Attractor of Beam Equation with Structural Damping under Nonlinear Boundary Conditions. (22nd February 2015) - Wang, Danxia; Zhang, Jianwen; Wang, Yinzhu; Zhang, Sufang
Abstract : Simultaneously, considering the viscous effect of material, damping of medium, and rotational inertia, we study a kind of more general Kirchhoff-type extensible beam equationu t t - u x x t t + u x x x x - σ ( ∫ 0 l ‍ ( u x ) 2 d x ) u x x - ϕ ( ∫ 0 l ‍ ( u x ) 2 d x ) u x x t = q ( x ), i n [ 0, L ] × R + with the structural damping and the rotational inertia term. Little attention is paid to the longtime behavior of the beam equation under nonlinear boundary conditions. In this paper, under nonlinear boundary conditions, we prove not only the existence and uniqueness of global solutions by prior estimates combined with some inequality skills, but also the existence of a global attractor by the existence of an absorbing set and asymptotic compactness of corresponding solution semigroup. In addition, the same results also can be proved under the other nonlinear boundary conditions.