Regularity and stability of the semigroup associated with some interacting elastic systems I: A degenerate damping case

In this paper, we examine regularity and stability issues for two damped abstract elastic systems. The damping involves the average velocity and a fractional power $\theta$, with $\theta$ in $[-1,1]$, of the principal operator. The matrix operator defining the damping mechanism for the coupled system is degenerate. First, we prove that for $\theta$ in $(1/2,1]$, the underlying semigroup is not analytic, but is differentiable for $\theta$ in $(0,1)$; this is in sharp contrast with known results for a single similarly damped elastic system, where the semigroup is analytic for $\theta$ in $[1/2,1]$; this shows that the degeneracy dominates the dynamics of the interacting systems, preventing analyticity in that range. Next, we show that for $\theta$ in $(0,1/2]$, the semigroup is of certain Gevrey classes. Finally, we show that the semigroup decays exponentially for $\theta$ in $[0,1]$, and polynomially for $\theta$ in $[-1,0)$. To prove our results, we use the frequency domain method, which relies on resolvent estimates. Optimality of our resolvent estimates is also established. Several examples of application are provided.