Uniform $L^{\infty}$-Boundedness of Global Attractors for Reaction-Diffusion Equations with Neumann boundary condition in Uniformly Perturbed Non-Smooth Domains
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 $L^\infty(Ω_μ)$; 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, $|Ω_μ\triangle Ω_0| \to 0$, 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 $s<1$.