Open Access

Blowup for degenerate and singular parabolic system with nonlocal source

Boundary Value Problems20062006:21830

https://doi.org/10.1155/BVP/2006/21830

Received: 23 January 2006

Accepted: 7 April 2006

Published: 13 August 2006

Abstract

We deal with the blowup properties of the solution to the degenerate and singular parabolic system with nonlocal source and homogeneous Dirichlet boundary conditions. The existence of a unique classical nonnegative solution is established and the sufficient conditions for the solution that exists globally or blows up in finite time are obtained. Furthermore, under certain conditions it is proved that the blowup set of the solution is the whole domain.

[1234567891011121314151617181920]

Authors’ Affiliations

(1)
Department of Mathematics, Sichuan University

References

  1. Budd C, Dold B, Stuart A: Blowup in a partial differential equation with conserved first integral. SIAM Journal on Applied Mathematics 1993,53(3):718-742. 10.1137/0153036MathSciNetView ArticleMATHGoogle Scholar
  2. Budd C, Galaktionov VA, Chen J: Focusing blow-up for quasilinear parabolic equations. Proceedings of the Royal Society of Edinburgh. Section A 1998,128(5):965-992. 10.1017/S0308210500030018MathSciNetView ArticleMATHGoogle Scholar
  3. Chan CY, Chan WY: Existence of classical solutions for degenerate semilinear parabolic problems. Applied Mathematics and Computation 1999,101(2-3):125-149. 10.1016/S0096-3003(98)10002-4MathSciNetView ArticleMATHGoogle Scholar
  4. Chan CY, Liu HT: Global existence of solutions for degenerate semilinear parabolic problems. Nonlinear Analysis 1998,34(4):617-628. 10.1016/S0362-546X(97)00599-3MathSciNetView ArticleMATHGoogle Scholar
  5. Chan CY, Yang J: Complete blow-up for degenerate semilinear parabolic equations. Journal of Computational and Applied Mathematics 2000,113(1-2):353-364. 10.1016/S0377-0427(99)00266-6MathSciNetView ArticleMATHGoogle Scholar
  6. Chen YP, Liu Q, Xie CH: Blow-up for degenerate parabolic equations with nonlocal source. Proceedings of the American Mathematical Society 2004,132(1):135-145. 10.1090/S0002-9939-03-07090-4MathSciNetView ArticleMATHGoogle Scholar
  7. Chen YP, Xie CH: Blow-up for degenerate, singular, semilinear parabolic equations with nonlocal source. Acta Mathematica Sinica 2004,47(1):41-50.MathSciNetMATHGoogle Scholar
  8. Dunford N, Schwartz JT: Linear Operators. Part II: Spectral Theory. Self Adjoint Operators in Hilbert Space. John Wiley & Sons, New York; 1963:ix+pp. 859–1923+7.MATHGoogle Scholar
  9. Floater MS: Blow-up at the boundary for degenerate semilinear parabolic equations. Archive for Rational Mechanics and Analysis 1991,114(1):57-77. 10.1007/BF00375685MathSciNetView ArticleMATHGoogle Scholar
  10. Friedman A, McLeod B: Blow-up of positive solutions of semilinear heat equations. Indiana University Mathematics Journal 1985,34(2):425-447. 10.1512/iumj.1985.34.34025MathSciNetView ArticleMATHGoogle Scholar
  11. Laddle GS, Lakshmikantham V, Vatsala AS: Monotone Iterative Techniques for Nonlinear Differential Equations. Pitman, Massachusetts; 1985.Google Scholar
  12. Ladyženskaja OA, Solonikiv VA, Ural'ceva NN: Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs. Volume 23. American Mathematical Society, Rhode Island; 1967:xi+648.Google Scholar
  13. Mclachlan NW: Bessel Functions for Engineers. 2nd edition. Clarendon Press, Oxford University Press, London; 1955.MATHGoogle Scholar
  14. Ockendon H: Channel flow with temperature-dependent viscosity and internal viscous dissipation. Journal of Fluid Mechanics 1979, 93: 737-746. 10.1017/S0022112079002007View ArticleMATHGoogle Scholar
  15. Pao CV: Nonlinear Parabolic and Elliptic Equations. Plenum, New York; 1992:xvi+777.MATHGoogle Scholar
  16. Qi Y-W, Levine HA: The critical exponent of degenerate parabolic systems. Zeitschrift für Angewandte Mathematik und Physik 1993,44(2):249-265. 10.1007/BF00914283MathSciNetView ArticleMATHGoogle Scholar
  17. Samarskii AA, Galaktionov VA, Kurdyumov SP, Mikhailoi AP: Blow-up in Qusilinear Parabolic Equations. Nauka, Moscow; 1987.Google Scholar
  18. Souplet P: Blow-up in nonlocal reaction-diffusion equations. SIAM Journal on Mathematical Analysis 1998,29(6):1301-1334. 10.1137/S0036141097318900MathSciNetView ArticleMATHGoogle Scholar
  19. Souplet P: Uniform blow-up profiles and boundary behavior for diffusion equations with nonlocal nonlinear source. Journal of Differential Equations 1999,153(2):374-406. 10.1006/jdeq.1998.3535MathSciNetView ArticleMATHGoogle Scholar
  20. Wang MX, Wang YM: Properties of positive solutions for non-local reaction-diffusion problems. Mathematical Methods in the Applied Sciences 1996,19(14):1141-1156. 10.1002/(SICI)1099-1476(19960925)19:14<1141::AID-MMA811>3.0.CO;2-9MathSciNetView ArticleMATHGoogle Scholar

Copyright

© Zhou et al. 2006

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.