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

Abstract

Solutions u(x, t) of the inequality □u ≥ A|u|^p for x ε R₃, 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 ≤ t₀ -|x - x⁰|, if the “initial data” u(x, 0), u_t(x, 0) have their support in the ball|x - x⁰| ≤ t₀. 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.

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