Stability characterization for single local Kelvin–Voigt damping with degeneracy parameter α_i ≥ 1

Ascertain the asymptotic stability properties (e.g., exponential or polynomial decay or lack of uniform stability) of the semigroup e^{tA} corresponding to the Timoshenko beam system (equation (1.1)) with exactly one local Kelvin–Voigt damping (D1(x)=0 or D2(x)=0) when the degeneracy parameter α_i = sup_{0<x≤1} x|a_i′(x)|/a_i(x) of the nonzero damping coefficient satisfies α_i ≥ 1, a regime not covered by assumption (H2).

Background

The analysis in the paper covers the weakly degenerate case where the damping coefficient a_i satisfies (H2), implying 0 ≤ α_i < 1, and establishes t^{-1/2} decay for single local Kelvin–Voigt damping. The behavior in the stronger degeneracy regime α_i ≥ 1 is not addressed by the present assumptions.

Existing literature provides stability results for systems with both damping terms active and for various regularity conditions, but the single-damping case with α_i ≥ 1 remains uncharacterized, motivating a precise determination of the decay type and rate in this regime.