We consider several Cauchy problems for the wave equation with some perturbation.
First of all, we consider the wave equation with a metric perturbation, that is, we consider the d'Alembert operator in the Schwarzschild metric (which is a model for a static black hole). Because of the sign changing properties of the solution to this equation, it is not trivial to establish the global existence or the blow-up of the solution depending on the power of the nonlinearity. However, introducing suitable weighted average functions and proving some modified versions of the well-known Kato lemma, we are able to provide two blow-up results, one in the case of small data far from the black hole, and one when the initial data are close to the black hole but large (even this case is not trivial at all). In both cases, we restrict ourselves to radial solutions and power p less than 1+sqrt(2).
We treat even the case of a linear wave equation with a potential-like perturbation. We consider a small electromagnetic potential depending on space and time with optimal decay properties, and null initial data. Under these assumptions, we can prove optimal dispersive estimates and in particular a 1/t decay in time. The proof exploits the gauge invariancy of the electromagnetic potential, which allows a suitable integral representation of the solution.
The thesis also reviews some important known results concerning the previous problems and deals with related and open problems.