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

Abstract

We prove that solution of defocusing semilinear wave equation in ${\mathbb{R}}^{1+3}$ with pure power nonlinearity is uniformly bounded for all $\frac{3}{2}<p\le 2$ 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.

Not for publication, the result has been combined in arXiv:1908.00607