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