Asymptotic stability of stationary solutions for the Kirchhoff equation

Abstract

This paper considers nonlinear Kirchhoff equation with Kelvin–Voigt damping. This model is used to describe the transversal motion of a stretched string. The existence of smooth stationary solutions of nonlinear Kirchhoff equation has been studied widely. In the present contribution, we prove that a class of stationary solutions is asymptotic stable by overcoming the “loss of derivative” phenomenon causing from the Kirchhoff operator. The key point is to find a suitable weighted function when we carry out the energy estimate for the linearized equation.