Existence and stabilization of a Kirchhoff moving string with a distributed delay in the boundary feedback

In this article, we are concerned by the study of the effect of a distributed time-delay in boundary stabilization of a moving string. The model adopted here is nonlinear and of "Kirchhoff" type. We prove the well-posedness of the system by exploiting of the Faedo-Galerkin method. We prove that the solution of the system approaches the equilibrium in an exponential manner in the energy norm. To this end we request that the delayed term be dominated by the damping term. It is also shown that the presence of a strong damping with some restrictive conditions makes system exponentially stable even in the presence of the delay term. This is established through the multiplier technique.