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

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Abstract

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, βc, 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 Pβc, 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<β̄(k), where β̄(k)βc(k) and the equality holds only for small values of k.

Keywords

Extensible elastic beam
Absorbing set
Exponential stability
Global attractor

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