#### Vol. 15, No. 1, 2022

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On the global behaviors for defocusing semilinear wave equations in $\mathbb{R}^{1+2}$

### Dongyi Wei and Shiwu Yang

Vol. 15 (2022), No. 1, 245–272
##### Abstract

We study the asymptotic decay properties for defocusing semilinear wave equations in ${ℝ}^{1+2}$ with pure power nonlinearity. By applying new vector fields to the 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 Brezis–Gallouet–Wainger-type of logarithmic Sobolev embedding, we show that the solution decays pointwise with sharp rate ${t}^{-1∕2}$ when $p>\frac{11}{3}$ and with rate ${t}^{-\left(p-1\right)∕8+𝜖}$ for all $1. This in particular implies that the solution scatters in energy space when $p>2\sqrt{5}-1$.

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