  We show that the limit infimum, as time $\,t\,$ goes to infinity, of any
uniformly bounded in time $H^{3/2+}\cap L^1$ solution to the Intermediate Long
Wave equation converge to zero locally in an increasing-in-time region of space
of order $\,t/\log(t)$. Also, for solutions with a mild $L^1$-norm growth in
time is established that its limit infimum converge to zero, as time goes to
infinity. This confirms the non existence of breathers and other solutions for
the ILW model moving with a speed "slower" than a soliton. We also prove that
in the far field linearly dominated region, the $L^2$ norm of the solution also
converges to zero as time approaches infinity.
  In addition, we deduced several scenarios for which the initial value problem
associated to the generalized Benjamin-Ono and the generalized Intermediate
Long Wave equations cannot possess time periodic solutions (breathers).
  Finally, as it was previously demonstrated in solutions of the KdV and BO
equations, we establish the following propagation of regularity result : if the
datum $u_0\in H^{3/2+}(\mathbb R)\cap H^m((x_0,\infty))$, for some
$\;x_0\in\mathbb R,\,m\in Z^+,\,m\geq 2$, then the corresponding solution
$u(t,\cdot)$ of the Intermediate Long Wave equation belongs to
$H^m(\beta,\infty)$, for any $t>0$ and $\beta\in\mathbb R$.
