The study of the dispersive properties of the linear wave equation on black hole backgrounds has been intensively studied during the past decade. In this talk I will present results for a specific class of black holes, namely the extremal black holes, which are of particular interest in the physics community. We will show that certain derivatives of the scalar fields asymptotically blow up; this result is in stark contrast with the subextremal case where it has been shown that all derivatives decay. I will also present similar (finite-time) blow-up results for certain non-linear wave equations.