On the defocusing semilinear wave equations in three space dimension with small power

Abstract

By introducing new weighted vector fields as multipliers, we derive quantitative pointwise estimates for solutions of defocusing semilinear wave equation in $\mathbb{R}^{1+3}$ with pure power nonlinearity for all $1<p\leq 2$. Consequently, the solution vanishes on the future null infinity and decays in time polynomially for all $\sqrt{2}<p\leq 2$. This improves the uniform boundedness result of the second author when $\frac{3}{2}<p\leq 2$.