Abstract and Applied Analysis

Blow-Up Analysis for a Quasilinear Parabolic Equation with Inner Absorption and Nonlinear Neumann Boundary Condition

Zhong Bo Fang and Yan Chai

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We investigate an initial-boundary value problem for a quasilinear parabolic equation with inner absorption and nonlinear Neumann boundary condition. We establish, respectively, the conditions on nonlinearity to guarantee that u ( x , t ) exists globally or blows up at some finite time t * . Moreover, an upper bound for t * is derived. Under somewhat more restrictive conditions, a lower bound for t * is also obtained.

Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2014), Article ID 289245, 8 pages.

Dates
First available in Project Euclid: 6 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1412607222

Digital Object Identifier
doi:10.1155/2014/289245

Mathematical Reviews number (MathSciNet)
MR3206777

Zentralblatt MATH identifier
07022099

Citation

Fang, Zhong Bo; Chai, Yan. Blow-Up Analysis for a Quasilinear Parabolic Equation with Inner Absorption and Nonlinear Neumann Boundary Condition. Abstr. Appl. Anal. 2014, Special Issue (2014), Article ID 289245, 8 pages. doi:10.1155/2014/289245. https://projecteuclid.org/euclid.aaa/1412607222


Export citation

References

  • J. Bebernes and D. Eberly, Mathematical Problems from Combustion Theory, vol. 83 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1989.
  • C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, NY, USA, 1992.
  • J. L. Vazquez, The Porous Medium Equations: Mathematical Theory, Oxford University Press, Oxford, UK, 2007.
  • J. Filo, “Diffusivity versus absorption through the boundary,” Journal of Differential Equations, vol. 99, no. 2, pp. 281–305, 1992.
  • H. A. Levine and L. E. Payne, “Nonexistence theorems for the heat equation with nonlinear boundary conditions and for the porous medium equation backward in time,” Journal of Differential Equations, vol. 16, pp. 319–334, 1974.
  • B. Straughan, Explosive Instabilities in Mechanics, Springer, Berlin, Germany, 1998.
  • A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov, and A. P. Mikhailov, Blow-Up in Quasilinear Parabolic Equations, vol. 19 of de Gruyter Expositions in Mathematics, Walter de Gruyter, Berlin, Germany, 1995.
  • P. Quittner and P. Souplet, Superlinear Parabolic Problems: Blow-Up, Global Existence and Steady States, Birkhäuser Advanced Texts, Birkhäuser, Basel, Switzerland, 2007.
  • C. Bandle and H. Brunner, “Blowup in diffusion equations: a survey,” Journal of Computational and Applied Mathematics, vol. 97, no. 1-2, pp. 3–22, 1998.
  • V. A. Galaktionov and J. L. Vázquez, “The problem of blow-up in nonlinear parabolic equations,” Discrete and Continuous Dynamical Systems, vol. 8, no. 2, pp. 399–433, 2002.
  • H. A. Levine, “The role of critical exponents in blowup theorems,” SIAM Review, vol. 32, no. 2, pp. 262–288, 1990.
  • H. A. Levine, “Nonexistence of global weak solutions to some properly and improperly posed problems of mathematical physics: the method of unbounded Fourier coefficients,” Mathematische Annalen, vol. 214, pp. 205–220, 1975.
  • L. E. Payne, G. A. Philippin, and P. W. Schaefer, “Blow-up phenomena for some nonlinear parabolic problems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 10, pp. 3495–3502, 2008.
  • M. Marras and S. Vernier Piro, “On global existence and bounds for blow-up time in nonlinear parabolic problems with time dependent coefficients,” Discrete and Continuous Dynamical Systems, vol. 2013, pp. 535–544, 2013.
  • Y. Li, Y. Liu, and C. Lin, “Blow-up phenomena for some nonlinear parabolic problems under mixed boundary conditions,” Nonlinear Analysis: Real World Applications, vol. 11, no. 5, pp. 3815–3823, 2010.
  • Y. Li, Y. Liu, and S. Xiao, “Blow-up phenomena for some nonlinear parabolic problems under Robin boundary conditions,” Mathematical and Computer Modelling, vol. 54, no. 11-12, pp. 3065–3069, 2011.
  • C. Enache, “Blow-up phenomena for a class of quasilinear parabolic problems under Robin boundary condition,” Applied Mathematics Letters, vol. 24, no. 3, pp. 288–292, 2011.
  • J. Ding, “Global and blow-up solutions for nonlinear parabolic equations with Robin boundary conditions,” Computers & Mathematics with Applications, vol. 65, no. 11, pp. 1808–1822, 2013.
  • Y. Liu, “Blow-up phenomena for the nonlinear nonlocal porous medium equation under Robin boundary condition,” Computers & Mathematics with Applications, vol. 66, no. 10, pp. 2092–2095, 2013.
  • L. E. Payne, G. A. Philippin, and S. Vernier Piro, “Blow-up phenomena for a semilinear heat equation with nonlinear boundary conditon, I,” Zeitschrift für Angewandte Mathematik und Physik, vol. 61, no. 6, pp. 999–1007, 2010.
  • L. E. Payne, G. A. Philippin, and S. Vernier Piro, “Blow-up phenomena for a semilinear heat equation with nonlinear boundary condition, II,” Nonlinear Analysis: Theory, Methods & Applications, vol. 73, no. 4, pp. 971–978, 2010.
  • Y. Liu, “Lower bounds for the blow-up time in a non-local reaction diffusion problem under nonlinear boundary conditions,” Mathematical and Computer Modelling, vol. 57, no. 3-4, pp. 926–931, 2013.
  • Y. Liu, S. Luo, and Y. Ye, “Blow-up phenomena for a parabolic problem with a gradient nonlinearity under nonlinear boundary conditions,” Computers & Mathematics with Applications, vol. 65, no. 8, pp. 1194–1199, 2013.
  • Z. B. Fang, R. Yang, and Y. Chai, “Lower bounds estimate for the blow-up time of a slow diffusion equation with nonlocal source and inner absorption,” Mathematical Problems in Engineering, vol. 2014, Article ID 764248, 6 pages, 2014. \endinput