Well‐posedness and stability of a nonlinear time‐delayed dispersive equation via the fixed point technique: A case study of no interior damping

Abstract

The primary concern of this article is to establish the well‐posedness as well as the exponential stability of the zero solution to a nonlinear time‐delayed dispersive equation of order four in a bounded interval. The main ingredient of the proof is the exploitation of Schauder Fixed Point Theorem. This outcome considerably improves an earlier result of Ammari et al. (2021) in the sense that we prove that no interior damping control is required. Furthermore, our results remain valid regardless of the sign of the antidiffusion parameter. Numerical simulations are also given as an illustration of our theoretical result.