Poincaré's inequality and global solutions of a nonlinear parabolic equation

Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire

Volumn 16, Issue 3, 1999, Pages 335-371

  Souplet, Philippe ^a   Weissler, Fred B ^a  

^a UNIVERSITÉ PARIS 13   (France)

Author keywords

Blow up; Bounded solutions; Critical exponent; Exponential decay; Global existence; Gradient term; Nonlinear parabolic equations; Poincar inequality

Indexed keywords

EID: 0012103230     PISSN: 02941449     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0294-1449(99)80017-1     Document Type: Article

References (26)

1. R. A. ADAMS, Sobolev Spaces, Academic Press, New York, 1975.
2. J. AGUIRRE and M. ESCOBEDO, On the blow up of solutions for a convective reaction diffusion equation, Proc. Roy. Soc. Edinburgh., Vol. 123, 1993, pp. 433-460.
3. H. AMANN, Existence and regularity for semilinear parabolic evolution equations, Ann. Scuola Norm. Sup. Pisa, Vol. 11, 4, 1984, pp. 593-676.
4. N for a parabolic equation with a damping nonlinear gradient term, Progress in nonlinear differential equations 1992, N. G. Lloyd et al. Eds, Birkhäuser.
5. H. BREZIS and T. CAZENAVE, to appear.
6. F. E. BROWDER, On the spectral theory of elliptic operators, I, Math. Ann., Vol. 142, 1961, pp. 22-30.
7. M. CHIPOT and F. B. WEISSLER, Some blow up results for a nonlinear parabolic problem with a gradient term, SIAM J. Math. Anal., Vol. 20, 4, 1989, pp. 886-907.
8. M. ESCOBEDO, Personal communication.
9. M. FILA, Remarks on blow up for a nonlinear parabolic equation with a gradient term, Proc. Amer. Math. Soc., Vol. 111, 2, 1991, pp. 795-801.
10. A. FRIEDMAN, Partial Differential Equations, 1969, Holt, Rinehart and Winston, Inc., New York.
11. A. FRIEDMAN, Blow up of solutions of parabolic equations, Nonlinear diffusion equations and their equilibrium states, I 1988, W. M. Ni et al. Eds, Springer.
12. W. HAYMAN, Some bounds for principal frequency, Applicable Analysis, Vol. 7, 1978, pp. 247-254.
13. B. KAWOHL and L. A. PELETIER, Observations on blow up and dead cores for nonlinear parabolic equations, Math. Z., Vol. 202, 1989, pp. 207-217.
14. O. A. LADYZENSKAJA, V. A. SOLONNIKOV and N. N. URALCEVA, Linear and Quasilinear Equations of Parabolic Type, 1968, Translations of Mathematical Monographs, Amer. Math. Soc., Providence, RI.
15. T. LEE and W. NI, Global existence, large time behaviour and life span of solutions of a semilinear parabolic Cauchy problem, Trans. Amer. Math. Soc., Vol. 333, 1, 1992, pp. 365-378.
16. H. A. LEVINE, L. N. PAYNE, P. E. SACKS and B. STRAUGHAN, Analysis of convective reaction-diffusion equation (II), SIAM J. Math. Anal., Vol. 20, 1, 1989, pp. 133-147.
17. E. A. LIEB, On the lowest eigenvalue of the Laplacian for the intersection of two domains, Invent. Math., Vol. 74, 1983, pp. 441-448.
18. P. QUITTNER, Blow-up for semilinear parabolic equations with a gradient term, Math. Meth. Appl. Sc., Vol. 14, 1991, pp. 413-417.
19. P. QUITTNER, On global existence and stationary solutions for two classes of semilinear parabolic equations, Comment. Math. Univ. Carolinae, Vol. 34, 1, 1993, pp. 105-124.
20. R. OSSERMAN, A note on Hayman's theorem on the bass note of a drum, Comment. Math. Helvetici, Vol. 52, 1 1977, pp. 545-555.
21. P. SOUPLET, Résultats d'explosion en temps fini pour une équation de la chaleur non linéaire, C. R. Acad. Sc. Paris, Vol. 321, Série I, 1995, pp. 721-726.
22. P. SOUPLET, Finite time blow up for a nonlinear parabolic equation with a gradient term and applications, Math. Meth. Appl. Sc., Vol. 19, 1996, pp. 1317-1333.
23. P. SOUPLET, Geometry of unbounded domains, Poincaré inequalities and stability in semilinear parabolic equations, Communications in Partial Differential Equations, to appear.
24. P. SOUPLET, S. TAYACHI and F. B. WEISSLER, Exact self-similar blow-up of solutions of a semilinear parabolic equation with a nonlinear gradient term, Indiana Univ. Math. J., Vol. 48, 3, 1996, pp. 655-682.
25. P. SOUPLET and F. B. WEISSLER, Self-similar sub-solutions and blow-up for nonlinear parabolic equations, J. Math. Anal. Appl., Vol. 212, 1997, pp. 60-74.
26. F. B. WEISSLER, Semilinear evolution equations in Banach spaces, J. Functional Analysis, Vol. 32, 4, 1979, pp. 277-296.