Lack of exponential stability for thermoelastic Timoshenko and Bresse systems with full homogeneous Dirichlet boundary conditions

Abstract

In this paper we prove that the classical thermoelastic Timoshenko and Bresse systems with thermal coupling acting only on the bending moment are not exponentially stable under full homogeneous Dirichlet boundary conditions if the equal wave speed condition is not satisfied. Combined with their well-known converses, our results provide complete characterizations of exponential stability in terms of the coefficients of each system. To the best of our knowledge, these characterizations were previously known only under mixed boundary conditions. The proof is based on the introduction of a boundary correction term in the standard trigonometric ansatz so that the full homogeneous Dirichlet conditions are satisfied.