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

This work is focused on the longtime behavior of a non linear 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<\barβ(k), where \barβ(k) < β_c(k) and the equality holds only for small values of k.

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