Global classical solutions of semilinear wave equations on \({\mathbb{R}^3} \times \mathbb{T}\) with cubic nonlinearities

In this paper, we establish global classical solutions of semilinear wave equations with small compact supported initial data posed on the product space \({\mathbb{R}^3} \times \mathbb{T}\). The semilinear nonlinearity is assumed to be of the cubic form. The main ingredient here is the establishment of the L²–L^∞ decay estimates and the energy estimates for the linear problem, which are adapted to the wave equation on the product space. The proof is based on the Fourier mode decomposition of the solution with respect to the periodic direction, the scaling technique, and the combination of the decay estimates and the energy estimates.