On Spatially Uniform Behavior in Reaction-Diffusion PDE and Coupled ODE Systems

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. We then present numerical procedures for the verification of this Lyapunov inequality and illustrate them on models of several biochemical reaction networks. Finally, we derive an analog of this PDE result for the synchronization of a network of identical ODE models coupled by diffusion terms.