Volume 28, 2022
|Number of page(s)||20|
|Published online||07 July 2022|
Optimal semigroup regularity for velocity coupled elastic systems: a degenerate fractional damping case
Computer Science Department, Stanford University, Stanford, CA 94305, USA
2 Department of Mathematics and Statistics, University of Minnesota, Duluth, MN 55812, USA
3 Department of Mathematics and Statistics, Florida International University, Modesto Maidique Campus, Miami, FL 33199, USA
* Corresponding author: email@example.com
Accepted: 12 May 2022
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].
Mathematics Subject Classification: 35Q74 / 35B35 / 35B65 / 47D06
Key words: Regularity / stability / fractional damping / interacting elastic systems
© The authors. Published by EDP Sciences, SMAI 2022
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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