Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-26T14:52:51.308Z Has data issue: false hasContentIssue false
  • English
  • Français

Global existence and convergence to a singular steady state for a semilinear heat equation

Published online by Cambridge University Press:  14 November 2011

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

Synopsis

With certain initial and boundary conditions the solution u* to the semilinear heat equation ∂u*/∂t = ∂u* + λ * f(u*), where f is a positive superlinear function and λ is the supremum of the open spectrum for the steady state problem Δw + λf(w) = 0, is found to exist for all time and to be unbounded. Moreover u* approaches w* a singular steady state, as / tends to infinity.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 105 , Issue 1 , 1987 , pp. 289 - 305
Copyright
Copyright © Royal Society of Edinburgh 1987

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

1Amann, H.. Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Rev. 18 (1976), 620–709.CrossRefGoogle Scholar
2Ball, J. M.. Remarks on the blow-up and nonexistence theorems for nonlinear equations. Quart. J. Math. Oxford Ser. (2) 28 (1977), 473–486.CrossRefGoogle Scholar
3Baras, P. and Cohen, L.. Sur l'explosion totale apres T mix de la solution d'une equation de la chaleur semi-linéaire. C.R. Acad. Sci Paris Ser. 1 Math. 300 (1985), 295–298.Google Scholar
4Bebernes, J. W. and Kassoy, D. R.. A mathematical analysis of blow-up for thermal reactions—the spatially nonhomogeneous case. SIAM J. Appl. Math. 40 (1981), 476–484.CrossRefGoogle Scholar
5Bellout, H.. A criterion for blow-up of solutions to semilinear heat equations (to appear).Google Scholar
6Crandall, M. G. and Rabinowitz, P. H.. Bifurcation, perturbation of simple eigenvalues, and linearized stability. Arch. Rational Mech. Anal. 52 (1973), 161–180.CrossRefGoogle Scholar
7Friedman, A.. Partial differential equations of parabolic type (New Jersey: Prentice Hall, 1964).Google Scholar
8Friedman, A. and McLeod, J. B.. Blow-up of positive solutions of semilinear heat equations. Indiana Univ. Math. J. 34 (1985), 425–447.CrossRefGoogle Scholar
9Fujita, H.. On the nonlinear equation Δu + eu =0 and vt = Δv + ev. Bull. Amer. Math. Soc. 75 (1969), 132–135.CrossRefGoogle Scholar
10Joseph, D. D. and Lundgren, T. S.. Quasilinear Dirichlet problems driven by positive sources. Arch. Rational Mech. Anal. 9 (1973), 241–269.CrossRefGoogle Scholar
11Keller, H. B. and Cohen, D. S.. Some positone problems suggested by nonlinear heat generation. /. Math. Mech. 16 (1967), 1361–1376.Google Scholar
12Lacey, A. A.. Mathematical analysis of thermal runaway for spatially inhomogeneous reactions. SIAM J. Appl. Math. 43 (1983), 1350–1366.CrossRefGoogle Scholar
13Mignot, F., Murat, F. and Puel, J. P.. Variations d'un point de retournement par rapport au domaine. Comm. Partial Differential Equations 4(11) (1979), 1263–1297.CrossRefGoogle Scholar
14Mignot, F. and Puel, J. P.. Sur une classe de problèmes non linéaires avec non linéarité positive, croissante, convexe. Comm. Partial Differential Equations 4 (1980), 791–836.CrossRefGoogle Scholar
15Pao, C. V.. Nonexistence of global solutions and bifurcation analysis of a boundary-value problem of parabolic type. Proc. Amer. Math. Soc. 65 (1977), 245–251.CrossRefGoogle Scholar
16Sattinger, D. H.. Monotone methods in nonlinear elliptic and parabolic boundary value problems. Indiana Univ. Math. J. 21 (1972), 979–1000.CrossRefGoogle Scholar