Blow up of the solutions of nonlinear wave equation

We construct for every fixed n >= 2 the metric g(s) = h(1)( r)dt(2) - h(2)(r)dr(2) - k1(omega) d omega(2)(1) - ... -k(n-1)(omega)d omega(2)(n-1), where h(1)(r), h(2)(r), ki(omega), 1 <= 1 <= n - 1, are continuous functions, r = vertical bar x vertical bar, for which we consider the Cauchy problem (u(tt) - Delta(u))(gs) = f(u) + g(vertical bar x vertical bar), x is an element of R-n, n >= 2; u(1,x) = u(o) (x) is an element of L-2(R-n), u(t)(1,x) = u(1)(x) is an element of (H) over dot(-1) (R-n), where f is an element of C-1 (R-1), f(0) = 0, a vertical bar u vertical bar <= b vertical bar u vertical bar, g is an element of C (R+), g(r) >= 0, r = vertical bar x vertical bar, a and b are positive constants.When g(r) = 0, we prove that the above Cauchy problem has a nontrivial solution u(t,r) in the form u(t,r) = v(t)omega(r) for which lim(t -> 0) parallel to u parallel to(L2([0,infinity))) = infinity. Copyright (C) 2007.