Long Time Behaviour of Solutions to Nonlinear Wave Equations

G(u,u',u") = 0, (1) where u = u(x, x,..., x), and u', u" denote all the first and second partial derivatives of u. For simplicity we will assume here that both u and G are scalars and denote by ua, uab, the partial derivatives dau and respectively dahu; a,b =1,2, ..., n+1. Let u (x) be a given solution of (1). Our equation is said to be elliptic or hyperbolic at u°(x) according to whether the (n+l)x(n+^l) matrix, whose entries are Gab