Asymptotically stable stationary solutions to the reaction-diffusion equations

We assume that there exists an asymptotically stable stationary solution of a Galerkin approximation for the reaction-diffusion system. It is shown that there exists a nearby stationary solution of the full reaction-diffusion system provided the order of the Galerkin approximation is high enough. The Lyapunov second method is used to prove the asymptotic stability of the stationary solution.