Sharp Estimates for the Global Attractor of Scalar Reaction–Diffusion Equations with a Wentzell Boundary Condition

In this paper, we derive optimal upper and lower bounds on the dimension of the attractor $\mathcal{A}_{\mathrm{W}}$ for scalar reaction–diffusion equations with a Wentzell (dynamic) boundary condition. We are also interested in obtaining explicit bounds on the constants involved in our asymptotic estimates, and to compare these bounds to previously known estimates for the dimension of the global attractor $\mathcal{A}_{K}$, K∈{D,N,P}, of reaction–diffusion equations subject to Dirichlet, Neumann and periodic boundary conditions. The explicit estimates we obtain show that the dimension of the global attractor $\mathcal {A}_{\mathrm{W}}$ is of different order than the dimension of $\mathcal{A}_{K}$, for each K∈{D,N,P}, in all space dimensions that are greater than or equal to three.