GLOBAL EXISTENCE AND ENERGY DECAY OF SOLUTIONS TO A NONLINEAR TIMOSHENKO BEAM SYSTEM WITH A DELAY TERM

Abstract

We consider the Timoshenko system in boundeddomain with a delay term in the nonlinear internal feedback $$\begin{cases} \rho_{1} \varphi_{tt}(x,t) - K(\varphi_{x}+\psi)_{x}(x,t) = 0, \\ \rho_{2} \psi_{tt}(x,t) - b \psi_{xx}(x,t) + K(\varphi_{x}+\psi)(x,t) \\ \qquad\qquad + \mu_1 g_1(\psi_{t}(x,t)) + \mu_2 g_2(\psi_{t}(x,t-\tau)) = 0, \end{cases}$$ and prove the global existence of its solutions in Sobolev spaces by means of the energy method combined with the Faedo-Galerkin procedureunder a condition between the weight of the delay term in the feedback and the weight of the term without delay. Furthermore, we establish a decay rate estimate for the energy by introducing suitable Lyapunov functionals.