jinwan Park},
journal= {arXiv preprint arXiv:2007.06830},
year = {2021}
}
备注
34 pages