We study the semilinear wave equation with nonlinearity in the spatial variable x. This article is interested in proving an existence and uniqueness result for generalized solutions to an important class of semilinear wave equations in the spatial dimensions \(\mathbb {R}^{\textbf {d}}\) , \({\varvec{d}}\varvec{\in \{1, 2, 3\}}\) . For a nonlinearity \(\varvec{\mathcal {H}}\) with arbitrary growth and sign, but multiplied with a generalized function which depends on the time variable and parameter, it is shown that the initial value problem for the semilinear wave equation has a unique solution in a bounded type Colombeau algebra. The proof is based on a fixed point theorem on a closed subspace of this subalgebra which is endowed with an ultra-metric topology. We also show the association between classical and generalized solution to a semilinear wave equation. The results presented generalize and unify several existing result in the literature.
