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

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.