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

Stabilization results for delayed fifth order KdV-type equation in a bounded domain