Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Blow-up for degenerate parabolic equations with nonlocal source


Authors: Youpeng Chen, Qilin Liu and Chunhong Xie
Journal: Proc. Amer. Math. Soc. 132 (2004), 135-145
MSC (2000): Primary 35K55, 35K57, 35K65
DOI: https://doi.org/10.1090/S0002-9939-03-07090-4
Published electronically: May 9, 2003
MathSciNet review: 2021256
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This paper deals with the blow-up properties of the solution to the degenerate nonlinear reaction diffusion equation with nonlocal source $x^{q}u_{t}-(x^{\gamma}u_{x})_{x}=\int_{0}^{a}u^{p}dx$ in $(0,a)\times (0,T)$ subject to the homogeneous Dirichlet boundary conditions. The existence of a unique classical nonnegative solution is established and the sufficient conditions for the solution exists globally or blows up in finite time are obtained. Furthermore, it is proved that under certain conditions the blow-up set of the solution is the whole domain.


References [Enhancements On Off] (What's this?)

  • 1. C. Budd, J. Dold and A. Stuart, Blow-up in a partial differential equation with constrained first integral, SIAM J. Appl. Math., 53(1993), 718-742. MR 94a:35015
  • 2. C. Budd, V. A. Galaktionov and J. Chen, Focusing blow-up for qusilinear parabolic equations, Proc. Roy. Soc. Edinb., 128A(1998), 965-992. MR 99h:35084
  • 3. C. Y. Chan, W. Y. Chan, Existence of classical solutions for degenerate semilinear parabolic problems, Appl. Math. Comput., 101(1999), 125-149. MR 99k:35103
  • 4. C. Y. Chan, H. T. Liu, Global existence of solutions for degenerate semilinear parabolic equations, Nonlinear Anal., 34(1998), 617-628. MR 99e:35120
  • 5. C. Y. Chan, J. Yang, Complete blow up for degenerate semilinear parabolic equations, J. Comput. Appl. Math., 113(2000), 353-364. MR 2000m:35104
  • 6. N. Dunford, J. T. Schwartz, Linear Operators, Part II: Spectral Theory, Self Adjoint Operators in Hilbert Space, Interscience Publishers, New York, NY, 1963. MR 32:6181; reprint MR 90g:47001b
  • 7. M. S. Floater, Blow up at the boundary for degenerate semilinear parabolic equations, Arch. Rat. Mech. Anal., 114(1991), 57-77. MR 92m:35139
  • 8. A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs. NJ, 1964. MR 31:6062
  • 9. A. Friedman and B. Mcleod, Blow-up of positive solutions of semilinear heat equations, Indiana Univ. Math. J., 34(1985), 425-447. MR 86j:35089
  • 10. G. S. Ladde, V. Lakshmikantham, A. S. Vatsala, Monotone Iterative Techniques for Nonlinear Differential Equations, Pitman, Boston, MA, 1985. MR 88g:35006
  • 11. O. A. Ladyzenskaya, V. A. Solonnikov, N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Soc., Translation of Mathematical Monographs, Rhode Island, 1967. MR 39:3159b
  • 12. N. W. Mclachlan, Bessel Functions for Engineers, 2nd ed. Oxford at the Clarendon Press, London, England, 1955.
  • 13. H. Ockendon, Channel flow with temperature-dependent viscosity and internal viscous dissipation, J. Fluid Mech., 93(1979), 737-746.
  • 14. C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992. MR 94c:35002
  • 15. A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov and A. P. Mikhailoi, Blow-up in Quasilinear Parabolic Equations, Nauka, Moscow, 1987. MR 96b:35003 (review of translation of Russian orig.)
  • 16. P. Souplet, Blow-up in nonlocal reaction-diffusion equations, SIAM J. Math. Anal., 29 (6)(1998), 1301-1334. MR 99h:35104
  • 17. P. Souplet, Uniform blow-up profile and boundary behavior for diffusion equations with nonlocal nonlinear source, J. Differential Equations, 153(1999), 374-406. MR 2000e:35105
  • 18. M. X. Wang and Y. M. Wang, Properties of positive solutions for non-local reaction diffusion problems, Math. Methods in the Applied Sciences, 19(1996), 1141-1156. MR 97f:35103

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 35K55, 35K57, 35K65

Retrieve articles in all journals with MSC (2000): 35K55, 35K57, 35K65


Additional Information

Youpeng Chen
Affiliation: Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China
Address at time of publication: Department of Mathematics, Yancheng Teachers College, Yancheng 224002, Jiangsu, People’s Republic of China
Email: youpengchen@263.sina.net, youpchen@yahoo.com.cn

Qilin Liu
Affiliation: Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China

Chunhong Xie
Affiliation: Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China

DOI: https://doi.org/10.1090/S0002-9939-03-07090-4
Keywords: Degenerate nonlocal problem, classical solution, global existence, blow-up set
Received by editor(s): August 20, 2002
Published electronically: May 9, 2003
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society