Analysis of an abstract thermoelastic system and applications to nonplanar Bresse models

In this paper, we study a linear system with $N\ge 1$ hyperbolic equations coupled with $M\le N$ parabolic equations. Using the classical theory of semigroups of linear operators, we prove well-posedness and provide conditions under which strong stability implies polynomial or exponential stability. From a practical point of view, this means that, in the applications of the Borichev-Tomilov and Gearhart-Prüss theorems, the resolvent estimates do not need to be verified for systems which satisfy the conditions; instead, it is enough to check a certain matrix condition associated with the coefficients of the system. Our general formulation includes, as special cases, many known results on Bresse and Timoshenko systems. In particular, our condition for exponential stability generalizes the well-known equal wave speed condition and provides a systematic way to derive this type of condition from the coefficients of the system. As applications of our general approach, we prove new results on well-posedness and asymptotic behavior of nonplanar thermoelastic Bresse systems, that have not been previously studied, and related beam models.