### N I Kavallaris, A A Lacey, C V Nikolopoulos and D E Tzanetis

#### Abstract

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 $\lambda>\lambda^\ast$, where $\lambda^\ast$ 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 $||w(\cdot,\lambda)||_{\infty}\rightarrow \infty$ as $0<\lambda\rightarrow \lambda^*-$. Estimates of the blow-up time are obtained by using comparison methods. Also an asymptotic analysis is applied when $f(s)=e^s$, for $\lambda-\lambda^*\ll 1$, 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.