Export file:

Format

  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text

Content

  • Citation Only
  • Citation and Abstract

Blow-up for degenerate nonlinear parabolic problem

Department of Mathematics, Texas A&M University-Texarkana, Texarkana, TX 75503

Special Issues: Initial and Boundary Value Problems for Differential Equations

In this paper, we deal with the existence, uniqueness, and finite time blow-up of the solution to the degenerate nonlinear parabolic problem: uτ=(ξrumuξ)ξ/ξr + up for 0 < ξ < a, 0 < τ < Γ, u (ξ, 0) = u0 (ξ) for 0 ≤ ξa, and u (0, τ) = 0 = u (a, τ) for 0 < τ < Γ, where u0 (ξ) is a positive function and u0 (0) = 0 = u0 (a). In addition, we prove that u exists globally if a is small through constructing a global-exist upper solution, and uτ blows up in a finite time.
  Figure/Table
  Supplementary
  Article Metrics

References

1. C. Y. Chan, W. Y. Chan, Existence of classical solutions of nonlinear degenerate parabolic problems, Proc. Dynam. Systems Appl., 5 (2008), 85-91.

2. C. Y. Chan, C. S. Chen, A numerical method for semilinear singular parabolic quenching problems, Q. Appl. Math., 47 (1989), 45-57.    

3. W. Deng, Z. Duan, C. Xie, The blow-up rate for a degenerate parabolic equation with a non-local source, J. Math. Anal. Appl., 264 (2001), 577-597.    

4. V. A. Galaktionov, Boundary-value problem for the nonlinear parabolic equation ut = △uσ+1+uβ, Differ. Uravn., 17 (1981), 836-842.

5. J. Gratton, F. Minotti, S. M. Mahajan, Theory of creeping gravity currents of a non-Newtonian liquid, Phy. Rev. E., 60 (1999), 6960-6967.    

6. M. E. Gurtin, R. C. MacCamy, On the diffusion of biological populations, Math. Biosci., 33 (1977), 35-49.    

7. H. E. Huppert, The propagation of two-dimensional and axisymmetric viscous gravity currents over a rigid horizontal surface, J. Fluid Mech., 121 (1982), 43-58.    

8. H. A. Levine, P. E. Sacks, Some existence and nonexistence theorems for solutions of degenerate paraoblic equations, J. Differ. Equations, 52 (1984), 135-161.    

9. C. V. Pao, Nonlinear Parabolic and Elliptic Equations, New York: Plenum Press, 1992.

10. M. H. Protter, H. F. Weinberger, Maximum Principles in Differential Equations, New York:Springer-Verlag, 1984.

11. M. I. Roux, Numerical solution of nonlinear reaction diffusion processes, SIAM J. Numer. Anal.,37 (2000), 1644-1656.    

12. P. L. Sachdev, Self-Similarity and Beyond: Exact Solutions of Nonlinear Problems, Florida:Chapman and Hall/CRC, 2000.

13. P. E. Sacks, Global behavior for a class of nonlinear evolution equations, SIAM J. Math. Anal.,16 (1985), 233-250.    

14. A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov, et al. Blow-up in Quasilinear Parabolic Equations, New York: Walter de Gruyter, 1995.

15. A. D. Solomon, Melt time and heat flux for a simple PCM body, Sol. Energy, 22 (1979), 251-257.    

16. W. Walter, Differential and Integral Inequalities, New York: Springer-Verlag, 1970.

© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Article outline

Show full outline
Copyright © AIMS Press All Rights Reserved