Resumen de Well-Posedness and Exponential Stability of the Kuramoto-Sivashinsky Equation in a Bounded Interval and With a Constant Boundary Time-Delay
Boumediène Chentouf, Nejib Smaoui
In this article, we consider a one-dimensional dispersive equation known in literature as Kuramoto-Sivashinsky (KS) equation. It models numerous physical phenomenon such as turbulent states in a distributed chemical reaction, flame propagation and waves in a viscous fluid. Motivated by practical reasons, a constant time-delay is assumed to occur in the boundary and then the main purpose is to study the effect of the presence of this delay on the behavior of the solutions to the KS equation.
To do so, we first show that the problem is well-posed. Second, we prove that the solutions to the problem are exponentially stable in the energy space. Finally, our theoretical findings are ascertained by means of a thorough numerical analysis of the problem and several numerical simulations are provided. Our findings extend earlier results that neglected boundary-delay effects and, moreover, improve previous results by addressing a streamlined model under more natural boundary conditions.
