2021/02/25 by Kaïs Ammari, Ammari, K., Farhat Shel +3 · 2 citations
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics #Numerical methods in inverse problems #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2102.13217
openalex publication_date 2021/02/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 θ, with θ 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 θ in (1/2,1], the underlying semigroup is not analytic, but is differentiable for θ in (0,1); this is in sharp contrast with known results for a single similarly damped elastic system, where the semigroup is analytic for θ 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 θ in (0,1/2], the semigroup is of certain Gevrey classes. Finally, we show that the semigroup decays exponentially for θ in [0,1], and polynomially for θ 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.