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

TL;DR

This paper investigates the regularity and decay properties of semigroups associated with degenerate damped elastic systems, revealing how degeneracy affects analyticity, Gevrey class regularity, and decay rates across different damping parameters.

Contribution

It provides a detailed analysis of how degeneracy in damping influences semigroup regularity and stability, including new results on analyticity, Gevrey classes, and decay rates for interacting elastic systems.

Findings

Semigroup is not analytic for  in (1/2,1] but differentiable for  in (0,1).

Semigroup exhibits Gevrey class regularity for  in (0,1/2].

Exponential decay for  in [0,1], polynomial decay for  in [-1,0).

Abstract

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.