Time-delayed generalized Korteweg–de Vries-Burgers equation: Well-posedness and exponential decay

This study explores the generalized Korteweg-de Vries-Burgers equation incorporating delay feedback and a damping term. Using semigroup arguments and Lyapunov functional techniques, we establish the existence of a global solution when the exponent of the nonlinear term satisfies some growth conditions. Furthermore, we prove exponential stability estimates under suitable assumptions: first in the case of a positive damping coefficient, then within a more comprehensive framework, accommodating sign changes in both coefficients, i.e. for the damping and the delay feedback. In both cases, we adopt refined conditions on the delay feedback's coefficient, extending and enhancing existing results in the literature. In particular, our conditions are independent of the time delay size.