Global existence to a reaction–diffusion equation through a self-similar-like upper solution

✍ Scribed by Guy Bernard

• Publisher: John Wiley and Sons
• Year: 2009
• Tongue: English
• Weight: 167 KB
• Volume: 33
• Category: Article
• ISSN: 0170-4214
• DOI: 10.1002/mma.1235

✦ Synopsis

Abstract

A demonstration method is presented, which will ensure the existence of positive global solutions in time to the reaction–diffusion equation −u~t~+Δ__u__+u^p^=0 in ℝ^n^×[0, ∞), for exponents p⩾3 and space dimensions n⩾3. This method does not require the initial value to have a specific uniform smallness condition, but rather to satisfy a bell‐like form. The method is based on a specific upper solution, which models the diffusion process of the heat equation. The upper solution is not self‐similar, but does have a self‐similar‐like form. After transforming the reaction–diffusion problem into an equivalent one, whose initial value is uniformly very small, a local solution is obtained in the time interval [0, 1] by the use of this upper solution. This local solution is then extended to [0, ∞) through an infinite sequence of extensions. At each step, an appropriate change of variables will transform the extension into a problem nearly identical to the local problem in [0, 1]. These transformations exploit the diffusive and self‐similar‐like nature of the upper solution. Copyright
© 2009 John Wiley & Sons, Ltd.