On the Exponential Stability of a Nonlinear Kuramoto–Sivashinsky–Korteweg-de Vries Equation with Finite Memory

The aim of this paper is to investigate the stability problem for a nonlinear dispersive equation with memory. More precisely, the equation under consideration combines the well-known Korteweg-de Vries and Kuramoto–Sivashinsky equations, subject to the presence of a boundary memory term. The problem is shown to be well-posed for small initial data and provided that reasonable conditions hold for the parameters of the system and the memory kernel. In addition, we prove that the trivial solution is exponentially stable in spite of the memory effect. It is noteworthy that these outcomes are obtained under numerous instances of the physical parameters of the system.