Asymptotic Behavior of Solutions for the Time-Delayed Kuramoto-Sivashinsky Equation

Abstract

In this paper, we investigate the asymptotic behavior of the solutions for the Kuramoto-Sivashinsky equation with a time delay. We prove the global existence of solutions and energy decay. By using the Liapunov function method, we shall show that the solution is exponentially decay if the delay parameter τ is sufficiently small.

Stability results for the KdV equation with time-varying delay

“…We mention here some possible future research: the cases of mixed boundary and internal damping with time-varying delay, time-and spatially-varying delay as in Lhachemi, Prieur and Shorten (2021) or study the stabilization problem when the delay (constant or variable) is in the nonlinear term as in Liu (2002); Zhu (2014) for Burger's and Kuramoto-Sivashinsky equations, respectively.…”

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].…”

Stability Results for the Kdv Equation with Time-Varying Delay

“…Finally, numerical simulations were presented to illustrate the results obtained. We mention here some possible future research: the cases of mixed boundary and internal damping with timevarying delay, time-and spatially-varying delay as in [13] or study the stabilization problem when the delay (constant or variable) is in the nonlinear term as in [14,25] for Burger's and Kuramoto-Sivashinsky equations, respectively.…”