  We study the semilinear wave equation in Schwarzschild metric (3+1
dimensional space--time). First, we establish that the problem is locally
well--posed in $\cs H^\sigma$ for any $\sigma \geq 1$; then we prove the blow
up of the solution for every real $p \in ]1,1+\sqrt{2}[$ and non--negative
non--trivial initial data.
