Attractor For The Nonlinear Coupling Beam System With Structural Damping

Existence and asymptotic behavior are studied for the solution of the nonlinear coupling beam system with structural damping 、 under the initial conditions and two different boundary conditions. Here<Φ(ω1,ω2,ω3,ω4)={β+k(|ω1|+|ω2|)+ σ ∫0lω1ω3dx)}w4, Ω= (0,1), l is the length of the beam. The constants α1, α2, β, k, δ,γ are positive, and a is nonnegative. The existence, uniqueness and continuous depen-dence upon data of weak solution, strong solution and classical solution for the initial-boundary problem are proved by Galerkin method. Moreover, the global attractor is obtained by semigroup theory. The full text structure is as follows:The first chapter briefly introduces the domestic and international development, the main content and the meaning of this paper.The second chapter lists some basic knowledge used in this paper, including the basic function spaces, definitions and lemmas.The third chapter proves the existence, uniqueness and continuous dependence up-on data of weak solution, strong solution and classical solution for the initial problem (1)-(2) under the hinged boundary conditionsThe fourth chapter proves the existence,uniqueness and continuous dependence upon data of weak solution, strong solution and classical solution for the initial problem (1)-(2) under the clamped boundary conditions To our attention, the transform of the boundary conditions makes the vary of the spaces in which we discuss the problem.The fifth chapter proves the global attractor of the problem (1)-(3) when σ= 0.