Optimality of the t^{-1/2} decay for single local weakly degenerate Kelvin–Voigt damping

Determine whether the polynomial decay rate t^{-1/2} proved for the semigroup e^{tA} associated with the Timoshenko beam system (equation (1.1)) with exactly one local Kelvin–Voigt damping (i.e., D1(x)=0 or D2(x)=0) and a damping coefficient satisfying assumptions (H1)–(H2) is optimal in the sense that no faster uniform decay rate holds for all initial data in D(A).

Background

The paper studies the Timoshenko beam system with local Kelvin–Voigt damping acting either on the shear stress or the bending moment. Under the weak degeneracy assumptions (H1)–(H2) on the nonzero damping coefficient and with the other coefficient equal to zero, the authors prove that the associated semigroup is polynomially stable with decay rate t^{-1/2}.

Prior work has shown various decay behaviors depending on damping placement and regularity, including exponential or polynomial decay for systems with both dampings active. However, for the single local Kelvin–Voigt damping considered here, whether the established t^{-1/2} rate is the fastest possible remains unsettled.