Blow-up of solutions of nonlinear wave equations in three space dimensions.
Solutions u(x, t) of the inequality squareu >/= A|u|(p) for x epsilon R(3), t >/= 0 are considered, where square is the d'Alembertian, and A,p are constants with A > 0, 1 < p < 1 + radical2. 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 squareu = A|u|(p) with initial data of compact support vanish identically. On the other hand, for A > 0, p > 1 + radical2, global solutions of squareu = A|u|(p) exist, if the initial data are of compact support and "sufficiently" small.
John
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Courant Institute of Mathematical Sciences, New York University, New York, New York 10012.