Local and Global Existence results for the Characteristic Problem for Linear and Semi-linear Wave Equations.

The thesis concerns the well posedness of the Characteristic Initial Value Problem for the Semilinear Wave Equation, with initial data on a light cone. In the first part of the thesis, an explicit representation formula for the solution of the linear equation is given, extending the results known for the homogeneous equation and the trace on the time axis of the solution. Further, Energy Estimates are derived. In constructing such Estimates one encounters several difficulties due to the presence of a geometrical singularity at the tip of the cone. To manage the construction of the Energy Estimate, one introduces suitable Sobolev-like norms characterized by weights, which mitigates the difficulties in the origin. These Estimates are well posed only for functions which vanish of order high enough at the origin. This fact brings us to split the initial data in the sum of two terms. The first term consists of the Taylor polynomial of the initial datum, the second one consist of remainder regular function with the required vanishing order at the origin. An interesting phenomenon observed here is a gap of differentiability between the solution and the initial data. The solution obtained using the Energy method is still incomplete, because of the splitting of the initial data. This fact brings us to solve the problem for purely polynomial data. For this purpose, it is used a generalization of the well-known harmonic polynomials. The last part of the thesis is devoted to the semi-linear problem, for which the tools developed in the previous chapters are generalized.