With Polynomial Non-linear Three-dimensional Reaction Diffusion Equations The Long Time Behavior Of The Finite Difference Scheme

In this paper, a three-dimensional reaction-diffusion equation with a polynomial nonlinearity that possesses a global attractor is considered.The existence of an attractor is one of the most important characteristics for a dissipative system,the long-time dynamics is completely determined by the attractor of the system.At present,most studies are linked to the semi-discrete scheme and the one-dimensional spatial variable.But,there are very few studies about the full-discrete scheme and the higher dimensional spatial variable. Furthermore,the former can't be extended directly to the latter.As the result,the latter is focused by the reseachers nowadays.The dynamical properties of the discrete dynamical system which is generated by a finite difference scheme are analysed.First,we construct the finite difference scheme of the equation,the existence and the uniqueness of the solution for the scheme are proved.Then,the existence of an absorbing set in the space of L_h~2-functions and H_h~1-functions is drawn.Therefore,the existence of an attractor of the equation in three-dimensional space is obtained by using the conclusion in[1].Finally,long-time stability and convergence of the finite difference scheme is proved.