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>0k>0 and positive damping constant. Buckling of solutions occurs as the axial load exceeds the first critical value, βcβc, which turns out to increase piecewise-linearly with kk. Under hinged boundary conditions and for a general axial load PP, the existence of a global attractor, along with its characterization, is proved by exploiting a previous result on the extensible viscoelastic beam. As P≤βcP≤βc, the stability of the straight position is shown for all values of kk. But, unlike the case with null stiffness, the exponential decay of the related energy is proved if P<β̄(k), where β̄(k)≤βc(k) and the equality holds only for small values of kk.