<p>In this paper, we study the long-time dynamics of solutions to the defocusing semilinear wave equation <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\Box _g\phi =|\phi |^{p-1}\phi $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mo>□</mo> <mi>g</mi> </msub> <mi>ϕ</mi> <mo>=</mo> <msup> <mrow> <mo stretchy="false">|</mo> <mi>ϕ</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mi>p</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mi>ϕ</mi> </mrow> </math></EquationSource> </InlineEquation> on the Schwarzschild black hole spacetimes. For <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\frac{1+\sqrt{17}}{2}<p<5$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <msqrt> <mn>17</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> <mo><</mo> <mi>p</mi> <mo><</mo> <mn>5</mn> </mrow> </math></EquationSource> </InlineEquation> and compactly supported initial data, we show that the solution decays like <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$|\phi |\lesssim t^{-1+\epsilon }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">|</mo> <mi>ϕ</mi> <mo stretchy="false">|</mo> </mrow> <mo>≲</mo> <msup> <mi>t</mi> <mrow> <mo>-</mo> <mn>1</mn> <mo>+</mo> <mi>ϵ</mi> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> in the domain of outer communication. The proof relies on the <i>r</i>-weighted vector field method of Dafermos–Rodnianski together with the Strichartz estimates for linear waves by Marzuola–Metcalfe–Tataru–Tohaneanu.</p>

Asymptotic Decay for Defocusing Semilinear Wave Equations on Schwarzschild Spacetimes

期刊论文
发表日期：

2025年1月15日

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He Mei,
Dongyi Wei,
Shiwu Yang

摘要

In this paper, we study the long-time dynamics of solutions to the defocusing semilinear wave equation $\Box _g\phi =|\phi |^{p-1}\phi $ $□_{g} ϕ = {| ϕ |}^{p - 1} ϕ$ on the Schwarzschild black hole spacetimes. For $\frac{1+\sqrt{17}}{2}<p<5$ $\frac{1 + \sqrt{17}}{2} < p < 5$ and compactly supported initial data, we show that the solution decays like $|\phi |\lesssim t^{-1+\epsilon }$ $| ϕ | ≲ t^{- 1 + ϵ}$ in the domain of outer communication. The proof relies on the r-weighted vector field method of Dafermos–Rodnianski together with the Strichartz estimates for linear waves by Marzuola–Metcalfe–Tataru–Tohaneanu.