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Blow-up of solutions of nonlinear parabolic inequalities

Author: Steven D. Taliaferro
Journal: Trans. Amer. Math. Soc. 361 (2009), 3289-3302
MSC (2000): Primary 35K55, 35B40, 35R45
Published electronically: January 26, 2009
MathSciNet review: 2485427
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Abstract | References | Similar Articles | Additional Information

Abstract: We study nonnegative solutions $ u(x,t)$ of the nonlinear parabolic inequalities

$\displaystyle au^\lambda \le u_t - \Delta u \le u^\lambda $

in various subsets of $ {\bf R}^n\times {\bf R}$, where $ \lambda>\frac{n+2}{n}$ and $ a\in (0,1)$ are constants. We show that changing the value of $ a$ in the open interval $ (0,1)$ can dramatically affect the blow-up of these solutions.

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  • 1. D. Andreucci and E. DiBenedetto, On the Cauchy problem and initial traces for a class of evolution equations with strongly nonlinear sources, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 18, 363-441 (1991). MR 1145316 (92m:35146)
  • 2. D. Andreucci, M. A. Herrero, and J. J. L. Velázquez, Liouville theorems and blow up behaviour in semilinear reaction diffusion systems, Ann. Inst. H. Poincaré Anal. Non Linéaire 14, 1-53 (1997). MR 1437188 (98e:35088)
  • 3. M.-F. Bidaut-Véron, Initial blow-up for the solutions of a semilinear parabolic equation with source term. Équations aux dérivées partielles et applications, 189-198, Gauthier-Villars, Ed. Sci. Méd. Elsevier, Paris, 1998. MR 1648222 (99h:35082)
  • 4. Y. Giga and R. V. Kohn, Characterizing blowup using similarity variables, Indiana Univ. Math. J. 36 (1987), 1-40. MR 876989 (88c:35021)
  • 5. Y. Giga, S. Matsui, and S. Sasayama, Blow up rate for semilinear heat equations with subcritical nonlinearity, Indiana Univ. Math. J. 53 (2004), 483-514. MR 2060042 (2005g:35153)
  • 6. M. A. Herrero and J. J. L. Velázquez, Explosion de solutions d'équations paraboliques semilinéaires supercritiques, C. R. Acad. Sci. Paris Sér. I Math. 319 (1994), 141-145. MR 1288393 (95i:35037)
  • 7. O. Kavian, Remarks on the large time behaviour of a nonlinear diffusion equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 4 (1987), 423-452. MR 921547 (89b:35013)
  • 8. H. Matano and F. Merle, On nonexistence of type II blowup for a supercritical nonlinear heat equation, Comm. Pure Appl. Math. 57 (2004), 1494-1541. MR 2077706 (2005e:35115)
  • 9. F. Merle and H. Zaag, Optimal estimates for blowup rate and behavior for nonlinear heat equations, Comm. Pure Appl. Math. 51 (1998), 139-196. MR 1488298 (98k:35107)
  • 10. N. Mizoguchi, Type-II blowup for a semilinear heat equation, Adv. Differential Equations 9 (2004), 1279-1316. MR 2099557 (2005k:35204)
  • 11. P. Poláčik, P. Quittner, and P. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems. Part II: Parabolic equations, Indiana Univ. Math. J. 56 (2007), 879-908. MR 2317549
  • 12. P. Poláčik and E. Yanagida, On bounded and unbounded global solutions of a supercritical semilinear heat equation, Math. Ann. 327 (2003), 745-771. MR 2023315 (2005b:35133)
  • 13. P. Quittner and P. Souplet, Superlinear parabolic problems. Blow-up, global existence and steady states, Birkhäuser, Basel, 2007. MR 2346798 (2008f:35001)
  • 14. P. Quittner, P. Souplet, and M. Winkler, Initial blow-up rates and universal bounds for nonlinear heat equations, J. Differential Equations 196, 316-339 (2004). MR 2028111 (2005a:35143)
  • 15. P. Souplet, personal communication.
  • 16. S. Taliaferro, Local behavior and global existence of positive solutions of $ au^\lambda\le-\Delta u\le u^\lambda$, Ann. Inst. H. Poincaré Anal. Non Linéaire 19 (2002), 889-901. MR 1939089 (2003j:35110)
  • 17. S. Taliaferro, Isolated singularities of nonlinear parabolic inequalities, Math. Ann. 338 (2007), 555-586. MR 2317931
  • 18. L. Véron, Singularities of solutions of second order quasilinear equations. Pitman Research Notes in Mathematics Series, 353. Longman, Harlow, 1996. MR 1424468 (98b:35053)

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Additional Information

Steven D. Taliaferro
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368

Received by editor(s): September 4, 2007
Published electronically: January 26, 2009
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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