We study the Cauchy problem of the semilinear damped wave equation with polynomial nonlinearity, and establish the local and global existence of the solution for slowly decaying initial data. By employing L^p – L^q estimates for the linear problem and a fractional Leibniz rule in suitable homogeneous Besov spaces, we show the existence of the solution for initial data that may not belong to L^2 at the spatial infinity in general. The main novelty of our result is to construct the solution for initial data having the decay like Lʳ with r>2. A crucial point in our argument is to control the derivative loss from the high frequency part by appropriately choosing the function space.
Ikeda et al. (Sun,) studied this question.