This lecture will discuss a simple but versatile method to investigate the behavior at large distances or near the boundary of the solutions of PDEs. The method is potentially applicable to a large array of problems (elliptic, parabolic, linear, nonlinear, scalar, system). However, a basic assumption is that the PDE of interest must be accounted for by a Fredholm operator acting between appropriate Banach function spaces. Both the range and limitations of the method are better described in a completely abstract setting, but the lecture will mostly focus on examples and applications.