Uniform $L^{\infty }$-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 $\{{\Omega }_{\mu }{\}}_{\mu \in \Lambda }\subset {\mathbb{R}}^n$. 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 $L^q({\Omega }_{\mu })$ are absorbed, in finite time and uniformly in $\mu$, into a fixed ball of $L^{\infty }({\Omega }_{\mu })$; 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, $|{\Omega }_{\mu }△{\Omega }_0|\to 0$, we construct a framework of connecting maps and prove that the family of attractors is upper semicontinuous at $\mu =0$ in $L^2({\Omega }_{\mu })$ and hence, by interpolation, in $H^s({\Omega }_{\mu })$ for every $s<1$.

36 pages