Thermoelasticity with antidissipation

We provide a complete stability analysis for the abstract differential system made by an antidamped wave-type equation, coupled with a dissipative heat-type equation\begin{document}$ \begin{cases} u_{tt} + A u -\gamma u_t = p A^{\alpha} \theta \\ \theta_{t} + \kappa A^{\beta} \theta = - p A^{\alpha} u_t \end{cases} $\end{document}where \begin{document}$ A $\end{document} is a strictly positive selfadjoint operator on a Hilbert space, \begin{document}$ \gamma, \kappa>0 $\end{document} , and both the parameters \begin{document}$ \alpha $\end{document} and \begin{document}$ \beta $\end{document} can vary between \begin{document}$ 0 $\end{document} and \begin{document}$ 1 $\end{document} . The asymptotic properties of the associated solution semigroup are determined by the strength of the coupling, as well as the quantitative balance between the antidamping \begin{document}$ \gamma $\end{document} and the damping \begin{document}$ \kappa $\end{document} . Depending on the value of \begin{document}$ (\alpha, \beta) $\end{document} in the unit square, one of the following mutually disjoint situations can occur: either the related semigroup decays exponentially fast, or all the solutions vanish but not uniformly, or there exists a trajectory whose norm blows up exponentially fast as \begin{document}$ t\to\infty $\end{document} .