Critical diffusion exponents for self-similar blow-up solutions of a quasilinear parabolic equation with an exponential source

Royal Society of Edinburgh - Proceedings A

Volumn 126, Issue 2, 1996, Pages 413-441

  Budd, Chris J ^a   Galaktionov, Victor A ^a  

^a UNIVERSITY OF BATH   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 21344460450     PISSN: 03082105     EISSN: None     Source Type: Journal    
DOI: 10.1017/S0308210500022824     Document Type: Article

References (25)

1. M. M. Ad'jutov, Ju. A. Klokov and A. P. Mikhailov. Self-similar thermal structures with contracting half-width. Differential Equations 19 (1983), 809-15.
2. A. A. Andronov, E. A. Leontovich, I. I. Gordon and A. G. Majer. Qualitative Theory of Second-Order Dynamical Systems (New York: Wiley, 1972).
3. J. Bebernes and S. Bricher. Final time blowup profiles for semilinear parabolic equations via center manifold theory. SIAM J. Math. Anal. 23 (1992), 852-69.
4. J. Bebernes, S. Bricher and V. A. Galaktionov. Asymptotics of blow-up for weakly quasilinear parabolic problems. Nonlinear Anal. 23 (1994), 489-514.
5. J. Bebernes and D. Eberly. Mathematical Problems from Combustion Theory, Applied Mathematical Sciences 83 (Berlin: Springer, 1989).
6. J. Bebernes and W. Troy. On the non-existence of solutions to the Kassoy problem in dimension 1. SIAM J. Math. Anal. 18 (1987), 1157-62.
7. A. Bressan. On the asymptotic shape of blow-up. Indiana Univ. Math. J. 39 (1990), 947-60.
8. A. Bressan. Stable blow-up patterns. J. Differential Equations 98 (1992), 57-75.
9. C. Budd, J. W. Dold and V. A. Galaktionov. Self-similar blow-up for quasilinear parabolic equation with gradient diffusion and exponential source (Bristol University report AM-92-09, 1992).
10. C. Budd and V. A. Galaktionov. Blow-up problems with quasi-linear gradient diffusivity (Bristol University Report AM-95-01, 1995).
11. C. Budd and V. A. Galaktionov. Asymptotics of blow-up in quasilinear diffusion equations (in prep.).
12. L. Cesari. Asymptotic Behaviour and Stability Problems in Ordinary Differential Equations (New York: Academic Press, 1971).
13. J. W. Dold. Analysis of the early stage of thermal runaway. Quart. J. Mech. Appl. Math. 38 (1985), 361-87.
14. D. Eberly. Non-existence for the Kassoy problem in dimension 1 and 2. J. Math. Anal. Appl. 129 (1988), 401-8.
15. A. Friedman. Partial Differential Equations of Parabolic Type (Englewood Cliffs, NJ: Prentice Hall, 1964).
16. β. Differential Equations 21 (1985), 11-18.
17. V. A. Galaktionov, S. P. Kurdyumov, S. A. Posashkov and A. A. Samarskii. Quasilinear parabolic equation with complex spectriem of unbounded self-similar solutions. In Mathematical Modelling. Processes in Nonlinear Media, 142-82 (Mowscow, Nauka: 1986 (in Russian); English translation: CRC Press Int., 1992).
18. V. A. Galaktionov and S. A. Posashkov. Single point blow-up for N-dimensional quasilinear equation with gradient diffusion and source. Indiana Univ. Math. J. 40 (1991), 1041-60.
19. Y. Giga. On elliptic equations related to self-similar solutions for non-linear heat equations. Hiroshima Math. J. 16 (1986), 541-54.
20. A. Haraux and F. B. Weissler. Non-uniqueness for a semilinear initial value problem. Indiana Univ. Math. J. 31 (1982), 167-89.
21. M. A. Herrero and J. J. L. Velázquez. Blow-up profiles in one-dimensional semi-linear problems. Comm. Partial Differential Equations 17 (1992), 205-19.
22. M. A. Herrero and J. J. L. Velázquez. Blow-up behavior of one-dimensional semilinear parabolic equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 10 (1993), 131-89.
23. A. A. Samarskii, G. G. Elenin, N. V. Zmitrenko, S. P. Kurdyunov and A. P. Mikhailov. Combustion of a non-linear medium in the form of complex structures. Dokl. Akad. Nauk SSSR, Ser. Math. Phys. 237 (1977), 1330-3 (in Russian).
24. A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov and A. P. Mikhailov. Blow-up in Quasilinear Parabolic Equations (Moscow, Nauka: 1987 (in Russian); English translation; Berlin: Walter de Gruyter, 1995).
25. J. J. L. Velázquez, V. A. Galaktionov and M. A. Herrero. The space structure near a blow-up point for semilinear heat equations, a formal approach. J. Vychisl. Matem. Matem. Fiz. 31 (1991), 399-411.