2019/02/08 by Matteo Bonforte, Alessio Figalli, Bonforte, Matteo +1
Computer Science · Engineering · Mathematics · #35B40 #35J20 #35K55 #35K67 #35P30 #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #Analysis of PDEs (math.AP) #Differential Equations and Numerical Methods #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.1902.03189
openalex publication_date 2019/02/08 · openalex created_date 2022/07/29 · openalex updated_date 2026/07/28
We investigate the homogeneous Dirichlet problem for the Fast Diffusion\nEquation ut=\Δ um, posed in a smooth bounded domain \Ω\⊂\n\ℝN, in the exponent range ms=(N-2)+/(N+2)<m<1. It is known that\nbounded positive solutions extinguish in a finite time T>0, and also that\nthey approach a separate variable solution u(t,x)\∼ (T-t)1/(1-m)S(x), as\nt\→ T-. It has been shown recently that v(x,t)=u(t,x) ,(T-t)-1/(1-m)\ntends to S(x) as t\→ T-, uniformly in the relative error norm. Starting\nfrom this result, we investigate the fine asymptotic behaviour and prove sharp\nrates of convergence for the relative error. The proof is based on an entropy\nmethod relying on a (improved) weighted Poincar 'e inequality, that we show to\nbe true on generic bounded domains. Another essential aspect of the method is\nthe new concept of "almost orthogonality", which can be thought as a nonlinear\nanalogous of the classical orthogonality condition needed to obtain improved\nPoincar 'e inequalities and sharp convergence rates for linear flows.\n