Some remarks on spatial uniformity of solutions of reaction-diffusion PDEs

In this paper, we present a condition which guarantees spatial uniformity for the asymptotic behavior of the solutions of a reaction-diffusion partial differential equation (PDE) with Neumann boundary conditions in one dimension, using the Jacobian matrix of the reaction term and the first Dirichlet eigenvalue of the Laplacian operator on the given spatial domain. The estimates are based on logarithmic norms in non-Hilbert spaces, which allow, in particular for a class of examples of interest in biology, tighter estimates than other previously proposed methods.