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

We study the initial value problem$$ u_{tt} = u_{xx} + {\|u (\cdot,t) \| } ^{p}, \, - \infty{<}x{<}\infty,\ t{>}0$$$$ u(x,0) = f(x),u_{t}{(x,0)}\,{=}\,g(x), \, - \infty{<}x{<}\infty$$ where $ \|u(\cdot,t)\| = \int \nolimits ^ {\infty} _ {- \infty}\varphi(x) | u( x,t ) | {\rm{ d }} x$ with φ( x )⩾0 and $ \int \nolimits^{\infty} _ {-\infty} \varphi (x) \, {\rm{d}}x\,= 1$ . We show that solutions exist globally for 0< p ⩽1, while they blow up in finite time if p >1. We also present the growth rate at blow‐up. Copyright © 2004 John Wiley & Sons, Ltd.