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 ut=(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(iota) w(., (lambada))(infinity)(right arrow)(iota)infinity as 0<(lambda)(right arrow)(lambda)*-. Estimates of the blow-up time are obtained by using comparison methods. Also an asymptotic analysis is applied when f(s)=es, for (lambda)-(lambada)*<<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.