Buckling and longterm dynamics of a nonlinear model for the extensible beam

This work is focused on the longtime behavior of a nonlinear evolution problem describing the vibrations of an extensible elastic homogeneous beam resting on a viscoelastic foundation with stiffness k > 0 and positive damping constant. Buckling of solutions occurs as the axial load exceeds the first critical value, β c , which turns out to increase piecewise-linearly with k . Under hinged boundary conditions and for a general axial load P , the existence of a global attractor, along with its characterization, is proved by exploiting a previous result on the extensible viscoelastic beam. As P ≤ β c , the stability of the straight position is shown for all values of k . But, unlike the case with null stiffness, the exponential decay of the related energy is proved if P < β ̄ ( k ) , where β ̄ ( k ) ≤ β c ( k ) and the equality holds only for small values of k .