Localized energy estimates for wave equations on high-dimensional Schwarzschild space-times (Q2845751)

The localized energy estimate for the wave equation is known to be a fairly robust measure of dispersion. Recent analogs on the \((1+3)\)-dimensional Schwarzschild space-time have played a key role in a number of subsequent results, including a proof of Price's law. In this article, the authors explore similar localized energy estimates for wave equations on \((1+n)\)-dimensional hyperspherical Schwarzschild space-times.NEWLINENEWLINENEWLINEMore precisely, hyperspherical Schwarzschild space-times are \(M=\mathbb R\times (R,\infty )\times\mathbb S^{n-1}\), \(n\geq 3\), equipped with the metric \(ds^2=-wdt^2+w^{-1}dr^2+r^2 d\omega^2\), \(w=1-(R/r)^{n-2}\). We have Killing vector field \(\partial_t\), which yields the conserved energy NEWLINE\[NEWLINEE[\phi](t)=\int_{\mathbb S^{n-1}}\int_{r\geq R} \Bigl[w^{-1}(\partial_t\phi)^2+w(\partial_r\phi)^2+|{{\not\negmedspace\nabla}}\phi|^2\Bigr]r^{n-1}\, dr\, d\omega ,NEWLINE\]NEWLINE for solutions to the homogeneous wave equations \(\square_g \phi=\nabla^\alpha\nabla_\alpha \phi=0\). Defining localized energy norm by NEWLINE\[NEWLINE |\phi|_{LE}^2=\int_{M, t>0} \Bigl[c_r w (\partial_r\phi)^2 + c_\omega |{{\not\negmedspace\nabla}}\phi|^2+ c_0\phi^2\Bigr]r^{n-1}\, dr\, d\omega\, dtNEWLINE\]NEWLINE where \(1/c_r=r^{n}\Bigl(1-\log\Bigl(\frac{r-R}{r}\Bigr)\Bigr)^2\), \(c_\omega=\frac{1}{r} \Bigl(\frac{r-r_{ps}}{r}\Bigr)^2\), \(r/(r-R)=c_0 r^3\Bigl(1-\log\Bigl(\frac{r-R}{r}\Bigr)\Bigr)^4\). Here \(r_{ps}=\Bigl(\frac{n}{2}\Bigr)^{\frac{1}{n-2}}R\), which is the location of the photon sphere. The main result of this article then states that the following localized energy estimate holds NEWLINE\[NEWLINE\sup_{t\geq 0}E[\phi](t)+\|\phi\|_{LE}^2\leq C E[\phi](0)NEWLINE\]NEWLINE for any solutions to the homogeneous wave equations.