Poincaré's inequality and global solutions of a nonlinear parabolic equation
Annales de l'I.H.P. Analyse non linéaire, Volume 16 (1999) no. 3, p. 335-371
@article{AIHPC_1999__16_3_335_0,
     author = {Souplet, Philippe and Weissler, Fred B.},
     title = {Poincar\'e's inequality and global solutions of a nonlinear parabolic equation},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Gauthier-Villars},
     volume = {16},
     number = {3},
     year = {1999},
     pages = {335-371},
     zbl = {0924.35065},
     mrnumber = {1687278},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_1999__16_3_335_0}
}
Souplet, Philippe; Weissler, Fred B. Poincaré's inequality and global solutions of a nonlinear parabolic equation. Annales de l'I.H.P. Analyse non linéaire, Volume 16 (1999) no. 3, pp. 335-371. http://www.numdam.org/item/AIHPC_1999__16_3_335_0/

[Ad] R.A. Adams, Sobolev Spaces, Academic Press, New York, 1975. | MR 450957 | Zbl 0314.46030

[AE] 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. | MR 1226611 | Zbl 0801.35038

[Am] H. Amann, Existence and regularity for semilinear parabolic evolution equations, Ann. Scuola Norm. Sup. Pisa, Vol. 11, 4, 1984, pp. 593-676. | Numdam | MR 808425 | Zbl 0625.35045

[AW] L. Alfonsi and F.B. Weissler, blow-up in IRN for a parabolic equation with a damping nonlinear gradient term, Progress in nonlinear differential equations 1992, N. G. Lloyd et al. Eds, Birkhäuser. | MR 1167826 | Zbl 0795.35051

[BC] H. Brezis and T. Cazenave, to appear.

[B] F.E. Browder, On the spectral theory of elliptic operators, I, Math. Ann., Vol. 142, 1961, pp. 22-30. | MR 209909 | Zbl 0104.07502

[CW] 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. | MR 1000727 | Zbl 0682.35010

[E] M. Escobedo, Personal communication.

[F] 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. | MR 1052569 | Zbl 0768.35047

[Fr1] A. Friedman, Partial Differential Equations, 1969, Holt, Rinehart and Winston, Inc., New York. | MR 445088 | Zbl 0224.35002

[Fr2] 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. | MR 956073 | Zbl 0669.35047

[H] W. Hayman, Some bounds for principal frequency, Applicable Analysis, Vol. 7, 1978, pp. 247-254. | MR 492339 | Zbl 0383.35053

[KP] 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. | MR 1013085 | Zbl 0661.35053

[LSU] 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. | MR 241822 | Zbl 0174.15403

[LN] 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. | MR 1057781 | Zbl 0785.35011

[LPSS] 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. | MR 977493 | Zbl 0702.35126

[L] E.A. Lieb, On the lowest eigenvalue of the Laplacian for the intersection of two domains, Invent. Math., Vol. 74, 1983, pp. 441-448. | MR 724014 | Zbl 0538.35058

[Q1] P. Quittner, Blow-up for semilinear parabolic equations with a gradient term, Math. Meth. Appl. Sc., Vol. 14, 1991, pp. 413-417. | MR 1119238 | Zbl 0768.35049

[Q2] 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. | MR 1240209 | Zbl 0794.35089

[O] 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. | MR 459099 | Zbl 0374.52008

[S1] 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. | MR 1354713 | Zbl 0843.35044

[S2] 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. | MR 1412998 | Zbl 0858.35067

[S3] P. Souplet, Geometry of unbounded domains, Poincaré inequalities and stability in semilinear parabolic equations, Communications in Partial Differential Equations, to appear. | MR 1680893 | Zbl 0926.35064

[STW] 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. | MR 1422101 | Zbl 0990.35061

[SW] 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. | MR 1460184 | Zbl 0892.35011

[W] F.B. Weissler, Semilinear evolution equations in Banach spaces, J. Functional Analysis, Vol. 32, 4, 1979, pp. 277-296. | MR 538855 | Zbl 0419.47031