Global existence and decay estimates for the solution of a nonlinear Bresse system

Our main focus in the present paper is to study the global existence and asymptotic behavior of a nonlinear version of the Bresse system. As it has been already proved in Soufyane and Said-Houari (2014), the linear version of this system is of regularity-loss type. This regularity loss creates difficulties when dealing with the nonlinear problem since the dissipative property of the problem becomes very weak in the high frequency region and as a result the classical energy method fails. To overcome this difficulty, following Ide and Kawashima (2008) and Ikehata (2002), we use an energy method with negative weights to create an artificial damping which allows us to control the nonlinearity.