Examples of cylindrical shock wave conversion by focusing (Q1201225)

This expository article presents a selfcontained discussion of the distribution solutions of the wave equation \(u_{tt}-u_{xx}- u_{yy}=0\), with rotationally symmetric initial data locally in \(L^ p\), \(1<p<\infty\). The analysis is based on the classical integral representation of solutions of the Darboux equation. The results are then used to construct the unique continuation to the interior of the cone \(\{t>\sqrt{x^ 2+y^ 2}\}\) of a specific (shock wave) solution defined in \(\{t<\sqrt{x^ 2+y^ 2}\}\) and having a finite jump across the backward cone \(\{t=-\sqrt{x^ 2+y^ 2}<0\}\).