论文标题
Schwarzschild时空中的非线性波方程式的爆炸和寿命估计
Blow-up and lifespan estimate to a nonlinear wave equation in Schwarzschild spacetime
论文作者
论文摘要
我们在Schwarzschild时空中使用功率类型非线性和较小的初始数据研究半线性波方程。如果非线性指数$ p $满足$ 2 \ le P <1+ \ sqrt 2 $，我们建立了寿命估算的尖锐上限，而对于最精致的关键功率$ p = 1+\ sqrt2 $，我们表明寿命满足 \ [ t（\ e）\ le \ exp \ left（c \ e^{ - （2+ \ sqrt 2）} \ right）， \] 其最佳性尚待证明。关键的新颖性是，初始数据的紧凑支持可以接近事件范围。 By combining the global existence result for $p>1+\sqrt 2$ obtained by Lindblad et al.(Math. Ann. 2014), we then give a positive answer to the interesting question posed by Dafermos and Rodnianski(J. Math. Pures Appl. 2005, the end of the first paragraph in page $1151$): $p=1+\sqrt 2$ is exactly the critical power of $p$ separating稳定和爆炸。
We study the semilinear wave equation with power type nonlinearity and small initial data in Schwarzschild spacetime. If the nonlinear exponent $p$ satisfies $2\le p<1+\sqrt 2$, we establish the sharp upper bound of lifespan estimate, while for the most delicate critical power $p=1+\sqrt2$, we show that the lifespan satisfies \[ T(\e)\le \exp\left(C\e^{-(2+\sqrt 2)}\right), \] the optimality of which remains to be proved. The key novelty is that the compact support of the initial data can be close to the event horizon. By combining the global existence result for $p>1+\sqrt 2$ obtained by Lindblad et al.(Math. Ann. 2014), we then give a positive answer to the interesting question posed by Dafermos and Rodnianski(J. Math. Pures Appl. 2005, the end of the first paragraph in page $1151$): $p=1+\sqrt 2$ is exactly the critical power of $p$ separating stability and blow-up.
