manuscripta mathematica

, Volume 28, Issue 1, pp 235-268

First online:

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

  • Fritz JohnAffiliated withCourant Institute of Mathematical Sciences

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Let u(x,t) be a solution, □ u≧A|u|p for x∈IR3, t≧0 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 contained in the cone 0≦t≦t0−|x−x0|, if the “initial data” u(x,0), ut(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 ∥u∥ is “sufficiently small” in a suitable norm. For p=2 the time at which u becomes infinite is of order ∥u∥−2.