Well-posedness and stability results for the Korteweg–de Vries–Burgers and Kuramoto–Sivashinsky equations with infinite memory: A history approach

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

“…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.…”

“…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. Recently, in [34], Parada et al studied the stabilization problem of the KdV equation with either a boundary or distributed time-dependent delay.…”

“…We end the literature review by mentioning that the occurrence of a memory phenomenon in the Kawahara problem (2) could be explained in practice by the fact that numerous compressible and incompressible fluids are intrinsically viscoelastic and therefore the influence of the past values of the amplitude of the dispersive wave of the fluid is unavoidable [2,13,18,33].…”

“…where ξ(s) = q 1 p 1 (s + 1) p1−1 , ξ 0 = q 1 p 1 , with d 1 > 0, q 1 > 0 and p 1 ∈ (0, 1). (ii) The assumptions on g are classical and used in numerous papers where other types of models and problems are treated (see for instance [13] and the references therein). However, one could relax some of the conditions on g but at the expense of technical complexities.…”

Stability of Multi-dimensional Nonlinear Piezoelectric Beam with Viscoelastic Infinite Memory

“…freitas [7] studied the long time dynamics of a kind of nonlinear piezoelectric beams with fractional damping and thermal effects, and fre [8] considered a nonlinear piezoelectric system with delay effect, in which the global attractor and exponential attractors are studied. For other kinds of PDEs on such issue, we refer to pao [19] for a semilinear wave equations with viscoelastic damping and delay feedback and ch2 [4] for two nonlinear systems including Korteweg-de Vries-Burgers and Kuramoto-Sivashinsky equations with memory, and the references therein.…”