关于非线性反应扩散方程全局吸引子的整体与局部几何拓扑结构的研究
摘要
在这篇博士学位论文中,我们主要研究了下列非线性反应扩散方程全局吸引子的整体与局部几何拓扑结构,得到了对全局吸引子几何拓扑结构的新的描述.其中Ω(?)RN是有界光滑区域.
假定f：Ω×R→R满足Caratheodory条件：
i)对每一个s∈R,函数F(·,s)关于Ω是Lebesgue可测的;
ii)对几乎所有的x∈Q,函数f(x,·)关于R是连续可微的.
另外,假定存在正常数Ci,1≤i≤4和整数p≥2,f满足下列增长条件：|f(x,s)|≤C1|s|p-1＋C2,对所有的(x,s)∈Ω×R, sf(x,s)≤-C3|s|p＋C4,对所有的(x,s)∈Ω×R, f'(x,s)≤(?),对所有的(x,s)∈Ω×R.其中Q(?)RN是有界光滑区域,(?)是-△算子的一列特征值,j=1,2,
假定f(u)是C1函数且满足下列假设|f'(s)|≤C1|s|p-2＋C2,p≥2, f(0)=f'(0)=0, f'≥一(?).全文共分五章：
第一章,介绍无穷维动力系统的理论和应用的背景,全局吸引子问题的发展及研究进展情况,总结全局吸引子存在性、维数估计和惯性流形的已有的理论和方法以及动力系统几何拓扑理论方面已有的成果.
第二章,给出了本文用到的一些基础知识.
第三章,主要研究了半线性反应扩散方程I当外力项g∈God时,God是相空间L2(Q)中的稠密子集(正则值集合),全局吸引子的整体几何拓扑结构,得到了对全局吸引子的新的刻画,也就是说,方程I的全局吸引子是平衡点的Lipschitz连续的不稳定流形的并,在一定程度上克服了方程I在惯性流形不存在时对全局吸引子的几何结构的刻画所带来的困难,这能很好地反映半线性反应扩散方程I的全局吸引子的整体几何拓扑结构.
第四章,主要研究了在第三章中得到的全局吸引子的代数和拓扑结构,通过充分考虑全局吸引子自身所具有的性质,受文献[111]中关于建立Witten复形理论的启发,在我们所得到的半线性反应扩散方程I的全局吸引子(?)上建立了Witten同调群.并证明了(?)具有CW复形结构,得到了Witten同调群、胞腔同调以及奇异同调群之间的同构关系,这给出了奇异同调群的一种有效的计算方法.最后,结合全局吸引子的Morse过滤结构和相对同调群理论,我们给出了全局吸引子的相对同调群的刻画,得到了Morse等式.
第五章,主要研究了一类具有任意阶多项式增长的非线性反应扩散方程Ⅱ的全局吸引子的局部几何拓扑结构,即如果方程Ⅱ的线性化方程的谱和虚轴相交时,我们所考虑的非线性反应扩散方程Ⅱ将出现中心流形,我们得到了中心流形定理.
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.
假定f：Ω×R→R满足Caratheodory条件：
i)对每一个s∈R,函数F(·,s)关于Ω是Lebesgue可测的;
ii)对几乎所有的x∈Q,函数f(x,·)关于R是连续可微的.
另外,假定存在正常数Ci,1≤i≤4和整数p≥2,f满足下列增长条件：|f(x,s)|≤C1|s|p-1＋C2,对所有的(x,s)∈Ω×R, sf(x,s)≤-C3|s|p＋C4,对所有的(x,s)∈Ω×R, f'(x,s)≤(?),对所有的(x,s)∈Ω×R.其中Q(?)RN是有界光滑区域,(?)是-△算子的一列特征值,j=1,2,
假定f(u)是C1函数且满足下列假设|f'(s)|≤C1|s|p-2＋C2,p≥2, f(0)=f'(0)=0, f'≥一(?).全文共分五章：
第一章,介绍无穷维动力系统的理论和应用的背景,全局吸引子问题的发展及研究进展情况,总结全局吸引子存在性、维数估计和惯性流形的已有的理论和方法以及动力系统几何拓扑理论方面已有的成果.
第二章,给出了本文用到的一些基础知识.
第三章,主要研究了半线性反应扩散方程I当外力项g∈God时,God是相空间L2(Q)中的稠密子集(正则值集合),全局吸引子的整体几何拓扑结构,得到了对全局吸引子的新的刻画,也就是说,方程I的全局吸引子是平衡点的Lipschitz连续的不稳定流形的并,在一定程度上克服了方程I在惯性流形不存在时对全局吸引子的几何结构的刻画所带来的困难,这能很好地反映半线性反应扩散方程I的全局吸引子的整体几何拓扑结构.
第四章,主要研究了在第三章中得到的全局吸引子的代数和拓扑结构,通过充分考虑全局吸引子自身所具有的性质,受文献[111]中关于建立Witten复形理论的启发,在我们所得到的半线性反应扩散方程I的全局吸引子(?)上建立了Witten同调群.并证明了(?)具有CW复形结构,得到了Witten同调群、胞腔同调以及奇异同调群之间的同构关系,这给出了奇异同调群的一种有效的计算方法.最后,结合全局吸引子的Morse过滤结构和相对同调群理论,我们给出了全局吸引子的相对同调群的刻画,得到了Morse等式.
第五章,主要研究了一类具有任意阶多项式增长的非线性反应扩散方程Ⅱ的全局吸引子的局部几何拓扑结构,即如果方程Ⅱ的线性化方程的谱和虚轴相交时,我们所考虑的非线性反应扩散方程Ⅱ将出现中心流形,我们得到了中心流形定理.
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.
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