2011/11/24 by Kin Ming Hui, Sunghoon Kim, Hui, Kin Ming +1
Mathematics · #35K65 #Analysis of PDEs (math.AP) #FOS: Mathematics #Primary 35B40 Secondary 35K57 #math.AP #msc:35B40 #msc:35K57 #msc:35K65
paper · pdf · doi:10.48550/arxiv.1111.5692
12 pages
arxiv created 2011/11/24 · arxiv updated 2011/11/28
Let n≥ 3 and ψλ0 be the radially symmetric solution of Δlogψ+2βψ+βx⋅∇ψ=0 in Rn, ψ(0)=λ0, for some constants λ0>0, β>0. Suppose u0≥ 0 satisfies u0-ψλ0∈ L1(Rn) and u0(x)≈\frac2(n-2)β(log |x|)/(|x|2) as |x|→∞. We prove that the rescaled solution \widetildeu(x,t)=e2βtu(eβtx,t) of the maximal global solution u of the equation ut=Δlog u in Rn× (0,∞), u(x,0)=u0(x) in Rn, converges uniformly on every compact subset of Rn and in L1(Rn) to ψλ0 as t→∞. Moreover ‖\widetildeu(⋅,t)-ψλ0‖L1(Rn) ≤ e-(n-2)βt‖u0-ψλ0‖L1(Rn) for all t≥ 0.