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Fritz John pp 566-588 | Cite as

Blow-Up for Quasi-Linear Wave Equations in Three Space Dimensions

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

Abstract

We consider equations of the form
$$u = \phi (x,t,u,u',u'')$$
(1a)
for a function u = u(x 1,…, x n ,t) = u(x,t). Here □ denotes the d’Alembertian
$$\square = \frac{{{\partial ^2}}}{{\partial {t^2}}} - \vartriangle$$
(1b)
and u′ represents the vector of first derivatives, u″ that of second derivatives of u with respect to the x k and t. On linearizing (1a) shall go over into the classical wave equation □ u = 0; this means that Φ and its first derivatives with respect to u, u′, u″ shall vanish for u = u′ = u″ = 0.

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© Springer Science+Business Media New York 1985

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  • Fritz John

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