On the stability of the Bresse system with frictional damping

In this paper, we consider the Bresse system with frictional damping terms and prove some optimal decay results for the L²-norm of the solution and its higher order derivatives. In fact, if we consider just one damping term acting on the second equation of the solution, we show that the solution does not decay at all. On the other hand, by considering one damping term alone acting on the third equation, we show that this damping term is strong enough to stabilize the whole system. In this case, we found a completely new stability number that depends on the parameters in the system. In addition, we prove the optimality of the results by using eigenvalues expansions. We have also improved the result obtained recently in [12] for the two damping terms case and get better decay estimates. Our obtained results have been proved under some assumptions on the wave speeds of the three equations in the Bresse system.