Maximal Decay Rates and Asymptotic Behavior of Solutions in Nonlinear Elastic Structures

Abstract

In this paper we study the asymptotic behavior of systems arising in nonlinear elasticity. Of particular interest are problems related to quantitative description (in terms of the parameters in equations) of margin of stability achieved by implementation of a suitable nonlinear damping mechanism. To be more precise, we are interested in the following question: how and to which extent one can maximize the margin of stability by introducing a suitably large damping? Even in the case of a simple linear, constant coefficient problem with viscous damping, it is known that the maximum of decay’s rate depends not only on the damping coefficient (representing the dynamic properties of the model) but also on the static properties of the model (for instance: the first eigenvalue of the Laplacian-the case of wave equation). In fact, in a recent paper [4] it was shown that the exact decay rates in one-dimensional wave equation with viscous damping are determined by the location of the eigenvalue corresponding to the wave operator. The above result, for a one-dimensional linear wave equation, is rather technical and it is based on Riesz property of associated eigenfunction. This, in turn, requires the so called “gap condition” which is known to fail for higher dimensional problems.