Existence of solutions to the semilinear damped wave equation with non-$L^2$ slowly decaying data : polynomial nonlinearity case

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 not belonging to $L^2(\mathbb{R}^n)$ in general. Our approach is based on the $L^p$-$L^q$ estimates of linear solutions and the fractional Leibniz rule in suitable homogeneous Besov spaces.