We present a condition which guarantees spatial uniformity for the asymptotic
behavior of the solutions of a reaction-diffusion PDE with Neumann boundary
conditions. This condition makes use of the Jacobian matrix of the reaction
terms and the second Neumann eigenvalue of the Laplacian operator on the given
spatial domain, and replaces the global Lipschitz assumptions commonly used in
the literature with a less restrictive Lyapunov inequality.