Global Attractors and Their Dimensions for a Nonlinear Damped Wave Equation 

Author  JinBaoXia 
Tutor  YangZhiJian 
School  Zhengzhou University 
Course  Basic mathematics 
Keywords  Initial boundary value problem nonlinear damped wave equation dynamical systerm longtime behavior global attractor of solution 
CLC  O175.29 
Type  Master's thesis 
Year  2008 
Downloads  7 
Quotes  0 
In this paper,we are concercd with the long time behavior of solutions to the initial boundary value problem （IBVP） of the nonliner damped wave equationwhere x∈Ω,t∈R^{+},σ（s）=s（?）,s≥0,m≥1,Ωis a bounded domain in R^{N} with smooth boundary （?）Ωand v is the unit outward normal on （?）Ω, and the assumptions on g（u）, h（u_{t}） and f will be specified later.This paper consists of seven chapters. In the first chapter, we give an introduction and some signs used in the paper. In the second chapter, we study the existence and uniqueness of global solution to problem （1）（3）, In the third chapter, we investigate in X_{1} the existence of the global attractor A and its fractal and Hausdorff dimension , In the fourth chapter, we discuss the regularity of the solution in C(R^{+};V_{2+α})∩C^{1}（R^{+};V_{α}）（0≤α≤1）, In the fifth chapter, we will study the global attractor A and its fractal and Hausdorff dimension of in X_{2}, In the sixth chapter, we will study the global attractor A and its fractal and Hausdorff dimension in X_{3}, the seventh chapter, we will give some examples to show the cxisternce of the nonlinear functions g（u） and h（u_{t}）. The main results are the following:Theorem 1 Suppose that （H_{1}）g:V_{2}→V_{2}with0＜ρ＜2,G（s）=（?）,1≤m≤（?）（m＜∞）, and where （a）^{+} = max{0,a}, and（H_{2}）h=h_{1}+h_{2},h_{i}:V_{1}→V_{1}（i=1,2） and there exist constants 0＜δ_{1}＜1,θ_{1}∈（0,1/2）,β_{1}＞0 such thatThen problem （2.4）（2.5） admits a unique solution u∈C（R^{+};V_{2}）∩C^{1}（R^{+};H）and （u,u_{t}）depends continuously on initial data in X_{1}.Remark 1 （H_{1}） implies that for anyη＞0, there exist C_{η}and （?） such thatRemark 2 We denote the solution （u,u_{t}） in Theorem 1 by S（t）（u_{0}, u_{1}） = （u,u_{t}）. Then S（t） composses a C_{0} semigroup in X_{1}.Theorem 2 In addition to the assumptions of Theorem 1, if there exists 0＜δ_{2}＜1/2 andσ_{1}:0＜σ_{1}＜＜1 such that（H_{4}）（H_{5}）f∈V_{4σ11}and Then the continuous semigroup S（t） defined in Remark 2 possesses in X_{1} a global attractor which is connected and has finite fractal and Hausdorff dimension.Theorem 3 In addition to the assumptions of （H_{1}）（H_{2}） of Theorem 2.1, if also （H_{6}） The map G （see （4））:V_{2}→L^{1} and there exists constantsδ_{3}∈（0,1） such that（H_{7}）f∈V_{a1},（u_{0},u_{1}）∈V_{2+α}×V_{α}where0＜α≤1.Then problem （1.1）（1.3） admitsaunique solution C(R^{+};V_{2+α})∩C^{1}（R^{+};V_{α}） and （u, u_{t}） depends continuously on （u_{0},u_{1}） in X_{2}.Theorem 4 In addition to assumptions of Theorem 2, with 0＜α＜1, if there existσ_{2}: 0＜σ_{2}＜1αsuch that（H_{8}）f∈V_{1+α+σ2}Then the C_{0}semigroup S（t） defined in Remark 2.2 possesses in X_{2} a global attractor which is connected and has finite fractal and Hausdorff dimension.Theorem 5 Assumed that the assumptions of Theorem 3 hold forα=1,m≥2 and there existδ:0＜δ＜＜1such that（H_{9}）Then the C_{0} semigroup S（t） defined in Remark 2 possesses in X_{3} a global attractor whichis connected and has finite fractal and Hausdorff dimension.Remark 3 By assumption （H_{1}） we know thatm≥2 implies N≤4, especially m = 2as N=4.