Stability analysis of abstract systems of Timoshenko type

We consider an abstract system of Timoshenko type $\begin{cases} ρ_1{\ddot φ} + a A^{\frac12}(A^{\frac12}φ+ ψ) =0\\ ρ_2{\ddot ψ} + b A ψ+ a (A^{\frac12}φ+ ψ) - δA^γθ = 0\\ ρ_3{\dot θ} + c Aθ+ δA^γ{\dot ψ} =0 \end{cases}$ where the operator $A$ is strictly positive selfadjoint. For any fixed $γ\in\mathbb{R}$, the stability properties of the related solution semigroup $S(t)$ are discussed. In particular, a general technique is introduced in order to prove the lack of exponential decay of $S(t)$ when the spectrum of the leading operator $A$ is not made by eigenvalues only.

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