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Fritz John pp 532-565 | Cite as

Blow-Up of Solutions of Nonlinear Wave Equations in Three Space Dimensions

  • Fritz John
Chapter
Part of the Contemporary Mathematicians book series (CM)

Abstract

Let u(x,t) be a solution, □ u ≧ A|u|p for x ∈ ℝ3, t ≧ 0 where □ is the d’Alembertian, and A, p are constants with
$$A > O,1 < p < 1 + \sqrt 2$$
. It is shown that the support of u is contained in the cone 0 ≦ t ≦ tO - |x-xO|, if the “initial data” u(x,O), ut (x,0) have their support in the ball |x-xO| ≦ tO. In particular “global solutions” of u = A|u|p with initial data of compact support vanish identically. On the other hand for
$$A > O,1 > p > 1 + \sqrt 2 $$
global solutions of □ u = A |u|p exist, if the initial data are of compact support and ‖u‖-2 is “sufficiently small” in a suitable norm. For p = 2 the time at which u becomes infinite is of order ‖u‖ -2.

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Copyright information

© Springer Science+Business Media New York 1985

Authors and Affiliations

  • Fritz John
    • 1
  1. 1.Courant Institute of Mathematical SciencesNew YorkUSA

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