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 θ in [0, 1], The damping matrix is degenerate, which makes 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: The semigroup is of Gevrey class δ for every δ > 1/2θ, for each θ in (0, 1/2). The semigroup is analytic for θ = 1/2. The semigroup is of Gevrey class δ for every δ > 1/2(1 — θ), for each θ in (1/2, 1). Next, we analyze the point spectrum, and derive the optimality of our regularity results. We also prove that the semigroup is not differentiable for θ = 0 or θ = 1. Those results strongly improve upon some recent results presented in Ammari et al. [J. Evol. Equ. 21 (2021) 4973-5002].