Uniform L^∞-Boundedness of Global Attractors for Reaction-Diffusion Equations with Neumann boundary condition in Uniformly Perturbed Non-Smooth Domains

2026/07/31 by Antonio L. Pereira
Mathematics · #math.AP #math.DS #msc:35B41 #msc:35K91 #msc:58D25

paper · pdf

27 pages

arxiv created 2026/08/03 · arxiv updated 2026/08/04

Abstract

We consider a family of semilinear parabolic equations with homogeneous Neumann boundary conditions on a family of varying non-smooth domains \Ω_μ\_{μ∈ Λ} ⊂ ℝⁿ. Assuming only that the domains have uniformly bounded volumes, satisfy a uniform Jones condition, and possess uniform ellipticity bounds, we establish the well-posedness of the problem in an appropriate scale of fractional Banach spaces and prove the existence of global attractors. Using a Moser-Alikakos bootstrap iteration in tandem with the uniform Gronwall lemma and the uniform properties of the Jones extension operator, we show that the family of attractors is uniformly bounded in L^∞(Ω_μ). Finally, assuming the volume convergence of the domains, |Ω_μ\triangle Ω₀| → 0, we construct a framework of connecting maps to prove that the family of attractors is upper semicontinuous at μ= 0 in the strong H¹ topology.

Citations

• Continuity of attractors for a highly oscillatory family of perturbations of the square
• Continuity of attractors for a family of C¹ perturbations of the square