Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-26T07:19:35.747Z Has data issue: false hasContentIssue false
  • English
  • Français

Global blow-up of a nonlinear heat equation

Published online by Cambridge University Press:  14 November 2011

A. A. Lacey
A. A. Lacey
Affiliation:
Department of Mathematics, Heriot-Watt University, Riccarton, Edinburgh EH14 4AS, Scotland
Get access Rights & Permissions [Opens in a new window]

Synopsis

Solutions to semilinear parabolic equations of the form ut = Δu + f(u), x in Ω, which blow up at some finite time t* are investigated for “slowly growing” functions f. For nonlinearities such as f(s) = (2 +s)(ln(2 +s))1+b with 0 < b < l,u becomes infinite throughout Ω as t→t* −. It is alsofound that for marginally more quickly growing functions, e.g. f(s) = (2 + s)(ln(2 +s))2, u is unbounded on some subset of Ω which has positive measure, and is unbounded throughout Ω if Ω is a small enough region.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 104 , Issue 1-2 , 1986 , pp. 161 - 167
Copyright
Copyright © Royal Society of Edinburgh 1986

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Baras, P. and Cohen, L.. Complete blow-up after Tmax for the solution of a semilinear heat equation. C. R. Acad. Sci. Paris Sér. I Math. 300 (1985), 295–298.Google Scholar
2Friedman, A. and McLeod, B.. Blow-up of positive solutions of semilinear heat equations. Indiana Univ. Math. J. 34 (1985), 425–447.Google Scholar
3Lacey, A. A.. Mathematical analysis of thermal runaway for spatially inhomogeneous reactions. SIAMJ. Appl. Math. 43 (1983), 1350–1366.CrossRefGoogle Scholar
4Lacey, A. A.. The form of blow-up for nonlinear parabolic equations. Proc. Roy. Soc. Edinburgh Sect. A 98 (1984), 183–202.Google Scholar
5Mueller, C. E. and Weissler, F. B.. Single point blow-up for a general semilinear heat equation. Indiana Univ. Math. J. 34 (1985), 881–913.Google Scholar
6Weissler, F. B.. Single point blow-up for a semilinear initial value problem. J. Differential Equations 55 (1984), 204–224.Google Scholar