Global Attractors And Their Dimensions For A Nonlinear Damped Wave Equation

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 X₁ 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)∩C¹(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 X₂, In the sixth chapter, we will study the global attractor A and its fractal and Hausdorff dimension in X₃, 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 (H₁)g:V₂�'V_-2with0＜ρ＜2,G(s)=(?),1≤m≤(?)(m＜∞), and where (a)⁺ = max{0,a}, and(H₂)h=h₁+h₂,h_i:V₁�'V_-1(i=1,2) and there exist constants 0＜δ₁＜1,θ₁∈(0,1/2),β₁＞0 such thatThen problem (2.4)-(2.5) admits a unique solution u∈C(R⁺;V₂)∩C¹(R⁺;H)and (u,u_t)depends continuously on initial data in X₁.Remark 1 (H₁) implies that for anyη＞0, there exist C_ηand (?) such thatRemark 2 We denote the solution (u,u_t) in Theorem 1 by S(t)(u₀, u₁) = (u,u_t). Then S(t) composses a C₀- semigroup in X₁.Theorem 2 In addition to the assumptions of Theorem 1, if there exists 0＜δ₂＜1/2 andσ₁:0＜σ₁＜＜1 such that(H₄)(H₅)f∈V_4σ1-1and Then the continuous semigroup S(t) defined in Remark 2 possesses in X₁ a global attractor which is connected and has finite fractal and Hausdorff dimension.Theorem 3 In addition to the assumptions of (H₁)-(H₂) of Theorem 2.1, if also (H₆) The map G (see (4)):V₂�'L¹ and there exists constantsδ₃∈(0,1) such that(H₇)f∈V_a-1,(u₀,u₁)∈V_2+α×V_αwhere0＜α≤1.Then problem (1.1)-(1.3) admitsaunique solution C(R⁺;V_2+α)∩C¹(R⁺;V_α) and (u, u_t) depends continuously on (u₀,u₁) in X₂.Theorem 4 In addition to assumptions of Theorem 2, with 0＜α＜1, if there existσ₂: 0＜σ₂＜1-αsuch that(H₈)f∈V_-1+α+σ2Then the C₀-semigroup S(t) defined in Remark 2.2 possesses in X₂ 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(H₉)Then the C₀ semigroup S(t) defined in Remark 2 possesses in X₃ a global attractor whichis connected and has finite fractal and Hausdorff dimension.Remark 3 By assumption (H₁) we know thatm≥2 implies N≤4, especially m = 2as N=4.