Stability and Lyapunov Functions for Reaction-Diffusion Systems

It is shown for a large class of reaction-diffusion systems with Neumann boundary conditions that in the presence of a separable Lyapunov structure, the existence of an a priori Lʳ estimate, uniform in time, for some $r>0$, implies the L^∞-uniform stability of steady states. The results are applied to a general class of Lotka--Volterra systems and are seen to provide a partial answer to the global existence question for a large class of balanced systems with nonlinearities that are not bounded by any polynomial.