  We consider the \emph{KdV} equation with an additional non-local perturbation
term defined through the Hilbert transform, also known as the OST-equation. We
prove that the solutions $u(t,x)$ has a pointwise decay in spatial variable:
$\vert u(t,x)\vert \lesssim \frac{1}{1 + |x|^{2}}$, provided that the initial
data has the same decaying and moreover we find the asymptotic profile of
$u(t,x)$ when $|x| \to +\infty$. Next, we show that decay rate given above is
optimal when the initial data is not a zero-mean function and in this case we
derive an estimate from below $\frac{1}{\vert x\vert^2} \lesssim \vert
u(t,x)\vert$ for $\vert x \vert$ large enough. In the case when the initial
datum is a zero-mean function, we prove that the decay rate above is improved
to $\frac{1}{1+\vert x \vert^{2+\varepsilon}}$ for $0<\varepsilon \leq 1$.
Finally, we study the local-well posedness of the OST-equation in the framework
of Lebesgue spaces.
