Asymptotic analysis and estimates of blow-up time for the radial symmetric semilinear heat equation in the open-spectrum case
✍ Scribed by N. I. Kavallaris; A. A. Lacey; C. V. Nikolopoulos; D. E. Tzanetis
- Publisher
- John Wiley and Sons
- Year
- 2007
- Tongue
- English
- Weight
- 303 KB
- Volume
- 30
- Category
- Article
- ISSN
- 0170-4214
- DOI
- 10.1002/mma.854
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✦ Synopsis
Abstract
We estimate the blow‐up time for the reaction diffusion equation u~t~=Δ__u__+ λ__f__(u), for the radial symmetric case, where f is a positive, increasing and convex function growing fast enough at infinity. Here λ>λ^*^, where λ^*^ is the ‘extremal’ (critical) value for λ, such that there exists an ‘extremal’ weak but not a classical steady‐state solution at λ=λ^*^ with
∥w(⋅, λ)∥~∞~→∞ as 0<λ→λ^*^−. Estimates of the blow‐up time are obtained by using comparison methods. Also an asymptotic analysis is applied when f(s)=e^s^, for λ−λ^*^≪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.