[For the entire collection see Zbl 0561.00014.]
Let (1) \(G(u,u',u'')=0\) be a nonlinear second order autonomous system where \(u=u(x^ 1,x^ 2,...,x^{n+1}),\) and u’,u” denote all the first and second partial derivatives of u. Assume here that both u and G are scalars and denote by \(u_ a,u_{ab}\) the partial derivatives \(\partial_ a u\) and respectively \(\partial^ 2_{ab} u\); \(a,b=1,2,...,n+1\). Assume that \(G(0,0,0)=0\) and that (1) is hyperbolic around the trivial solution \(u^ 0\equiv 0\). Without loss of generality consider the wave operator \(\partial^ 2_ 1+...+\partial^ 2_ n- \partial^ 2_ t=-\square,\) where we have ascribed to \(x_{n+1}\) the role of the time variable t. The equation (1) takes the form \[ (1')\quad \square u=F(u,u',u'') \] with F a smooth function of (u,u’,u”), independent of \(u_{tt}\), vanishing together with all its first derivatives at (0,0,0).
Associate to (1’) the pure initial value problem \[ (1'a)\quad u(x,0)=\epsilon f(x),\quad u_ t(x,0)=\epsilon g(x) \] where f,g are \(C^{\infty}\)-functions, decaying sufficiently fast at infinity (for simplicity, say \(f,g\in C_ 0^{\infty}({\mathbb{R}}^ n))\) and \(\epsilon\) is a parameter which measures the amplitude of the data. Given f,g and F we define the life span \(T_*=T_*(\epsilon)\) as the supremum over all \(T\geq 0\) such that a \(C^{\infty}\)-solution of (1’), (1’a) exists for all \(x\in R^ n\), \(0\leq t<T\). It is proved [the author and A. Majda, Commun. Pure Appl. Math. 33, 241-263 (1980; Zbl 0443.35040)] that \(T_*<\infty\) even if the genuine nonlinearity condition is violated.
If the original equation, or system, can be written in conservation form, i.e. in the case of the present paper, \[ F(u,u',u'')=\sum^{n+1}_{a=1}\partial_ a f^ a(u,u'), \] one can try to extend the solutions pass these breakdown points by introducing the concept of weak solutions. This was accomplished for every general first order system of conservation laws, in one space dimension, by the work of Olejnik, P. Lax and J. Glimm [cf. P. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves (1973; Zbl 0268.35062)].
The author restricts himself to classical solutions i.e., \(C^{\infty}\)- solutions, and obtains some nice results.