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

TL;DR

This paper develops new weighted vector field techniques to establish decay and vanishing properties of solutions to defocusing semilinear wave equations in three dimensions with small power nonlinearities, extending previous boundedness results.

Contribution

Introduction of novel weighted vector fields as multipliers to derive pointwise decay estimates for solutions with small power nonlinearities in three-dimensional wave equations.

Findings

Solutions vanish on future null infinity.

Solutions decay polynomially in time.

Results cover all nonlinear powers 1<p≤2.

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 $\frac{3}{2}<p\leq 2$.