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Abstract. 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 L r -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.

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Abstract. The existence of time-periodic solutions is proven for a large class of reactiondiffusion systems in which Dirichlet boundary data, diffusivities, and reaction rates are periodic with common period.