Uniform blow-up profile and boundary behaviour for a non-local reaction–diffusion equation with critical damping

✍ Scribed by Philippe Souplet

• Publisher: John Wiley and Sons
• Year: 2004
• Tongue: English
• Weight: 107 KB
• Volume: 27
• Category: Article
• ISSN: 0170-4214
• DOI: 10.1002/mma.567

✦ Synopsis

Abstract

We consider the Dirichlet problem for a non‐local reaction–diffusion equation with integral source term and local damping involving power non‐linearities. It is known from previous work that for subcritical damping, the blow‐up is global and the blow‐up profile is uniform on all compact subsets of the domain. In this paper, we address the critical case. It turns out that the blow‐up profile is still uniform. Also we compute the sharp blow‐up rate and we find that, unlike in the subcritical case, the blow‐up rate reveals the influence of the damping term.

Next, we give precise estimates of the solution in the boundary layer and show that the width of the boundary layer behaves like $\sqrt {T-t}$ as t approaches the blow‐up time T. Even in the subcritical case, this last property was known only for powers p<2. Here, we remove this restriction on p. Some other non‐local equations are also discussed. Copyright © 2004 John Wiley & Sons, Ltd.