Blow-up for the semilinear wave equation in the Schwarzschild metric

Abstract

We study the Cauchy problem for the semilinear wave equation in the Schwarzschild metric ($(3+1)$--dimensional space--time). First, we establish that the problem is locally well posed in $ \mathrm H^\sigma$ for any $\sigma \in [1,p+1)$; then we prove the blow--up of the solution in two cases: a)} $p \in (1,1+\sqrt{2})$ and small initial data supported far away from the black hole, b) $p \in (2,1+\sqrt{2})$ and large data supported near the black hole. In both cases, we also give an estimate from above for the lifespan of the solution.