Heriot-Watt Mathematics Report Series
HWM07-7, 13 Feb 2007
Asymptotic analysis and estimates of blow-up time for the radial symmetric semilinear heat equation in the open-spectrum case
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.
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