On the global behaviors for defocusing semilinear wave equations in ℝ1+2

Abstract

In this paper, we study the asymptotic decay properties for defocusing semilinear wave equations in $\mathbb{R}^{1+2}$ with pure power nonlinearity. By applying new vector fields to null hyperplane, we derive improved time decay of the potential energy, with a consequence that the solution scatters both in the critical Sobolev space and energy space for all $p>1+\sqrt{8}$. Moreover combined with Br\'{e}zis-Gallouet-Wainger type of logarithmic Sobolev embedding, we show that the solution decays pointwise with sharp rate $t^{-\frac{1}{2}}$ when $p>\frac{11}{3}$ and with rate $t^{ -\frac{p-1}{8}+\epsilon }$ for all $1<p\leq \frac{11}{3}$. This in particular implies that the solution scatters in energy space when $p>2\sqrt{5}-1$.