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Asymptotic behaviour and blow-up of some unbounded solutions for a semilinear heat equation

  • D. E. Tzanetis (a1)
Abstract

The initial-boundary value problem for the nonlinear heat equation u1 = Δu + λf(u) might possibly have global classical unbounded solutions, for some “critical” initial data . The asymptotic behaviour of such solutions is studied, when there exists a unique bounded steady state w(x;λ) for some values of λ We find, for radial symmetric solutions, that u*(r, t)→w(r) for any 0<r≤l but supu*(·, t) = u*(0, t)→∞, as t→∞. Furthermore, if , where is some such critical initial data, then û = u(x, t; û0) blows up in finite time provided that f grows sufficiently fast.

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Copyright
COPYRIGHT: © Edinburgh Mathematical Society 1996
References
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1. Amann, H., Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces SIAM Rev. 18 (1976), 620–709.
2. Bebernes, J. and Eberly, D., Mathematical Problems from Combustion Theory (Springer-Verlag, New York, 1989).
3. Friedman, A. and McLeod, B., Blow-up of positive solutions of semilinear heat equations, Indiana Univ. Math. J. 34 (1985), 425–447.
4. Fujita, H., On the nonlinear equations Δu + eu = 0 and ut = Δu + eu, Bull. Amer. Math. Soc. 75 (1969), 132–135.
5. Gidas, B., Ni, W.-M. and Nirenberg, L., Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), 209–243.
6. Joseph, D. D. and Lundgren, T. S., Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal. 49 (1973), 241–269.
7. Keller, H. B. and Cohen, D. S., Some positive problems suggested by nonlinear heat generation, J. Math. Mech. 16 (1967), 1361–1376.
8. Lacey, A. A., Mathematical analysis of thermal runaway for spatially inhomogeneous reactions, SIAM J. Appl. Math. 43 (1983), 1350–1366.
9. Lacey, A. A. and Tzanetis, D., Global existence and convergence to a singular steady state for a semilinear heat equation, Proc. Roy. Soc. Edinburgh Sect. A 105 (1987), 289–305.
10. Lacey, A. A. and Tzanetis, D. E., Global, unbounded solutions to a parabolic equation, J. Differential Equations 101 (1993), 80–102.
11. Ladyzhenskaja, O. A., Solonnikov, V. A. and Ural'ceva, N. N., Linear and quasilinear equations of parabolic type, in Trans. Math. Monographs 23 (Amer. Math. Soc., Providence, RI, 1968).
12. Matano, H., Asymptotic behavior and stability of solutions of semilinear diffusion equations, Publ. Res. Inst. Math. Sci. 15 (1979), 401–454.
13. SacksW.-M. Ni, P. E. W.-M. Ni, P. E. and Tavantzis, J., On the asymptotic behavior of solutions of certain quasilinear parabolic equations, J. Differential Equations 54 (1984), 97–120.
14. Ni, W.-M. and Sacks, P. E., The number of peaks of positive solutions of semilinear parabolic equations, SIAM J. Math. Anal. 16 (1985), 460–471.
15. Ni, W.-M. and Sacks, P. E., The singular behavior in nonlinear parabolic equations, Trans. Amer. Math. Soc. 287 (1985), 657–671.
16. Sattinger, D. H., Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J. 21 (1972), 979–1000.
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Proceedings of the Edinburgh Mathematical Society
  • ISSN: 0013-0915
  • EISSN: 1464-3839
  • URL: /core/journals/proceedings-of-the-edinburgh-mathematical-society
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