The Global And Local Geometric And Topological Structure Of Global Attractor For Nonlinear Reaction-Diffusion Equations

In this doctoral dissertation,we study the global and local geometric and topological structure for global attractor of the following nonlinear reaction-diffusion equations, whereΩis a smooth bounded domain of RN.Assume that f:Ω×R→R satisfies the Caratheory conditions:i) For each s∈R,the function f(.,s)is Lebesgue measurable inΩ；ii) For almost every x∈Ω,the function f(x,.)is continuously differentiable functions in R.Assume further that there are positive constants Ci,for 1≤i≤4,and integer p≥2, f satisfies the following conditions:|f(x,s)|≤C1|s|p-1＋C2, for(x,s)∈Ω×R,sf(x,s)≤一C3|s|p＋C4, for(x,s)∈Ω×R,f'(x,s)≤(?), for(x,s)∈Ω×R.whereΩis a smooth bounded domain of RN,λj are eigenvalues of一△,j= 1,2,Assume that f(u)is C1 function satisfying the following hypotheses, |f'(s)|≤C1|s|p-2＋C2,p≥2, f(0)=f'(0)= 0, f'≥-(?).This thesis consists of five chapters.In Chapter 1, we introduce the background of the theory and its applica-tions of infinite dimensional dynamical systems, and the evolution of global attractor, and then, the method and theory of the existence of global attrac-tor, dimensional estimate, inertial manifold, and the basic theory of geometry and topology of dynamical systems are listed in this chapter.In Chapter 2, some preliminary results and definitions that we will used in this thesis are presented.In Chapter 3, we mainly study the global geometric and topological struc-ture of the corresponding global attractor for semilinear reaction-diffusion equationsⅠwhen the forcing term g belongs to God, where God is an open and dense subset (regular value set) of the phase space L2(Ω), that is, the global attractor for equationsⅠcan be decomposed into the union of Lipschitz continuous manifold of equilibrium points. In some sense, we overcome some difficulties to describe the geometric structure of global attractor when the in-ertial manifold does not exist for equationsⅠ. In this way our decomposition for global attrator of equationsⅠgives a good description on the geometric and topological structure of global attractor for semilinear reaction-diffusion equationsⅠ.In Chapter 4, we mainly study the algebraic and topological structure of global attractor obtained in Chapter 3. We are motivated by the refer-ence [111], in which Witten gave a beautiful method to establish Witten com-plex. Based on the theory of Witten complex, we establish Witten homology group on the global attractor (?) of the reaction-diffusion equations I when g∈God, and prove the global attractor (?) possesses the structure of CW complex. We derive the isomorphism relation between any two of Witten ho-mology group, cell homology group and singular homology group, which give an efficient tool to calculate the singular homology group. Last, by using the structure of Morse filtration and the theory of relative homology group, we give a description of relative homology group to the global attractor (?) and obtain the corresponding Morse's equation. In Chapter 5, we mainly study the local geometric and topological struc-ture of global attractor for nonlinear reaction-diffusion equationsⅡwith a polynomial growth nonlinearity of arbitrary order, that is, if the spectrum of the linearized equation of equationsⅡmeet the imaginary axis, then evolution equationsⅡwill occur the center manifold. So we derive the center manifold theorem for the equationsⅡ.