Asymptotic analysis and estimates of blow-up time for the radial symmetric semilinear heat equation in the open-spectrum case

We estimate the blow-up time for the reaction diffusion equation u _t = ?u+?f(u), for the radial symmetric case, where f is a positive, increasing and convex function growing fast enough at infinity. Here ?>?*, where ?* is the 'extremal' (critical) value for ?, such that there exists an 'extremal' weak but not a classical steady-state solution at ? = ?* with ||w(?)||₈ ? infin; as 0<? ? ?*-. Estimates of the blow-up time are obtained by using comparison methods. Also an asymptotic analysis is applied when f(s) = e^s, for ? - ?* «1, regarding the form of the solution during blow-up and an asymptotic estimate of blow-up time is obtained. Finally, some numerical results are also presented. Copyright © 2007 John Wiley & Sons, Ltd.