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. 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 $$L^r$$ 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.