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

TL;DR

This paper proves local and global existence of solutions for a semilinear damped wave equation with polynomial nonlinearity, even when initial data decay slowly and are not in L^2, using advanced functional analysis techniques.

Contribution

It establishes existence results for solutions with non-$L^2$ initial data, expanding understanding of damped wave equations with polynomial nonlinearities.

Findings

Proves local and global existence for non-$L^2$ initial data.

Uses $L^p$-$L^q$ estimates and fractional Leibniz rule in Besov spaces.

Handles slowly decaying initial data outside traditional $L^2$ framework.

Abstract

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.