Abstract

We study the semilinear wave equation with power type nonlinearity and small initial data in Schwarzschild spacetime. If the nonlinear exponent $p$ satisfies $2\le p\le1+\sqrt 2$, we establish the blow-up result and lifespan estimate. 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): $p=1+\sqrt 2$ is exactly the critical power of $p$ separating stability and blow-up. This is a joint work with Prof. Yi Zhou.