Two approaches for the stability of a nonlocal time-delayed fourth-order dispersive model

Abstract

The primary concern of the current article is to investigate the well-posedness and stability problem of a nonlinear dispersive equation of order four and with a nonlocal time-delayed term. In order to deal with such a problem, two different approaches are used. The first approach consists in using the Lyapunov method to prove the well-posedness of the system and the exponential stability of the zero solution. The latter is established, provided that the time-delay has small values and an interior damping control acts on the equation. In turn, the second approach permits to get the well-posedness outcome as well as another stability finding by invoking Schauder Theorem. The advantage of the second method is the lack of need for any damping control. To conclude, our findings show that the dispersive equation under consideration possesses a unique stable solution despite the occurrence of a nonlocal time-delay term in the nonlinearity of the equation, with or without the presence of a damping control. These results extent and complement those of [1,2,10,21,28], where no local term takes place.