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
Fredy Maglorio Sobrado Suárez, Filomena Barbosa Rodrigues Mendes
TL;DR
This paper analyzes a coupled Kirchhoff-Love plate and membrane-like electrical network with higher-order coupling and fractional damping $(-\Delta)^{{\theta}}v_t$. Using semigroup theory and spectral analysis, it proves well-posedness and establishes exponential decay of the semigroup $S(t)=e^{\mathbb{B}t}$ for all $0\leq \theta \leq 1$, while revealing a dichotomy in analyticity: there is no analyticity for $\theta<1$ but analyticity holds at $\theta=1$. Moreover, for $0<\theta<1$ the semigroup is Gevrey of sharp class $s>1/\theta$, with sharpness shown via resolvent growth arguments. These results precisely quantify the regularity and decay properties of the coupled system, informing stability and damping design in applied plate-network models.
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 $γΔu_t$ and the Laplacian of the potential electric field $γΔv_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 $(-Δ)^θv_t$ between the potential electric $θ=0$ (frictional damping) and the Laplacian of the electric potential for $θ=1$ (damping Kelvin Voigt). We show that $S(t)=e^{\mathbb{B}t}$ is not analytic for $θ\in [0, 1[$ and analytic for $θ=1$, however $S(t)=e^{\mathbb{B}t}$ decays exponentially for $0\leq θ\leq 1$ and $S(t)$ is of Gevrey sharp class $s>\frac{1}θ$ when the parameter $θ$ lies in the interval $]0,1[$.