Asymptotic behavior for coupled Kirchhoff/Euler-Bernoulli plates with fractional damping

In this paper, we study the asymptotic behavior of a coupled system of Kirchhoff plate and Euler-Bernoulli plate with fractional damping. The dissipation mechanism is given by the fractional damping $ (-\triangle)^{2\theta_1}u_t $ or $ (-\triangle)^{2\theta_2}v_t $, $ \theta_i\in [0,1] $, $ i = 1,2 $, which is acting on Kirchhoff plate and Euler-Bernoulli plate respectively. These dampings act either on both plate equations or only on one plate equation, and we show that the system decays to zero either exponentially or polynomially. By comparing the three situations, we find some surprising results. In the presence of both fractional dampings, the stability result of system is determined only by the fractional damping $ (-\triangle)^{2\theta_1}u_t $ acting on Kirchhoff plate, while the system is exponentially stable for $ \theta_1\in[1/2,1] $ and is polynomially stable with the rate $ t^{-1/(2-4\theta_1)} $ for $ \theta_1\in[0,1/2) $. In the presence of only one fractional damping, we show that the decay rate of fractional damping acting only on the Euler-Bernoulli equation is more slowly than that of fractional damping acting only on the Kirchhoff equation. Furthermore we prove that these decay rates are optimal.