Blow-up of solutions of nonlinear wave equations in three space dimensions

Solutions u(x, t) of the inequality □ u ≥ A | u | p for x ε R 3 , t ≥ 0 are considered, where □ is the d'Alembertian, and A,p are constants with A > 0, 1 < p < 1 + √2. It is shown that the support of u is compact and contained in the cone 0 ≤ t ≤ t 0 -| x - x 0 |, if the “initial data” u(x , 0), u t (x , 0) have their support in the ball| x - x 0 | ≤ t 0 . In particular, “global” solutions of □ u = A | u | p with initial data of compact support vanish identically. On the other hand, for A > 0, p > 1 + √2, global solutions of □ u = A | u | p exist, if the initial data are of compact support and “sufficiently” small.