快速扩散方程有限点爆破解的存在性与大时间性态
偏微分方程分析
2018-05-04 v2
摘要
设 Ω ⊂ R n \Omega\subset\R^n Ω ⊂ R n a 1 , a 2 , … , a i 0 ∈ Ω a_1,a_2,\dots,a_{i_0}\in\Omega a 1 , a 2 , … , a i 0 ∈ Ω Ω ^ = Ω ∖ { a 1 , a 2 , … , a i 0 } \widehat{\Omega}=\Omega\setminus\{a_1,a_2,\dots,a_{i_0}\} Ω = Ω ∖ { a 1 , a 2 , … , a i 0 } R n ^ = R n ∖ { a 1 , a 2 , … , a i 0 } \widehat{R^n}=\R^n\setminus\{a_1,a_2,\dots,a_{i_0}\} R n = R n ∖ { a 1 , a 2 , … , a i 0 } Ω ^ × ( 0 , ∞ ) \widehat{\Omega}\times (0,\infty) Ω × ( 0 , ∞ ) R n ^ × ( 0 , ∞ ) \widehat{R^n}\times (0,\infty) R n × ( 0 , ∞ ) u t = Δ u m u_t=\Delta u^m u t = Δ u m u > 0 u>0 u > 0 u u u 0 < m < n − 2 n 0<m<\frac{n-2}{n} 0 < m < n n − 2 n ≥ 3 n\geq 3 n ≥ 3 0 ≤ u 0 ∈ L l o c p ( Ω ^ ∖ { a 1 , ⋯ , a i 0 } ) 0\le u_0\in L^p_{loc}(\widehat{\Omega}\setminus\{a_1,\cdots,a_{i_0}\}) 0 ≤ u 0 ∈ L l oc p ( Ω ∖ { a 1 , ⋯ , a i 0 }) u 0 ∈ L l o c p ( R n ^ ) u_0\in L^p_{loc}(\widehat{R^n}) u 0 ∈ L l oc p ( R n ) p > n ( 1 − m ) 2 p>\frac{n(1-m)}{2} p > 2 n ( 1 − m ) x ≈ a i x\approx a_i x ≈ a i γ i > 2 1 − m , λ i > 0 \gamma_i>\frac{2}{1-m},\lambda_i>0 γ i > 1 − m 2 , λ i > 0 i = 1 , 2 , … , i 0 i=1,2,\dots,i_0 i = 1 , 2 , … , i 0 u 0 ( x ) ≥ λ i ∣ x − a i ∣ − γ i u_0(x)\ge \lambda_i|x-a_i|^{-\gamma_i} u 0 ( x ) ≥ λ i ∣ x − a i ∣ − γ i t > 0 t>0 t > 0 i = 1 , ⋯ , i 0 i=1,\cdots,i_0 i = 1 , ⋯ , i 0 x → a i x\to a_i x → a i u ( x , t ) → ∞ u(x,t)\to\infty u ( x , t ) → ∞ a 1 , a 2 , … , a i 0 a_1,a_2,\dots,a_{i_0} a 1 , a 2 , … , a i 0 μ 0 > 0 \mu_0>0 μ 0 > 0 Ω ^ \widehat{\Omega} Ω R n ^ \widehat{R^n} R n u 0 ≥ μ 0 u_0\ge\mu_0 u 0 ≥ μ 0 γ 1 > n − 2 m \gamma_1>\frac{n-2}{m} γ 1 > m n − 2 u u u t → ∞ t\to\infty t → ∞ Ω ^ \widehat{\Omega} Ω R n ^ \widehat{R^n} R n μ 0 > 0 \mu_0>0 μ 0 > 0 Ω ^ \widehat{\Omega} Ω R n ^ \widehat{R^n} R n u 0 ≥ μ 0 u_0\ge\mu_0 u 0 ≥ μ 0 x ≈ a i x\approx a_i x ≈ a i 2 1 − m < γ i ≤ γ i ′ < n − 2 m \frac{2}{1-m}<\gamma_i\le\gamma_i'<\frac{n-2}{m} 1 − m 2 < γ i ≤ γ i ′ < m n − 2 λ i > 0 \lambda_i>0 λ i > 0 λ i ′ > 0 \lambda_i'>0 λ i ′ > 0 i = 1 , 2 , … , i 0 i=1,2,\dots,i_0 i = 1 , 2 , … , i 0 λ i ∣ x − a i ∣ − γ i ≤ u 0 ( x ) ≤ λ i ′ ∣ x − a i ∣ − γ i ′ \lambda_i|x-a_i|^{-\gamma_i}\le u_0(x)\le \lambda_i'|x-a_i|^{-\gamma_i'} λ i ∣ x − a i ∣ − γ i ≤ u 0 ( x ) ≤ λ i ′ ∣ x − a i ∣ − γ i ′ u u u Ω ^ ∖ { a 1 , a 2 , … , a i 0 } \widehat{\Omega}\setminus\{a_1,a_2,\dots,a_{i_0}\} Ω ∖ { a 1 , a 2 , … , a i 0 } R n ^ \widehat{R^n} R n K K K C 2 ( K ) C^2(K) C 2 ( K ) t → ∞ t\to\infty t → ∞
引用
@article{arxiv.1712.05515,
title = {Existence and large time behaviour of finite points blow-up solutions of the fast diffusion equation},
author = {Kin Ming Hui and Sunghoon Kim},
journal= {arXiv preprint arXiv:1712.05515},
year = {2018}
}
备注
38 pages