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.