Exponential Decay and Lack of Analyticity for the System of the Kirchhoff Love Plates and Membrane Like Electric Network Equation with Fractional Partial Damping.

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 $\theta\in[0,1]$, however $S(t)$ decays exponentially for $0\leq\theta\leq 1$.