Existence and non-existence of global solutions of a non-local wave equation

We study the initial value problem u(tt) = u(xx) + parallel tou((.),t)parallel to(p), -infinity < x < infinity, t > 0 u(x, 0) = f(x), u(t)(x, 0) = g(x), -infinity < x < infinity where parallel tou((.),t)parallel to = integral(-infinity)(infinity) phi(x)\u(x,t)\dx with phi(x)greater than or equal to0 and integral(-infinity)(infinity) phi(x) dx = 1. We show that solutions exist globally for 0 < p less than or equal to 1, while they blow up in finite time if p > 1. We also present the growth rate at blow-up. Copyright (C) 2004 John Wiley Sons, Ltd.