Generalized solutions to a semilinear wave equation

Semilinear wave equations in space dimension n⩽9 with singular data and various types of nonlinearities are considered. We employ the framework of the algebra GL2 of generalized functions. In the general case, the nonlinear term is regularized with respect to a small parameter ε such that it becomes globally Lipschitz for each such ε. This produces a family of Cauchy problems, to which we construct a family of solutions which in turn determines an element of GL2 which serves as a generalized solution to the original equation. For cubic nonlinear terms the equation is directly solved in GL2 without regularizations. Finally, in certain cases, it is shown that the solution to the regularized equation coincides with the solution to the nonregularized one.