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

Philippe Souplet; Fred B. Weissler

Annales de l'I.H.P. Analyse non linéaire (1999)

  • Volume: 16, Issue: 3, page 335-371
  • ISSN: 0294-1449

How to cite

top

Souplet, Philippe, and Weissler, Fred B.. "Poincaré's inequality and global solutions of a nonlinear parabolic equation." Annales de l'I.H.P. Analyse non linéaire 16.3 (1999): 335-371. <http://eudml.org/doc/78468>.

@article{Souplet1999,
author = {Souplet, Philippe, Weissler, Fred B.},
journal = {Annales de l'I.H.P. Analyse non linéaire},
keywords = {global existence; blow-up; exponential decay; gradient term},
language = {eng},
number = {3},
pages = {335-371},
publisher = {Gauthier-Villars},
title = {Poincaré's inequality and global solutions of a nonlinear parabolic equation},
url = {http://eudml.org/doc/78468},
volume = {16},
year = {1999},
}

TY - JOUR
AU - Souplet, Philippe
AU - Weissler, Fred B.
TI - Poincaré's inequality and global solutions of a nonlinear parabolic equation
JO - Annales de l'I.H.P. Analyse non linéaire
PY - 1999
PB - Gauthier-Villars
VL - 16
IS - 3
SP - 335
EP - 371
LA - eng
KW - global existence; blow-up; exponential decay; gradient term
UR - http://eudml.org/doc/78468
ER -

References

top
  1. [Ad] R.A. Adams, Sobolev Spaces, Academic Press, New York, 1975. Zbl0314.46030MR450957
  2. [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. Zbl0801.35038MR1226611
  3. [Am] H. Amann, Existence and regularity for semilinear parabolic evolution equations, Ann. Scuola Norm. Sup. Pisa, Vol. 11, 4, 1984, pp. 593-676. Zbl0625.35045MR808425
  4. [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 equations1992, N. G. Lloyd et al. Eds, Birkhäuser. Zbl0795.35051MR1167826
  5. [BC] H. Brezis and T. Cazenave, to appear. 
  6. [B] F.E. Browder, On the spectral theory of elliptic operators, I, Math. Ann., Vol. 142, 1961, pp. 22-30. Zbl0104.07502MR209909
  7. [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. Zbl0682.35010MR1000727
  8. [E] M. Escobedo, Personal communication. 
  9. [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. Zbl0768.35047MR1052569
  10. [Fr1] A. Friedman, Partial Differential Equations, 1969, Holt, Rinehart and Winston, Inc., New York. Zbl0224.35002MR445088
  11. [Fr2] A. Friedman, Blow up of solutions of parabolic equations, Nonlinear diffusion equations and their equilibrium states, I1988, W. M. Ni et al. Eds, Springer. Zbl0669.35047MR956073
  12. [H] W. Hayman, Some bounds for principal frequency, ApplicableAnalysis, Vol. 7, 1978, pp. 247-254. Zbl0383.35053MR492339
  13. [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. Zbl0661.35053MR1013085
  14. [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. Zbl0174.15403MR241822
  15. [LN] T. Lee and W. Ni, Globalexistence, 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. Zbl0785.35011MR1057781
  16. [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. Zbl0702.35126MR977493
  17. [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. Zbl0538.35058MR724014
  18. [Q1] P. Quittner, Blow-up for semilinear parabolic equations with a gradient term, Math. Meth. Appl. Sc., Vol. 14, 1991, pp. 413-417. Zbl0768.35049MR1119238
  19. [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. Zbl0794.35089MR1240209
  20. [O] R. Osserman, A note on Hayman's theorem on the bass note of a drum, Comment. Math. Helvetici, Vol. 52, 11977, pp. 545-555. Zbl0374.52008MR459099
  21. [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. Zbl0843.35044MR1354713
  22. [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. Zbl0858.35067MR1412998
  23. [S3] P. Souplet, Geometry of unbounded domains, Poincaré inequalities and stability in semilinear parabolic equations, Communications in Partial Differential Equations, to appear. Zbl0926.35064MR1680893
  24. [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. Zbl0990.35061MR1422101
  25. [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. Zbl0892.35011MR1460184
  26. [W] F.B. Weissler, Semilinear evolution equations in Banach spaces, J. Functional Analysis, Vol. 32, 4, 1979, pp. 277-296. Zbl0419.47031MR538855

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.