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

摘要

We estimate the blow-up time for the reaction diffusion equation u(t) = Delta u + lambda f (u), for the radial symmetric case, where f is a positive, increasing and convex function growing fast enough at infinity. Here lambda >lambda*, where lambda* is the 'extremal' (critical) value for lambda, such that there exists an 'extremal' weak but not a classical steady-state solution at lambda=lambda* with parallel to w(., lambda)parallel to(infinity)->infinity as 0 lambda*-. Estimates of the blow-up time are obtained by using comparison methods. Also an asymptotic analysis is applied when f (s) = e(s), for lambda-lambda* 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 (c) 2007 John Wiley & Sons, Ltd.