Exact self-similar blow-up of solutions of a semilinear parabolic equation with a nonlinear gradient term

Indiana University Mathematics Journal

Volumn 45, Issue 3, 1996, Pages 655-682

  Souplet, Philippe ^a   Tayachi, Slim ^a   Weissler, Fred B ^b  

^a FACULTÉ DES SCIENCES DE TUNIS   (Tunisia)

^b UNIVERSITÉ PARIS 13   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0000126964     PISSN: 00222518     EISSN: None     Source Type: Journal    
DOI: 10.1512/iumj.1996.45.1197     Document Type: Article

References (41)

1. n for a parabolic equation with a damping nonlinear gradient term, Nonlinear diffusion equations and their equilibrium states, 3 (Gregynog, 1989), pp. 1-20, Progr. Nonlinear differential equations Appl., 7, Birkhäuser, Boston, MA, (1992).
2. n for a parabolic equation with a damping nonlinear gradient term, Nonlinear diffusion equations and their equilibrium states, 3 (Gregynog, 1989), pp. 1-20, Progr. Nonlinear differential equations Appl., 7, Birkhäuser, Boston, MA, (1992).
3. N, Indiana Univ. Math. J. 30 (1981), 141-157.
4. M. BERGER and R. V. KOHN, A rescaling algorithm for the numerical calculation of blowing-up solutions, Comm. Pure Appl. Math. 41 (1988), 841-863.
5. H. BREZIS, L. A. PELETIER and D. TERMAN, A very singular solution of the heat equation with absorption, Arch. Rat. Mech. Analysis 95 (1986), 185-209.
6. M. CHIPOT and F. B. WEISSLER, Some blowup results for a nonlinear parabolic equation with a gradient term, SIAM J. Math. Anal. 20 (1989), 886-907.
7. E. A. CODDINGTON and N. LEVINSON, Theory of Ordinary Differential Equations, MacGraw-Hill, New York, (1955).
8. M. FILA, Remarks on blow up for a nonlinear parabolic equation with a gradient term, Proc. Amer. Math. Soc. 111 (1991), 795-801.
9. M. FILA and P. QUITTNER, Radial positive solutions for a semilinear elliptic equation with a gradient term, Adv. math. Sci. Appl. 2 (1993), 39-45.
10. A. FRIEDMAN and J. B. MCLEOD, Blow-up of positive solutions of semilinear heat equations, Indiana Univ. Math. J. 34 (1985), 425-447.
11. 1+α, J. Fac. Sci. Univ. Tokyo Sect. I 13 (1966), 109-124.
12. Y. GIGA, On elliptic equations related to self-similar solutions for nonlinear heat equations, Hiroshima Math. J. 16 (1986), 539-552.
13. Y. GIGA and R. V. KOHN, Asymptotically self-similar blow-up of semilinear heat equations, Comm. Pure Appl. Math. 38 (1985), 297-319.
14. Y. GIGA and R. V. KOHN, Characterizing blowup using similarity variables, Indiana Univ. Math. J. 36 (1987), 1-40.
15. Y. GIGA and R. V. KOHN, Nondegeneracy of blowup for semilinear heat equations, Comm. Pure Appl. Math. 42 (1989), 845-884.
16. A. HARAUX and F. B. WEISSLER, Non-uniqueness for a semilinear initial value problem, Indiana Univ. Math. J. 31 (1982), 167-189.
17. P. HARTMAN, Ordinary Differential Equations, 2nd ed., Birkhäuser, Boston-Bäsel-Stuttgart, (1982).
18. M. A. HERRERO and J. J. L. VELÁZQUEZ, Blow-up behaviour of one-dimensional semilinear parabolic equations, Ann. Inst. H. Poincaré 10 (1993), 131-189.
19. M. A. HERRERO and J. J. L. VELÁZQUEZ, Flat blow up in one-dimensional semilinear heat equation, Differential Integral Equations 5 (1992), 973-997.
20. M. A. HERRERO and J. J. L. VELÁZQUEZ, Blow-up profiles in one-dimensional, semilinear parabolic problems, Comm. Partial Differential Equations 17(1 & 2) (1992), 205-219.
