Engery Decay Of Solution And Blow Up Of Solution For A Class Of Nonlinear Wave Equations With Viscosity

This paper consists of four chapters. The first chapter is the introduction. In the secondchapter,we will study engery decay of solution for the initial boundary value problem ofa class of nonlinear wave equations with viscosity. In the third chapter, we will study theexistence and uniqueness of the local solution for the above mentioned equation . In thefour chapter, we will study blow up of solution to the initial boundary value problem forthe above mentioned equation and give sufficient conditions of blow up of solution.In the second chapter, we discuss engery decay of solution of the following initialboundary value problem for a nonlinear wave equation with viscosity :whereμ＞0,δ＞0,p≥1,q＞1are constants,Ω=(0,1),Q_T=Ω×(0,T).u(x,t)denotesthe unknown function.σ(s)is a given nonlinear function, u₀(x)and u₁(x) are given initial value functions, subcripts x and t indicate partial derivatives. The main results are thefollowing:Theorem 1 Suppose the following conditions hold:(1)σ(s)∈C¹[0,∞) and there exist constants K_i＞0(i=1,2,3) such that 0≤σ(v²)≤K₁,|σ'(v²)|v²â‰¤K₂(σ'(s)=(?)σ(s)) and |σ(v₁²)-σ(v²)||v|+|σ(v²)-σ(v₁²)||v₁|≤K₃|v-v₁| forv,v₁∈R;(2)1＜q＜∞,1≤p＜∞andδ＞0;where K₄＞0andμ＞0are constants.If q≤p, then the problem (1)-(3)has a unique global generalized solution u(x,t) satisfyingandfor any w∈L²(0,T;H₀¹(Ω))∩L^p+1(Q_T).We employ the following lemma to get engery decay of solution of the problem (1)-(3).Lemma 1[9] Let h(t) : R⁺â†'R⁺ be a nonincreasing and derivative funtion.andassume that there exist constants r, a > 0 such thatthenwhere C only depended on h(0) is constant.Theorem 2 Suppose that the assumptions of theorem 1 are satisfied and u(x. t) is the generalized solution of the problem (1)-(3), If p = q = 5,σ(v²)v²ï¼ž(?) andμ＞0,thenwhere C＞0 only dependen E(0) is constant.Remark 1 The funtionσ(v²) satisfyingσ(v²)v²ï¼ž(?) exists.For exampleσ(v²)= (?).In the third chapter, we will study the existence and uniqueness of the local solutionfor the problem (1)-(3). Firstly, we will prove the existence and uniquness of solution forthe following linear initial boundary value problem:The result is given as follows:Theorem 3 Suppose that u₀∈H²(Ω),u₁∈H¹(Ω),f∈C([0.T];L²(Ω)),then the problem(1) -(3)has a unique generalized solution u∈C([0,T];H²(Ω)),u_t∈C([0.T];H¹(Ω))∩H²(Q_T).u_tt∈L²(Q_T), and the following estimation holdswhere C(T) is a constant depended on T.We apply the contraction mapping principle to prove the problem (1)-(3)admitting anunique local generalized solution, the result is the following:Theorem 4 Suppose that u₀∈H²(Ω),u₁∈H¹(Ω),σ∈C²(R),p≥1,q＞1,μ＞0,δ＞0,p, q,μandδare constants. If T is enough small for M, then the mapping from P(M, T)to P(M.T) is strictly contractive. In the four chapter, the blow up of solution to the initial boundary value problem (1)-(3) is proved. The result is the following:Theorem 5 Suppose the following conditions hold:(1)δ＞0,μ＞0,1≤p＜2,andq＞(?)(2)sσ(s)≤K(?)(s),(?)(s)≤-α|s|^?. where (?)(s)=(?),K＞2,α＞0 are constants,Then the solution of the problem(1)-(3) must blow up in finite time (?),i.e..whentâ†'(?),where, (?)...