高维对数扩散方程的大时间行为
偏微分方程分析
2011-11-28 v1
摘要
设 n ≥ 3 n\ge 3 n ≥ 3 ψ λ 0 \psi_{\lambda_0} ψ λ 0 Δ log ψ + 2 β ψ + β x ⋅ ∇ ψ = 0 \Delta\log\psi+2\beta\psi+\beta x\cdot\nabla\psi=0 Δ log ψ + 2 β ψ + β x ⋅ ∇ ψ = 0 R n R^n R n ψ ( 0 ) = λ 0 \psi(0)=\lambda_0 ψ ( 0 ) = λ 0 λ 0 > 0 \lambda_0>0 λ 0 > 0 β > 0 \beta>0 β > 0 u 0 ≥ 0 u_0\ge 0 u 0 ≥ 0 u 0 − ψ λ 0 ∈ L 1 ( R n ) u_0-\psi_{\lambda_0}\in L^1(R^n) u 0 − ψ λ 0 ∈ L 1 ( R n ) ∣ x ∣ → ∞ |x|\to\infty ∣ x ∣ → ∞ u 0 ( x ) ≈ 2 ( n − 2 ) β log ∣ x ∣ ∣ x ∣ 2 u_0(x)\approx\frac{2(n-2)}{\beta}\frac{\log |x|}{|x|^2} u 0 ( x ) ≈ β 2 ( n − 2 ) ∣ x ∣ 2 l o g ∣ x ∣ u t = Δ log u u_t=\Delta\log u u t = Δ log u R n × ( 0 , ∞ ) R^n\times (0,\infty) R n × ( 0 , ∞ ) u ( x , 0 ) = u 0 ( x ) u(x,0)=u_0(x) u ( x , 0 ) = u 0 ( x ) u u u u ~ ( x , t ) = e 2 β t u ( e β t x , t ) \widetilde{u}(x,t)=e^{2\beta t}u(e^{\beta t}x,t) u ( x , t ) = e 2 β t u ( e β t x , t ) t → ∞ t\to\infty t → ∞ R n R^n R n L 1 ( R n ) L^1(R^n) L 1 ( R n ) ψ λ 0 \psi_{\lambda_0} ψ λ 0 t ≥ 0 t\ge 0 t ≥ 0 ∥ u ~ ( ⋅ , t ) − ψ λ 0 ∥ L 1 ( R n ) ≤ e − ( n − 2 ) β t ∥ u 0 − ψ λ 0 ∥ L 1 ( R n ) \|\widetilde{u}(\cdot,t)-\psi_{\lambda_0}\|_{L^1(R^n)} \le e^{-(n-2)\beta t}\|u_0-\psi_{\lambda_0}\|_{L^1(R^n)} ∥ u ( ⋅ , t ) − ψ λ 0 ∥ L 1 ( R n ) ≤ e − ( n − 2 ) β t ∥ u 0 − ψ λ 0 ∥ L 1 ( R n )
引用
@article{arxiv.1111.5692,
title = {Large time behaviour of higher dimensional logarithmic diffusion equation},
author = {Kin Ming Hui and Sunghoon Kim},
journal= {arXiv preprint arXiv:1111.5692},
year = {2011}
}
备注
12 pages