The Initial Boundary Value Problems For A Class Of Damped Nonlinear Wave Equations Of Fourth Order

In this paper,we study the following initial boundary value problem for a class of damp nonlinear wave equations of fourth order:whereα,β> 0,γ< 0 are constants,u(x,t) denotes the unknown function,φ(s), f(s) and g(s) are given nonlinear functions,u₀(x) and u₁(x) are given initial value functions,subcripts x and t indicate partial derivatives.This paper consists of three chapters. The first chapter is the introduction. In the second chapter, we will study the existence and uniqueness of the local classical solution and the global classical solution for the initial boundary value problem (1)-(3) of a class of damped nonlinear wave equations of fourth order . In the third chapter, we will use the Komornik inequality to study the asymptotic behavior of the solution to the problem (1)-(3).Then the blow up of the solution to the problem (1)-(3) are proved by means of the cancavity method. The main results are the following:Theorem 1 Suppose that (?),then the initial boundary value problem(1)-(3) has a unique solution (?),where [0, T₀)is the maximal time interval of existence of u(x,t). Moreover,if then T₀ =∞.Theorem 2 Suppose that (?),且|f'(s)|≤c₀ (c₀ is a constant); g∈C¹(R) and g'(s) is bounded blow ,i.e.,there is a constant b₀,such that g'(s)≥b₀,(?)s∈R andwhere A, B > 0 are positive constants ,then the initial boundary value problem (1)-(3) has a unique global classical solution.Theorem 3 Assume that the following conditions are satisfied:(1)G(s)≥0, 2G(s)≤g(s)s, (?)s∈R;(2)ψ(s)≥0, 2ψ(s)≤φ(s)s, (?)s∈R;, then the classical solution of the problem (l)-(3) has the estimatewhere (?),Theorem 4 Assume that (?),there is a constantη> 0 such thatwhereThen the classical solution u(x,t) of the initial boundary value problem (1)-(3) blows up in finite time if one of the following conditions holds: (1)E(0)< 0;(2)E(0) = 0,(?);(3)E(0)>0,(?),where E(0) = (?).