2011/02/20 by Martino Prizzi, Prizzi, Martino
Computer Science · Engineering · Mathematics · #35B41 #35K57 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics #Nonlinear Differential Equations Analysis #Stability and Controllability of Differential Equations #math.AP #math.DS #msc:35B41 #msc:35K57
paper · pdf · doi:10.48550/arxiv.1102.4062
20 pages
arxiv created 2011/02/20 · openalex publication_date 2011/02/20 · arxiv updated 2011/02/22 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28
Under fairly general assumptions, we prove that every compact invariant set \mathcal I of the semiflow generated by the semilinear reaction diffusion equation ut+β(x)u-Δu&=f(x,u),&&(t,x)∈[0,+∞[×Ω, u&=0,&&(t,x)∈[0,+∞[×∂Ω equation* in H10(Ω) has finite Hausdorff dimension. Here Ω is an arbitrary, possibly unbounded, domain in \R3 and f(x,u) is a nonlinearity of subcritical growth. The nonlinearity f(x,u) needs not to satisfy any dissipativeness assumption and the invariant subset \mathcal I needs not to be an an attractor. If Ω is regular, f(x,u) is dissipative and \mathcal I is the global attractor, we give an explicit bound on the Hausdorff dimension of \mathcal I in terms of the structure parameter of the equation.