Optimal semigroup regularity for velocity coupled elastic systems: a degenerate fractional damping case

Abstract

In this note, we consider an abstract system of two damped elastic systems. The damping involves the average velocity and a fractional power of the principal operator, with power $\theta$ in $[0,1]$. The damping matrix is degenerate, which makes the the regularity analysis more delicate. First, using a combination of the frequency domain method and multipliers technique, we prove the following regularity for the underlying semigroup: \begin{itemize} \item The semigroup is of Gevrey class $\delta$ for every $\delta>1/2\theta$, for  each $\theta$ in $(0,1/2)$. \item The semigroup is analytic for $\theta=1/2$. \item The semigroup is of Gevrey class $\delta$ for every $\delta>1/2(1-\theta)$, for  each $\theta$ in $(1/2,1)$.\end{itemize}  Next, we analyze the point spectrum, and derive the optimality of our regularity results. We also prove that the semigroup is not differentiable for $\theta=0$ or $\theta=1$. Those results strongly improve upon some recent results presented in \cite{ast}.