Pointwise decay for semilinear wave equations in ${\mathbb{R}}^{!+3}$

In this paper, we use Dafermos-Rodnianski’s new vector field method to study the asymptotic pointwise decay properties for solutions of energy subcritical defocusing semilinear wave equations in ${\mathbb{R}}^{1+3}$. We prove that the solution decays as quickly as linear waves for $p>\frac{1+\sqrt{17}}{2}$, covering part of the sub-conformal case, while for the range $2<p\leq \frac{1+\sqrt{17}}{2}$,thesolutionstilldecayswithrateatleast$t^{-\frac{1}{3}}$. As a consequence, the solution scatters in energy space when $p>2.3542$. We also show that the solution is uniformly bounded when $p>\frac{3}{2}$.