Decay estimates for the Cauchy problem for the damped extensible beam equation

Abstract

The extensible beam equation proposed by Woinovsky-Krieger [13] is a fourth order dispersive equation with nonlocal nonlinear terms. In this paper we study the Cauchy problem of the extended model by Ball who proposed the following model with external and structural damping terms:For η > 0 this represents a Kelvin-Voigt damping. We show the unique global existence of solution for this problem and give a precise description of the decay of solutions in time.

Singular limits in the Cauchy problem for the damped extensible beam equation

“…For the physical background of this model we refer to [1]. For the initial value problem the authors proved in [8] global existence and decay estimates for the solution as follows: …”