Long-time behavior of a model of extensible beams with nonlinear boundary dissipations

This paper is concerned with the long-time behavior of a model of extensible beams u t t + u x x x x − M ( ∫ 0 L | u x | 2 d x ) u x x = h , defined in a bounded interval ( 0 , L ) , where M is a continuous nonnegative function and h is a static load. To this problem little is known with nonlinear boundary conditions. One considers the case where a nonlinear forcing interacts with the shear force at the boundary. Then the existence of a global attractor is proved with a sole boundary damping.