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 potential electric field $\gamma\Delta v_t $, here only one of the equations is conservative, and the other has dissipative properties. The mechanism was dissipative is given by an intermediate damping $(-\Delta)^\theta v_t$ between the potential electric $\theta=0$ (frictional damping) and the Laplacian of the electric potential for $\theta=1$ (damping Kelvin Voigt). 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 sharp class $s>\frac{1}{\theta}$ when the parameter $\theta$ lies in the interval $]0,1[$.