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

We study the asymptotic decay properties for defocusing semilinear wave equations in 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 + 8. Moreover, combined with Brezis–Gallouet–Wainger-type of logarithmic Sobolev embedding, we show that the solution decays pointwise with sharp rate t12 when p > 11 3 and with rate t(p1)8+𝜖 for all 1 < p 11 3 . This in particular implies that the solution scatters in energy space when p > 25 1.

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asymptotic behavior, defocusing semilinear wave equation, energy subcritical
Mathematical Subject Classification
Primary: 35L05
Received: 4 March 2020
Revised: 9 March 2020
Accepted: 15 September 2020
Published: 16 March 2022
Dongyi Wei
School of Mathematical Sciences
Peking University
Shiwu Yang
Beijing International Center for Mathematical Research
Peking University