Carleman estimate with second large parameter for second order hyperbolic operators in a Riemannian manifold and applications in thermoelasticity cases

In this article we prove a Carleman estimate with second large parameter for a second order hyperbolic operator in a Riemannian manifold M. Our Carleman estimate holds in the whole cylindrical domain M x (0, T) independent of the level set generated by a weight function if functions under consideration vanish on boundary partial derivative(M x (0, T)). The proof is direct by using calculus of tensor fields in a Riemannian manifold. Then, thanks to the dependency of the second larger parameter, we prove Carleman estimates also for (i) a coupled parabolic-hyperbolic system (ii) a thermoelastic plate system (iii) a thermoelasticity system with residual stress.