A sharp estimate and change on the dimension of the attractor for Allen-Cahn equations

2008/02/13 by Nikos I. Karachalios, I. Karachalios, Karachalios, Nikos I. +2
Computer Science · Engineering · Mathematics · #35B40 #35B41 #35K57 #37L30 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics #Nonlinear Partial Differential Equations #Stability and Controllability of Differential Equations #math.AP #math.DS #msc:35B40 #msc:35B41 #msc:35K57 #msc:37L30

paper · pdf · doi:10.48550/arxiv.0802.1860

9 pages

openalex publication_date 2008/02/13 · arxiv created 2008/02/14 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

We consider the semilinear reaction diffusion equation ∂_tϕ-νΔϕ-V(x)ϕ+f(ϕ)=0, ν>0 in a bounded domain Ω⊂ℝ^N. We assume the standard Allen-Cahn-type nonlinearity, while the potential V is either the inverse square potential V(x)=δ|x|^-2 or the borderline potential V(x)=δdist(x,∂Ω)^-2, δ≥ 0 (thus including the classical Allen-Cahn equation as a special case when δ=0). In the subcritical cases δ=0, N≥ 1 and 0

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