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Green’s function approach to a large time behavior problem for a coupled system of 1-D wave and plate equations

We study a system composed of a 1-D damped wave equation and a 1-D undamped plate equation, in which the energy of the two parts can be transmitted through the boundary. We show that the spectrum of the infinitesimal generator of the semigroup associated to a system equivalent to the concerned system is contained in the open left half complex plane, and prove by establishing a resolvent estimate on the afore-mentioned infinitesimal generator that the energy of the system under consideration decays at a logarithmic rate. Green’s function and Young’s inequality are the two main ingredients in proving the afore-mentioned resolvent estimate.