Well-Posedness and Energy Decay for KdV-B Equations with Time-Dependent Delays

Abstract.

In this paper, we study the well-posedness and exponential stability of the generalized Korteweg–de Vries–Burgers equation incorporating a damping term and a time-dependent delay feedback. The delay function is examined under two distinct frameworks: one in which it is assumed to be merely continuous, and another in which it satisfies additional regularity conditions, while remaining bounded in both cases. Under suitable assumptions on the delay feedback, we establish the global existence of solutions. Moreover, we derive exponential decay estimates in both delay settings. Notably, in the case where the delay function exhibits higher regularity, exponential stability is obtained without requiring any smallness conditions on the time delay size.