Stability of the KdV equation in the case of mixed internal and boundary dampings with time-dependent delay

The aim of this paper is to consider the nonlinear Korteweg-de Vries equation with internal feedback without delay and a boundary feedback with time-dependent delay. We study the well-posedness of the system under some assumptions on the length of the spatial domain and on the delay using semigroups theory and we study the exponential stability of the equation considering a Lyapunov functional approach.