Well-posedness and Energy Decay of Solutions to a Nonlinear Bresse System with Delay Terms

Abstract

We consider the Bresse system in bounded domain with delay terms in the nonlinear internal feedbacks

$$\begin{aligned} \left\{ {\begin{array}{l}\rho _{1}\varphi _{tt}-Gh (\varphi _{x}+\psi +l\omega )_{x}-lEh(\omega _{x}-l\varphi )+\mu _{1}g_1(\varphi _{t}(x,t)) +\mu _{2}g_2(\varphi _{t}(x,t-\tau _{1}))=0\\ \rho _{2}\psi _{tt}-EI\psi _{xx}+Gh(\varphi _{x}+\psi +l\omega ) +\widetilde{\mu _{1}}\widetilde{g_1}(\psi _{t}(x,t)) +\widetilde{\mu _{2}}\widetilde{g_2}(\psi _{t}(x,t-\tau _{2}))=0\\ \rho _{1}\omega _{tt}-Eh(\omega _{x}-l\varphi )_{x}+lGh(\varphi _{x}+\psi +l\omega ) +\widetilde{\widetilde{\mu _{1}}}\widetilde{\widetilde{g_1}}(\omega _{t}(x,t)) +\widetilde{\widetilde{\mu _{2}}}\widetilde{\widetilde{g_2}} (\omega _{t}(x,t-\tau _{3}))=0\\ \end{array}}\right. \end{aligned}$$

and prove the global existence of its solutions in Sobolev spaces by means by means of the energy method combined with the Faedo–Galerkin procedure under a condition between the weight of the delay terms in the feedbacks and the weight of the terms without delay. Furthermore, we study the asymptotic behavior of solutions using multiplier method and some weighted integral inequalities.