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

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Fritz John

Part of the book series: Contemporary Mathematicians ((CM))

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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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Authors and Affiliations

  1. Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, NY, 10012, USA

    Fritz John

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  1. Fritz John
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Editors and Affiliations

  1. Mathematik, ETH-Zentrum, CH-8092, Zürich, Germany

    Jürgen Moser

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© 1985 Springer Science+Business Media New York

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John, F. (1985). Blow-Up of Solutions of Nonlinear Wave Equations in Three Space Dimensions. In: Moser, J. (eds) Fritz John. Contemporary Mathematicians. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-5406-5_38

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  • DOI: https://doi.org/10.1007/978-1-4612-5406-5_38

  • Publisher Name: Birkhäuser, Boston, MA

  • Print ISBN: 978-1-4612-5408-9

  • Online ISBN: 978-1-4612-5406-5

  • eBook Packages: Springer Book Archive

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