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

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.