论文标题
Kerr黑洞背景上半线性波方程的爆炸
Blow-up for semilinear wave equations on Kerr black hole backgrounds
论文作者
论文摘要
我们检查了黑洞背景上半连续波方程的解决方案，并给出了约翰定理吹来部分的类似物，并在Schwarzschild和Kerr Black Hole背景上使用$ f_p（u）= | u |^{p} $。关于Schwarzschild的情况，我们构建了一类小数据，因此该解决方案沿着发出的零锥体爆炸，该圆锥均适用于$ f_p（u）= | u | u |^{p} $和焦点非线性$ f_p（u f_p（u）= | u | | u |^p-1} u $。证明表明，黑洞对奇异性的形成没有任何基本影响，在远离Cauchy Horizon $ r = r _- $或奇异性$ r = 0 $的地区。我们的方法也足够强大，可以适应一般渐近平坦的时空歧管，可能是紧凑型域的外部，并带有空间尺寸$ n \ ge 2 $。典型的例子包括外部域，渐近的欧几里得空间，Reissner-Nördström太空时期和Kerr-Newman太空时光。
We examine solutions to semilinear wave equations on black hole backgrounds and give a proof of an analog of the blow up part of the John theorem, with $F_p(u)=|u|^{p}$, on the Schwarzschild and Kerr black hole backgrounds. Concerning the case of Schwarzschild, we construct a class of small data, so that the solution blows up along the outgoing null cone, which applies for both $F_p(u)=|u|^{p}$ and the focusing nonlinearity $F_p(u)=|u|^{p-1}u$. The proof suggests that the black hole does not have any essential influence on the formation of singularity, in the region away from the Cauchy horizon $r=r_-$ or the singularity $r=0$. Our approach is also robust enough to be adapted for general asymptotically flat space-time manifolds, possibly exterior to a compact domain, with spatial dimension $n\ge 2$. Typical examples include exterior domains, asymptotically Euclidean spaces, Reissner-Nördström space-times, and Kerr-Newman space-times.
