Dynamic stability of equilibrium states of the extensible beam

Abstract

In this paper an equation describing the dynamic behavior of a nonlinear beam with viscous damping is treated. In particular it is shown that when the trivial solution is the only equilibrium solution then all solutions, regardless of intial data, decay exponentially to the trivial solution. In those cases where nontrivial equilibrium solutions in addition to the trivial solution are possible it is shown that the nontrivial solution corresponding to the ’lowest buckled mode’ is locally stable, i.e. dynamic solutions with initial data ’close’ to the lowest buckled mode decay to this equilibrium solution. Estimates are obtained for the various decay rates.