Study On The Attractors And Dimensional Estimates Of Two Nonlinear Evolution Equations

In this paper, we review the work in infinite dimensional dynamical systems during past years and give an overview of this active field up to now. Based on current knowledge, we mainly discussed two problems as follows.(1) We consider the global attractors for the following pattern formation equationsIn paper [19], the author deals with the caseΩ(?)R~2, g (u ) = u~3 +βu~2- (r + 1)u.We now generalize the nonlinearity to a larger category and can get the conclusion that the global attractor exists after a series sophisticated estimates by the means of interpolation inequity and embedding theorems in Sobolev spaces. Furthermore, by using Sobolev-Lieb-Thirring inequity, the dimensional estimate of the global attractor is obtained in the case n is less than three.(2) We consider the existence of global attractor for P-Laplace equation in unbounded domain R nAccording to the method used by Wang[12], the theory of monotone operator and Gateaux differentiable functional and a series of prior estimates, we prove that the semigroups were asymptotically compact and the global attractor exists.