A Gevrey class semigroup, exponential decay and Lack of analyticity for a system formed by a Kirchhoff-Love plate equation and the equation of a membrane-like electric network with indirect fractional damping

Abstract: The emphasis in this paper is on the Coupled System of a Kirchhoff-Love Plate Equation with the Equation of a Membrane-like Electrical Network, where the coupling is of higher order given by the Laplacian of the displacement velocity $\gamma\Delta u_t$ and the Laplacian of the electric potential field $\gamma\Delta v_t $, here only one of the equations is conservative and the other has dissipative properties. The dissipative mechanism is given by an intermediate damping $(-\Delta)^\theta v_t$ between the electrical damping potential for $\theta=0$ and the Laplacian of the electric potential for $\theta=1$. We show that $S(t)=e^{\mathbb{B}t}$ is not analytic for $\theta\in[0, 1)$ and analytic for $\theta=1$, however $S(t)=e^{\mathbb{B}t}$ decays exponentially for $0\leq \theta\leq 1$ and $S(t)$ is of Gevrey class $s> \frac{2+\theta}{\theta}$ when the parameter $\theta$ lies in the interval $(0,1)$.