Indirect stabilization of locally weakly coupled second‐order evolution systems of hyperbolic type

Abstract

The purpose of this work is to investigate the stabilization of locally weakly coupled second‐order evolution equations of hyperbolic type, where only one of the two equations is directly damped. As such system cannot be exponentially stable, we are interested in polynomial energy decay rates. Our main contributions concern strong abstract and polynomial stability properties based on stability properties of two auxiliary problems: the sole damped equation and the second equation with a damping related to the coupling operator. The main novelty is that the polynomial energy decay rates are obtained in different important situations not covered before; let us mention the case of a coupling operator not partially coercive and not necessarily bounded. The main tools are the use of the frequency domain approach combined with new multipliers technique based on the solutions of the resolvent equations of the two auxiliary problems mentioned before. Finally, our abstract framework is illustrated by several concrete examples not treated before.