Well‐posedness and exponential stability of the Kawahara equation with a time‐delayed localized damping

Abstract

The aim of this article is to investigate the well‐posedness and stability problems of the so‐called Kawahara equation under the presence of an interior localized delayed damping. The system is shown to be well‐posed. Furthermore, we prove that the trivial solution is exponentially stable in spite of the delay effect. Specifically, local and semi‐global stability results are established according to the properties of the spatial distribution of the delay term.

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

“…Regarding the main contribution of this paper, we can claim that we go one step further in the study of the stabilization problem for the fifth-order Kortewegde-Vries type system. Compared to the recent works [3,6,9,11], where damping mechanisms and delay controls are used, this paper closes the gap since it is the first work to treat exponential stability using only infinite memory. It is also noteworthy that the current paper shows that a memory term plays a role of a damping control in the sense that it leads to the stability of the system without any additional damping such as a(x)u used in [3,11,39] to get the stability property of the system.…”

“…Compared to the recent works [3,6,9,11], where damping mechanisms and delay controls are used, this paper closes the gap since it is the first work to treat exponential stability using only infinite memory. It is also noteworthy that the current paper shows that a memory term plays a role of a damping control in the sense that it leads to the stability of the system without any additional damping such as a(x)u used in [3,11,39] to get the stability property of the system. Finally, note that our results remain valid if a 1 = 0 and hence the drift term ∂ x u(x, t) can be omitted.…”

“…The last studies on the stabilization of the Kawahara equation deal with a localized time-delayed interior control. In [9,11], under suitable assumptions on the time delay coefficients, the authors were able to prove that solutions of the Kawahara system are exponentially stable. The results were obtained using the Lyapunov approach and a compactness-uniqueness argument.…”

Asymptotic behavior of the linearized compressible barotropic Navier‐Stokes system with a time varying delay term in the boundary or internal feedback

“…For semilinear wave type equation, see [35]; for nonlinear wave equation with switching delay, see [28]. In the constant time delay case, we refer to the other related works: [36] for Schrodinger equation, [37] for KdV equation with boundary time-delay, [38] for KdV equation with interior delay feedback, [39] KdV equation with star shaped network, [40][41][42] for Kawahara equation with boundary and interior time delay feedback, respectively, [43] for KdV-Burger equation, Kuramoto-Sivashinsky equation with the time delay in the nonlinear term [44], Benjamin-Bona-Mahony equation [45], and microbeam equation [46], and for other evolution equation with time delay feedback, see [47].…”