2010/09/30 by Rolando Magnanini, Shigeru Sakaguchi, Magnanini, Rolando +1 · 1 citation
Computer Science · Mathematics · #35B40 #35K55 #35K60 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Numerical Methods #FOS: Mathematics #Nonlinear Partial Differential Equations #math.AP #msc:35B40 #msc:35K55 #msc:35K60
paper · pdf · doi:10.48550/arxiv.1009.6131
25 pages, no figures. Added some details to introduction. A couple of small changes. To appear in Journal Diff. Eqs
openalex publication_date 2010/09/30 · arxiv created 2011/08/09 · arxiv updated 2011/08/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let Ω be a domain in \mathbb RN, where N ≥ 2 and ∂Ω is not necessarily bounded. We consider nonlinear diffusion equations of the form ∂t u= Δϕ(u). Let u=u(x,t) be the solution of either the initial-boundary value problem over Ω, where the initial value equals zero and the boundary value equals 1, or the Cauchy problem where the initial data is the characteristic function of the set \mathbb RN∖ Ω. We consider an open ball B in Ω whose closure intersects ∂Ω only at one point, and we derive asymptotic estimates for the content of substance in B for short times in terms of geometry of Ω. Also, we obtain a characterization of the hyperplane involving a stationary level surface of u by using the sliding method due to Berestycki, Caffarelli, and Nirenberg. These results tell us about interactions between nonlinear diffusion and geometry of domain.