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

✍ Scribed by Azmy S. Ackleh; Keng Deng

Publisher: John Wiley and Sons
Year: 2004
Tongue: English
Weight: 87 KB
Volume: 27
Category: Article
ISSN: 0170-4214
DOI: 10.1002/mma.565

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✦ Synopsis

Abstract

We study the initial value problem

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.

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