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 $\cs H^σ$ for any $σ ≥ 1$; then we prove the blow up of the solution for every real $p 𝟄 ]1,1+\sqrt{2}[$ and non--negative non--trivial initial data. ; Comment: some typos are corrected and some references are added