Polynomial stability of transmission system for coupled Kirchhoff plates

Abstract

In this paper, we study the asymptotic behavior of transmission system for coupled Kirchhoff plates, where one equation is conserved and the other has dissipative property, and the dissipation mechanism is given by fractional dam** $(-\Delta )^{2\theta }v_t$ with $\theta \in [\frac{1}{2},1]$. By using the semigroup method and the multiplier technique, we obtain the exact polynomial decay rates, and find that the polynomial decay rate of the system is determined by the inertia/elasticity ratios and the fractional dam** order. Specifically, when the inertia/elasticity ratios are not equal and $\theta \in [\frac{1}{2},\frac{3}{4}]$, the polynomial decay rate of the system is $t^{-1/(10-4\theta )}$. When the inertia/elasticity ratios are not equal and $\theta \in [\frac{3}{4},1]$, the polynomial decay rate of the system is $t^{-1/(4+4\theta )}$. When the inertia/elasticity ratios are equal, the polynomial decay rate of the system is $t^{-1/(4+4\theta )}$. Furthermore it has been proven that the obtained decay rates are all optimal. The obtained results extend the results of Oquendo and Suárez (Z Angew Math Phys 70(3):88, 2019) for the case of fractional dam** exponent $2\theta$ from [0, 1] to [1, 2].