Asymptotic analysis and estimates of blow‐up time for the radial symmetric semilinear heat equation in the open‐spectrum case |
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| Authors: | N. I. Kavallaris A. A. Lacey C. V. Nikolopoulos D. E. Tzanetis |
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| Affiliation: | 1. Department of Mathematics, School of Applied Mathematical and Physical Sciences, National Technical University of Athens, Zografou Campus, Athens 157 80, GreeceDepartment of Mathematics, School of Applied Mathematical and Physical Sciences, National Technical University of Athens, Zografou Campus, Athens 157 80, Greece===;2. School of Mathematical and Computer Sciences, Heriot‐Watt University, Riccarton, Edinburgh EH14 4AS, U.K.;3. Department of Mathematics, University of the Aegean, Karlovasi, Samos 83200, Greece;4. Department of Mathematics, School of Applied Mathematical and Physical Sciences, National Technical University of Athens, Zografou Campus, Athens 157 80, Greece |
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| Abstract: | We estimate the blow‐up time for the reaction diffusion equation ut=Δu+ λf(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)=es, 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. |
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| Keywords: | reaction diffusion equation blow‐up time estimates boundary‐layer theory |
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