Galerkin approximations to the nonautonomous periodic reaction–diffusion equations

Abstract

REACTION-diffusion equations play an important role in modelling many chemical and biological phenomena. They have been studied in recent years by many researchers. A common topic of investigation is the behavior of solutions as the time tends to infinity. Owing to the structure of the nonlinear term and the magnitude of the diffusion a vast variety of different phenomena occur, which are hard to study by analytical methods. The idea is to replace the original reaction-diffusion equation by some more simple one, say, a finite-dimensional Galerkin approximation, which captures some properties of the reaction-diffusion equation as t + 00. For the Navier-Stokes equations Constantin et al. [l] showed that if a Galerkin equation of high order has an asymptotically stable stationary solution, then there exists a nearby asymptotically stable stationary solution to the Navier-Stokes equation. Kloeden [2], using Lyapunov’s second method, obtained some results from [l], considerably simplifying their proofs. Titi [3] obtains the existence of a stable time-periodic solution to the NavierStokes equation assuming the existence of a stable time-periodic solution to the Galerkin education of sufficiently high order. Dikansky considered reaction-diffusion equations and in [4, 51 he obtained the existence of an asymptotical stable stationary solution and in [6] an asymptotically orbitally stable time-periodic solution to an autonomous system from the assumption that a Galerkin equation of high order possesses such solutions. The purpose of this article is to show that the existence of an asymptotically stable periodic solution to a nonautonomous periodic Galerkin equation of sufficiently high order implies the existence of a nearby asymptotically stable periodic solution to the boundary value problem for the reaction-diffusion system. The idea of the proof is as follows. One projects the reactiondiffusion system on subspaces PL, and QL2, where PL, is a finite-dimensional subspace, spanned by the first m eigenfunctions of the operator DA with corresponding boundary conditions and Q = I P. The Q-part of the solution q, q E QL2, corresponding to the eigenfrequences 5 A,:, represents the small structures. This allows the technique for perturbed ordinary differential equations to be used (see [7, 81).