Uniform bound for solutions of semilinear wave equations in $\mathbb{R}^{1+3}$

TL;DR

This paper establishes uniform bounds for solutions of defocusing semilinear wave equations in 3+1 dimensions with certain power nonlinearities, using weighted energy estimates, marking a novel result for small power nonlinearities.

Contribution

It provides the first global asymptotic bounds for solutions with small power nonlinearities in the specified range, employing the $r$-weighted energy method.

Findings

Solutions are uniformly bounded for all $p$ in $(rac{3}{2}, 2]$.

The $r$-weighted energy estimate is effective for analyzing small power nonlinearities.

First demonstration of global asymptotic behavior for these solutions.

Abstract

We prove that solution of defocusing semilinear wave equation in $\mathbb{R}^{1+3}$ with pure power nonlinearity is uniformly bounded for all $\frac{3}{2}<p\leq 2$ with sufficiently smooth and localized data. The result relies on the $r$-weighted energy estimate originally introduced by Dafermos and Rodnianski. This appears to be the first result regarding the global asymptotic property for the solution with small power $p$ under 2.