In this work, we investigate an initial boundary value problem related to the nonlinear hyperbolic equation utt +uxxxx+αuxxxxt = f (ux)x, for f(s) =|s|ρ +|s|σ,1 < ρ,σ, α > 0. Under suitable conditions, we prove the existence of global solutions and the exponential decay of energy. The nonlinearity f (s) introduces some obstacles in the process of obtaining a priori estimates and we overcome this difficulty by employing an argument due to Tartar (1978). The exponential decay is obtained via an integral inequality introduced by Komornik (1994)