Initial Boundary Value Problem For An Nonlinear Beam Equation With Structural Damping

In this paper we study the existence and the asymptotic behavior of global weak solution of the following initial boundary value problems for an beam equation with structural damping:where are constants, are positive integers.In Chapter 2,by constructing the modified potential well W associated with (1)-(3) and using a new Gronwall type integral inequality,we obtain the global weak solution for the problem (1)-(3) with r>0,by applying Galerkin method. The main resut is the following:Theorem 1 Suppose that andthen the problem (1)-(3) has global weak solution.In Chapter 3, we get the asymptotic behavior of the global weak solution of the problem(1)-(3) with r>0 by using Nakao difference inequality. The main reslut reads as:Theorem 2 Suppose that the problem (1) - ( 3 ) has global weak solution, andwhere [t-1] + = max{t-1,0} ,and k, (E(0)) are positive constant depending on || uox|| and || MI || .In Chapter 4 , we prove the existence and uniqueness of the global weak solution for the problem (1)-(3) with r<0 by Galerkin method. The main result is the following:Theorem 3 Suppose that r<0, , then the problem (1)-(3) has the unique global weak solution.