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

TL;DR

This paper investigates the exponential stability of the Kawahara equation with a distributed memory damping mechanism, demonstrating energy decay under certain kernel conditions.

Contribution

It introduces a novel damping approach using distributed memory to establish exponential stability of the Kawahara equation.

Findings

Solutions exhibit exponential decay under specific kernel assumptions

Energy method effectively proves stability and decay estimates

Provides conditions for the memory kernel to ensure stability

Abstract

This article will deal with the stabilization problem for the higher-order dispersive system, commonly called the Kawahara equation. To do so, we introduce a damping mechanism via a distributed memory term in the equation to prove that the solutions of the Kawahara equation are exponentially stable, provided that specific assumptions on the memory kernel are fulfilled. This is possible thanks to the energy method that permits to provide a decay estimate of the system energy.