Stability analysis of abstract systems of Timoshenko type

We consider an abstract system of Timoshenko type $$ \begin{cases} \rho_1{{\ddot \varphi}} + a A^{\frac12}(A^{\frac12}\varphi + \psi) =0\\ \rho_2{{\ddot \psi}} + b A \psi + a (A^{\frac12}\varphi + \psi) - \delta A^\gamma {\theta} = 0\\ \rho_3{{\dot \theta}} + c A\theta + \delta A^\gamma {{\dot \psi}} =0 \end{cases} $$ where the operator $A$ is strictly positive selfadjoint. For any fixed $\gamma\in\mathbb{R}$, 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.