Attractors For Nonlinear Parabolic Models With Delays

This thesis is a survey of the recent results in investigating the attractors problems of nonlinear parabolic equations with delays. We mainly overview the results of this kind of problems and concentrate on some important problems, such as autonomous systems, non-autonomous systems and 2D Navier-Stokes equations.This thesis consists of 3 Chapters. In the first Chapter, by reviewing the theory of dynamic systems and attractors and their development, we introduce the existence problems of attractors for nonlinear parabolic equations with delays.In chapter 2, we introduce some basic definitions and important theories including basic inequalities,semi-group theory,process theory and attractor theory.In chapter 3, we introduce the method of proving the existence of attractors for the nonlinear parabolic system with delays in different systems.We first consider the autonomous systemWe will list four sufficient conditions to the system and corresponding results.(1)The delay is state selective delay, i.e.,Assume thatξis bounded and Lipschitz continuous in the second and third coordinate.b: R→R is locally Lipschitz and satisfies│b(w)│≤C1│w│+C2, C1≥0, C2≥0, w∈R.fΩ-Ω→Ris bounded,then the system has a global attractor.(2)The delay is discrete state-dependent delay,ⅰ.e.,Assume thatη:H→[0, r]is locally Lipschitz and b:R→Ris locally Lipschitz and bounded, i.e.,there exists a constant C,such that│b(w)│≤C, w∈R,f:Ω-Ω→Ris bounded, then the system has a trajectory attractor. (3)The delay is discrete state-dependent delay i.e.,Assume thatη(·):C([-r,0];L2(Ω))→[0,r]and it satisfies the following "ignores" assumption "ignores" assumption Assume that there existsη0>0,(?)θ(-η0,0],such thatη"ignores"values ofφ(θ),i.e. (?)φ1,φ2∈C,(?)θ∈[-r,-η0],φ1(θ)=φ2(θ) thenη(φ3)=η(φ2).then the system has a global attractor.(4)The delay is state-dependent delay,i.e.,We assume that B andηsatisfies the following assumption:(H.B)The operator Bis Lipschitz:Assume that (H.B), (H.η)hold, then we can obtain the existence of global attractor by constructing a wider space.In non-autonomous system, we consider the following equation with external forceWe mainly consider the external force belong to different spaces and introduce the re-sults to the existence of attractors.(a)Assume that g∈Lloc2(R,L2(Ω))is translation compact, Fis locally Lipschitz, and there exists k1, k2,k3≥0,(?)ξ∈LH2,η∈H,such thatthen the system has a uniform attractor.(b)Assume thatgisα-translation exponential,i.e.(?)t∈R,there exists C0,α,such that∫t-1t│g(s)│2 ds≤C0eα│t│,then the system has a pullback attractor.(c))Assume thatgis integral absolutely continuous and translation bounded.ⅰ.e.And F is Lipschitz,then the system has a pullback attractor.Finally we introduce a 2D-Navier-Stokes equationBy constructing Hilbert spaces, we formulate the operator form of the system. Then we can obtain the existence of attractor by using semi-group theory. We consider two different problems including 2D-Navier-Stokes equation with delays on unbounded domains and 2D-Navier-Stokes equation with an infinite delay. We will find that pullback and global attractors lead to the same object when the system becomes autonomous in the second problems.