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, \beta_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<\beta_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<\bar\beta(k), where \bar\beta(k) < \beta_c(k) and the equality
holds only for small values of k.Comment: 14 pages, 2 figure
