Exponential decay for a locally damped fifth-order equation posed on the line

We prove the exponential decay of the energy related to a locally damped fifth-order equation posed on the whole real line with the initial datum from a bounded set of L2. A local smoothing effect in H2 is established, which is essential to obtain the necessary a priory estimates. Moreover, it is shown that arguments used in the article can be applied to prove the exponential decay rate of solutions for the Korteweg–de Vries equation with a similar localized damping term provided the initial data are uniformly bounded in L2. This last fact improves some previous results.