Indirect stabilization of weakly coupled Kirchhoff plate and wave equations with frictional damping

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:474

Article

Hajej, Ahmed¹  Hajjej, Zayd²  Tebou, Louis³ 

[1] Univ Cergy Pontoise, CNRS, UMR 8088, Dept Math, 2 Ave Adolphe Chauvin, F-95302 Cergy Pontoise, France

[2] Univ Gabes, Fac Sci Gabes, Dept Math, Gabes 6029, Tunisia

[3] Florida Int Univ, Dept Math & Stat, Modesto Maidique Campus, Miami, FL 33199 USA

关键词: Kirchhoff plate equation;    Wave equation;    Weakly coupled equations;    Frictional damping;    Free boundary conditions;   

DOI  :  10.1016/j.jmaa.2019.01.046

来源: Elsevier

【 摘 要 】

We investigate the indirect stabilization of a weakly coupled system consisting of a Kirchhoff plate equation involving free boundary conditions and the wave equation with Dirichlet boundary conditions in a bounded domain. The distributed damping is frictional and appears in one of the equations only. First, we consider the case where the damping occurs in the wave equation, and using the frequency domain method combined with an interpolation inequality, we derive a polynomial decay estimate for the associated semigroup. Afterwards, we consider the case where the frictional damping occurs in the Kirchhoff plate equation, and we use the same technique to derive a polynomial decay estimate for the underlying semigroup. That latter polynomial decay estimate is similar to the one obtained earlier by different authors in the case of a coupling between a frictionally damped Euler Bernoulli plate equation and an undamped wave equation, which is quite surprising since the operator defining the damping in the Kirchhoff plate equation is compact. (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

Free