Local asymptotic stability for polyharmonic Kirchhoff systems†

Abstract

In this article we treat the question of local asymptotic stability for polyharmonic Kirchhoff systems, governed by time-dependent source forces and nonlinear damping terms. Even the simplest case (𝒫₃ ) studied here includes several systems of great interest, as the physical models for vibrating beams of the Woinowsky–Krieger type. This article extends and generalizes some local stability results of Autuori et al. [Asymptotic stability for anisotropic Kirchhoff systems, J. Math. Anal. Appl. 352 (2009), pp. 149–165] to higher order systems, and also of Nakao and Zhu [Decay rate of solutions of a wave equation with damping and external force, Nonlinear Anal. 46 (2001), pp. 335–345; An attractor for a nonlinear dissipative wave equation of Kirchho type, J. Math. Anal. Appl. 353 (2009), pp. 652–659].

Keywords:

• polyharmonic operator
• Kirchhoff systems
• local asymptotic stability
• qualitative properties of solutions

AMS Subject Classifications::

• Primary 35B40
• 35L75
• 35B35
• Secondary 35L55
• 35Q99

Acknowledgement

This research was supported by the Italian MIUR project titled Metodi Variazionali ed Equazioni Differenziali alle Derivate Paziali non Lineari.