Generalized solutions to a semilinear wave equation (Q1776947)

The authors study generalized solutions to semilinear wave equations with singular data and various types of nonlinearities in space dimension \(n \leq 9\). The generalized functions setting employed for this purpose is the space \({\mathcal G}_{L^2}\), a version of Colombeau's algebra based on asymptotic (with respect to a regularization parameter \(\varepsilon\)) \(L^2\)-estimates. The basic strategy is to regularize the nonlinear term and analyze the solutions of the resulting family of Cauchy problems using Sobolev-type embedding and interpolation theorems. For dimensions \(7,8,9\), more precise results, based on work by [\textit{H. Pecher}, Math. Z. 161, 9--40 (1978; Zbl 0384.35039); Math. Z. 150, 159--183 (1976; Zbl 0318.35054)] are required. In the case of cubic nonlinearities also solutions without regularizations are considered and compatibility results with the regularized equations are derived.