2010/12/09 by Kin Ming Hui, Sunghoon Kim, Hui, Kin Ming +2
Computer Science · Mathematics · #35K65 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations #Primary 35B40 Secondary 35K57 #math.AP #msc:35B40 #msc:35K57 #msc:35K65
paper · pdf · doi:10.48550/arxiv.1012.1915
The introduction is re-written and some more references are added, 26 pages
openalex publication_date 2010/12/09 · arxiv created 2012/09/25 · arxiv updated 2012/09/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let u be the solution of ut=Δlog u in \RN× (0,T), N=3 or N≥ 5, with initial value u0 satisfying Bk1(x,0)≤ u0≤ Bk2(x,0) for some constants k1>k2>0 where Bk(x,t) =2(N-2)(T-t)+N/(N-2)/(k+(T-t)+2/(N-2)|x|2) is the Barenblatt solution for the equation. We prove that the rescaled function \4u(x,s)=(T-t)-N/(N-2)u(x/(T-t)-1/(N-2),t), s=-log (T-t), converges uniformly on \RN to the rescaled Barenblatt solution \4Bk0(x)=2(N-2)/(k0+|x|2) for some k0>0 as s→∞. We also obtain convergence of the rescaled solution \4u(x,s) as s→∞ when the initial data satisfies 0≤ u0(x)≤ Bk0(x,0) in \RN and |u0(x)-Bk0(x,0)|≤ f(|x|)∈ L1(\RN) for some constant k0>0 and some radially symmetric function f.