Thermoelasticity with antidissipation

Abstract

<p style='text-indent:20px;'>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</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \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} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ A $\end{document}</tex-math></inline-formula> is a strictly positive selfadjoint operator on a Hilbert space, <inline-formula><tex-math id="M2">\begin{document}$ \gamma, \kappa>0 $\end{document}</tex-math></inline-formula>, and both the parameters <inline-formula><tex-math id="M3">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ \beta $\end{document}</tex-math></inline-formula> can vary between <inline-formula><tex-math id="M5">\begin{document}$ 0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ 1 $\end{document}</tex-math></inline-formula>. The asymptotic properties of the associated solution semigroup are determined by the strength of the coupling, as well as the quantitative balance between the antidamping <inline-formula><tex-math id="M7">\begin{document}$ \gamma $\end{document}</tex-math></inline-formula> and the damping <inline-formula><tex-math id="M8">\begin{document}$ \kappa $\end{document}</tex-math></inline-formula>. Depending on the value of <inline-formula><tex-math id="M9">\begin{document}$ (\alpha, \beta) $\end{document}</tex-math></inline-formula> in the unit square, one of the following mutually disjoint situations can occur: either the related semigroup decays exponentially fast, or all the solutions vanish but not uniformly, or there exists a trajectory whose norm blows up exponentially fast as <inline-formula><tex-math id="M10">\begin{document}$ t\to\infty $\end{document}</tex-math></inline-formula>.</p>