Regularity of the semigroups associated with some damped coupled elastic systems II: a nondegenerate fractional damping case

TL;DR

This paper investigates the regularity of semigroups associated with damped coupled elastic systems involving nondegenerate fractional damping, establishing conditions for analyticity and Gevrey class regularity.

Contribution

It extends previous work by analyzing nondegenerate fractional damping cases, proving analyticity for higher fractional powers and Gevrey regularity for lower ones.

Findings

Semigroup is analytic for $1/2 \,\leq\, \mu, \theta \leq 1$.

Semigroup belongs to certain Gevrey classes when $\min(\mu, \theta) \in (0, 1/2)$.

Provides applications illustrating theoretical results.

Abstract

In this paper, we examine regularity issues for two damped abstract elastic systems; the damping and coupling involve fractional powers $\mu, \theta$, with $0 \leq \mu , \theta \leq 1$, of the principal operators. The matrix defining the coupling and damping is nondegenerate. This new work is a sequel to the degenerate case that we discussed recently in \cite{kfl}. First, we prove that for $1/2 \leq \mu , \theta \leq 1$, the underlying semigroup is analytic. Next, we show that for $\min(\mu,\theta) \in (0,1/2)$, the semigroup is of certain Gevrey classes. Finally, some examples of application are provided.