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. Two examples of application are provided.
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Ammari, K., Shel, F. & Tebou, L. Regularity and stability of the semigroup associated with some interacting elastic systems I: a degenerate damping case. J. Evol. Equ. 21, 4973–5002 (2021). https://doi.org/10.1007/s00028-021-00738-7
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DOI: https://doi.org/10.1007/s00028-021-00738-7