  Studied here is the Kawahara equation, a fifth order Korteweg-de Vries type
equation, with time-delayed internal feedback. Under suitable assumptions on
the time delay coefficients we prove that solutions of this system are
exponentially stable. First, considering a damping and delayed system, with
some restriction of the spatial length of the domain, we prove that the
Kawahara system is exponentially stable for $T> T_{\min}$. After that,
introducing a more general delayed system, and by introducing suitable
energies, we show using Lyapunov approach, that the energy of the Kawahara
equation goes to zero exponentially, considering the initial data small and a
restriction in the spatial length of the domain. To remove these hypotheses, we
use the compactness-uniqueness argument which reduces our problem to prove an
observability inequality, showing a semi-global stabilization result.
