Uniform L∞-Boundedness of Global Attractors for Reaction-Diffusion Equations with Neumann boundary condition in Uniformly Perturbed Non-Smooth Domains
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\in Λ} \subset \mathbb{R}^n $. Assuming only that the domains have uniformly bounded volumes, bounded away from zero, and satisfy a uniform Jones condition, together with the usual structural requirements on the linear and nonlinear terms, 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 conjunction with the uniform Gronwall lemma and the uniform properties of the Jones extension operator, we show that bounded sets of Lq(Ωμ) are absorbed, in finite time and uniformly in μ, into a fixed ball of L∞(Ωμ); the iteration uses only first-order Sobolev embeddings, which is what makes it available on domains where the domain of the elliptic operator is not contained in H2. Finally, assuming the volume convergence of the domains, |Ωμ△ Ω0| → 0, we construct a framework of connecting maps and prove that the family of attractors is upper semicontinuous at μ= 0 in L2(Ωμ) and hence, by interpolation, in Hs(Ωμ) for every s<1.
Related
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BibTeX
@misc{pereira2026uniform,
title = {{Uniform $L^\infty$-Boundedness of Global Attractors for Reaction-Diffusion Equations with Neumann boundary condition in Uniformly Perturbed Non-Smooth Domains}},
author = {Antonio L. Pereira},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.08061},
eprint = {2607.08061},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.08061},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Uniform L^∞-Boundedness of Global Attractors for Reaction-Diffusion Equations with Neumann boundary condition in Uniformly Perturbed Non-Smooth Domains
AU - Antonio L. Pereira
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2607.08061
UR - https://arxiv.org/abs/2607.08061
AB - We consider a family of semilinear parabolic equations with homogeneous Neumann boundary conditions on a family of varying non-smooth domains $\{Ω_μ\}_{μ\in Λ} \subset \mathbb{R}^n $. Assuming only that the domains have uniformly bounded volumes, bounded away from zero, and satisfy a uniform Jones condition, together with the usual structural requirements on the linear and nonlinear terms, 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 conjunction with the uniform Gronwall lemma and the uniform properties of the Jones extension operator, we show that bounded sets of L^q(Ω_μ) are absorbed, in finite time and uniformly in μ, into a fixed ball of L^∞(Ω_μ); the iteration uses only first-order Sobolev embeddings, which is what makes it available on domains where the domain of the elliptic operator is not contained in H^2. Finally, assuming the volume convergence of the domains, |Ω_μ△ Ω_0| → 0, we construct a framework of connecting maps and prove that the family of attractors is upper semicontinuous at μ= 0 in L^2(Ω_μ) and hence, by interpolation, in H^s(Ω_μ) for every s<1.
ER -