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

Wei, Dongyi; Yang, Shiwu

doi:10.2140/apde.2022.15.245

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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

DOI: 10.2140/apde.2022.15.245

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 $t^{- (p - 1) ∕ 8 + 𝜖}$ for all $1 < p \leq \frac{11}{3}$ . This in particular implies that the solution scatters in energy space when $p > 2 \sqrt{5} - 1$ .

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Keywords

asymptotic behavior, defocusing semilinear wave equation, energy subcritical

Mathematical Subject Classification

Primary: 35L05

Milestones

Received: 4 March 2020

Revised: 9 March 2020

Accepted: 15 September 2020

Published: 16 March 2022

Authors

Dongyi Wei
School of Mathematical Sciences Peking University Beijing China

Shiwu Yang
Beijing International Center for Mathematical Research Peking University Beijing China

Vol. 15, No. 1, 2022