Existence Of Global Attractor For Partly Dissipative Reaction Diffusion System In Unbounded Domain

Dynamical system is an active branch of mathematics. It is an important researchobject as well as an important research tool of nonlinear science. After its developmentfor about half a century,mathematicians have already established its basic theory frame-works. One major problem that dynamical system deals with is the asymptotical behaviorof the evolution operator,which we also call the dynamical complexity. As for the quali-tative theory of partial equations,the critial work is to obtain the prior estimates for thesolutions while time is large enough.In this paper, we will analyse the dissipative mechanism of the partly dissipativereaction-di?usion dynamical system on Rn and study its asymptotical behavior. Firstly,we obtain the uniform prior estimates with large time of the solutions'norm with thehelp of inner production and classical inequality theory. Secondly,we approximate theunbounded domain with bounded balls and prove that there exists a ball large enoughsuch that the approximation error of the solutions'norm is arbitrary small uniformly forlarge time. lastly,we prove the asymptotical compactness of the evolution operator by thetheorem of Sobolev compact embedding in bounded domain and establish the existenceof the global attractor of the partly dissipative reaction-di?usion dynamical system inthis paper.