21. M. A. HERRERO and J. J. L. VELÁZQUEZ, Generic behaviour of one-dimensional blow up patterns, Annali Sc. Norm. Sup. Pisa XIX. 3 (1992), 381-450.
22. L. M. HOCKING, K. STEWARTSON and J.T. STUART, A nonlinear instability burst in plane parallel flow, J. Fluid Mech. 51 (1972), 705-735.
23. B. KAWOHL and L. A. PELETIER, Observations on blow up and dead cores for nonlinear parabolic equations, Math. Z. 202 (1989), 207-217.
24. T. Y. LEE and W.-M. NI, Global existence, large time behavior and life span of solutions of a semilinear parabolic Cauchy problem, Trans Amer. Math. Soc. 333 (1992), 365-378.
25. C. E. MUELLER and F. B. WEISSLER, Single point blow-up for general semilinear heat equation, Indiana Univ. Math. J. 34 (1985), 881-913.
26. L. A. PELETIER, D. TERMAN and F. B. WEISSLER, On the equation Δu + 1/2x.▽u + f(u) = 0, Arch. Rat. Mech. Analysis 94 (1986), 83-99.
27. Y.-W. QI, The asymptotics of blow-up solutions to a degenerate parabolic equation, ZAMP 42 (1991), 480-496.
28. Y.-W. QI, The existence and asymptotic behavior of similarity solutions to a quasi-linear parabolic equation, Nonlinear diffusion equations and their equilibrium states, 3 (Gregynog, 1989), pp. 451-469, Progr. Nonlinear differential equations Appl., 7, Birkhäuser, Boston, MA, (1992).
29. Y.-W. QI, The existence and asymptotic behavior of similarity solutions to a quasi-linear parabolic equation, Nonlinear diffusion equations and their equilibrium states, 3 (Gregynog, 1989), pp. 451-469, Progr. Nonlinear differential equations Appl., 7, Birkhäuser, Boston, MA, (1992).
30. P. QUITTNER, Blow-up for semilinear parabolic equations with a gradient term, Math. Methods Appl. Sciences 14 (1991), 413-417.
31. P. QUITTNER, On global existence and stationary solutions for two classes of semilinear parabolic equations, Comment. Math. Univ. Carolinae 34, 1 (1993), 105-124.
32. J. SERRIN and H. ZOU, Existence and nonexistence for ground states of quasi-linear elliptic equations, Arch. Rat. Mech. Analysis 121, (1992), 101-130.
33. Ph. SOUPLET, Résultats d'explosion en temps fini pour une équation de la chaleur non linéaire, C. R. Acad. Sc. Paris 321, Série I (1995), 721-726.
34. Ph. SOUPLET, Finite time blow up for a nonlinear parabolic equation with a gradient term and applications, Math. Methods Appl. Sciences, in press.
35. S. TAYACHI, Forward self-similar solutions of a semilinear parabolic equation with a nonlinear gradient term, Differential Integral Equations 9 (1996), 1107-1117.
36. S. TAYACHI, Solutions autosimilaires d'équations semi-linéaires paraboliques avec terme de gradient non linéaire, Doctoral Thesis, Université Paris-Nord, (1996).
37. J. J. L. VELÁZQUEZ, Blow up for semilinear parabolic equations, pp. 131-145, Research in applied mathematics, Recent advances in partial differential equations, Editors M. A. Herrero and E. Zuazua, John Wiley & Sons, (1994).
38. J. J. L. VELÁZQUEZ, Classification of singularities for blowing up solutions in higher dimensions, Trans. Amer. Math. Soc. 338 (1993), 441-464.
39. F. VOIROL, Etudes de quelques équations elliptiques fortement non linéaires, Doctoral Thesis, Université de Metz, (1994).
40. F. B. WEISSLER, Single point blow-up for a semilinear initial value problem, J. Differential Equations 55 (1984), 204-224.
41. ∞ blow-up estimate for a nonlinear heat equation, Comm. Pure Appl. Math. 38 (1985), 291-295.