We study the asymptotic behavior of the solutions of the time-delayed higher-order dispersive nonlinear differential equation \(\begin{aligned} u_t(x,t)+Au(x,t) +\lambda _0(x) u(x,t)+\lambda (x) u(x,t-\tau )=0 \end{aligned}\) where \(\begin{aligned} Au=(-1)^{j+1}\partial _x^{2j+1}u+(-1)^m\partial _x^{2m}u+ \frac{1}{p+1}\partial _xu^{p+1} \end{aligned}\) with \(m\le j\) and \(1\le p<2j\) . Under suitable assumptions on the time delay coefficients, we prove that the system is exponentially stable if the coefficient of the delay term is bounded from below by a suitable positive constant, without any assumption on the sign of the coefficient of the undelayed feedback. Additionally, in the absence of delay, general results of stabilization are established in \(H^s({\mathbb {R}})\) for \(s\in [0,2j+1]\) . Our results generalize several previous theorems for the Korteweg–de Vries-type delayed systems in the literature.
