We consider a damped plate model with rotational forces in a bounded domain. The plate is either clamped or hinged. The rotational forces and damping involve the spectral fractional Laplacian with powers \(\theta \) in [0, 1] and \(\delta \) in [0, 2], respectively. Using the frequency domain approach, and appropriate interpolation inequalities, we show that the underlying semigroup is: (a) analytic for all \(\theta \) in [0, 1] and \(\delta \) in \([(2+\theta )/2,2]\), (b) not analytic for all \(\theta \) in [0, 1] and \(\delta \) in \((\theta ,(2+\theta )/2)\), (c) of Gevrey class \(\alpha >(2-\theta )/2(\delta -\theta )\) for all \(\theta \) in [0, 1] and \(\theta<\delta <(2+\theta )/2\). We also prove that for each admissible value of \(\theta \), the semigroup is exponentially stable for \(\delta \ge \theta \), and only polynomially stable, with rate \(O(t^{-{2-\theta \over 2(\theta -\delta )}})\), for \(\delta <\theta \), when \(\theta >0\). In particular, in the hinged plate case, we show that the polynomial decay rate is optimal.

我们考虑在有界域中具有旋转力的阻尼板模型。该板被夹紧或铰接。旋转力和阻尼涉及分别在 [0, 1] 和\(\delta \)在 [0, 2] 中具有幂\(\theta \)的谱分数拉普拉斯算子。使用频域的方法，以及适当的内插不等式，我们表明，潜在的半群是：（1）分析所有\（\ THETA \）在[0，1]和\（\三角洲\）在\（[（2- +\theta )/2,2]\) , (b) 不解析[0, 1] 中的所有\(\theta \)和\((\theta ,(2+\theta )中的\( \delta \) )/2)\) , (c) 的 Gevrey 类\(\alpha >(2-\theta )/2(\delta -\theta )\)对于[0, 1] 和\(\theta<\delta <(2+\theta )中的所有\(\theta \) )/2\)。我们还证明，对于\(\theta \) 的每个可容许值，半群对于\(\delta \ge \theta \)是指数稳定的，并且只有多项式稳定，速率\(O(t^{-{2 -\theta \over 2(\theta -\delta )}})\)，对于\(\delta <\theta \)，当\(\theta >0\)。特别是，在铰链板的情况下，我们表明多项式衰减率是最佳的。