Qualitative Analysis of the Dynamic for the Nonlinear Korteweg–de Vries Equation with a Boundary Memory

On the stability of the Kawahara equation with a distributed infinite memory

“…The nonlinear problem. The next result ensures the well-posedness of the system (2), which is represented by the problem (12). Theorem 3.3.…”

“…First, we shall focus on the third-order Korteweg-de Vries (KdV) equation. In the case when a memory term occurs, numerous stability results were obtained in [12,13] (see also the reference therein). Chentouf [12] considered the KdV equation with a boundary finite memory term in a bounded interval.…”

“…In the case when a memory term occurs, numerous stability results were obtained in [12,13] (see also the reference therein). Chentouf [12] considered the KdV equation with a boundary finite memory term in a bounded interval. Additionally, Chentouf and Guesmia [13] studied the stability problem for the KdV equation subject to the effect of a distributed infinite memory term.…”

Stability results for the KdV equation with time-varying delay

“…Recently, the problem of robustness with respect to constant time-delay of the KdV equation was studied in Baudouin, Crépeau and Valein (2019); Parada, Crépeau and Prieur (2022a); Valein (2022) using Lyapunov theory or deriving suitable observability inequalities. In the case where the KdV equation is in presence of memory terms, stability results were obtained in Chentouf (2021); Chentouf and Guesmia (2022). The stability of PDE's involving timevarying delays was analyzed in Nicaise, Valein and Fridman (2009) for one-dimensional heat and wave equations, in ; Nicaise, Pignotti and Valein (2011) for wave equations in domains in ℝ 𝑛 and in Fridman, Nicaise and Valein (2010) for general second-order evolution equations.…”