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

Abstract

AbstractA demonstration method is presented, which will ensure the existence of positive global solutions in time to the reaction–diffusion equation −ut+Δu+up=0 in ℝn×[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.