Sharp estimates for the global attractor of scalar reaction-diffusion equations with a Wentzell boundary condition

TL;DR

This paper establishes optimal bounds on the dimension of the global attractor for scalar reaction-diffusion equations with Wentzell boundary conditions, comparing these bounds with classical boundary conditions across various dimensions.

Contribution

It provides explicit bounds and asymptotic estimates for the attractor dimension under Wentzell boundary conditions, extending previous results for Dirichlet, Neumann, and periodic cases.

Findings

Attractor dimension differs in order from classical boundary conditions in dimensions ≥ 3.

Explicit bounds depend on constants explicitly calculated.

The results highlight the impact of boundary conditions on attractor complexity.

Abstract

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