**Abstract:**
We consider waveguides solutions to the Laplace equation
\begin{equation*}
\partial_t^2 u + \Delta_x u + W(t,x)\cdot\nabla u + V(t,x)u = 0
\end{equation*}
of the form $u(t,x)=\sin(\lambda t)Q(x)$, which are usually referred to as
{\it waveguides}.
These are special solutions to equations of the above form, which
surprisingly also appear in nonlinear cases, such as $V(t,x)= u^p(t,x)$.
They can be interpreted as evolutive solutions of an elliptic equation, and
our aim is to study their {\it dynamical} behavior.
Using unique continuation (at infinity) arguments, involving Carleman
estimates, we can describe the space-profile of waveguides, and in
particular the sharpest possible decay they can have without being null.
The results are obtained in collaboration with L. Escauriaza and L. Vega.