Grow-up of critical solutions for a non-local porous medium problem with Ohmic heating source

  • Evangelos A. Latos
  • Dimitrios E. TzanetisEmail author


We investigate the behaviour of solution uu(x, t; λ) at λ =  λ* for the non-local porous medium equation \({u_t = (u^n)_{xx} + {\lambda}f(u)/({\int_{-1}^1} f(u){\rm d}x)^2}\) with Dirichlet boundary conditions and positive initial data. The function f satisfies: f(s),−f ′ (s) > 0 for s ≥ 0 and s n-1 f(s) is integrable at infinity. Due to the conditions on f, there exists a critical value of parameter λ, say λ*, such that for λ > λ* the solution u = u(x, t; λ) blows up globally in finite time, while for λ ≥ λ* the corresponding steady-state problem does not have any solution. For 0 < λ < λ* there exists a unique steady-state solution w = w(x; λ) while u = u(x, t; λ) is global in time and converges to w as t → ∞. Here we show the global grow-up of critical solution u* =  u(x, t; λ*) (u* (x, t) → ∞, as t → ∞ for all \({x\in(-1,1)}\).

Mathematics Subject Classification (2000)

Primary 35K55 Secondary 35B05 


Non-local parabolic problems Porous medium Grow-up of solutions 


  1. 1.
    Bebernes, J.W., Eberly, D.: Mathematical problems from combustion theory. Appl. Math. Sci. 83. Springer, Berlin (1989)Google Scholar
  2. 2.
    Bebernes J.W., Lacey A.A.: Global existence and finite–time blow–up for a class of non-local parabolic problems. Adv. Differ. Equ. 2, 927–953 (1997)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Bebernes J.W., Talaga P.: Non-local problems modelling shear banding. Commun. Appl. Nonlinear Anal. 3, 79–103 (1996)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Bebernes J.W., Li C., Talaga P.: Single-point blow-up for non-local parabolic problems. Physica D 134, 48–60 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Caglioti E., Lions P.-L., Marchioro C., Pulvirenti M.: A special class of stationary flows for two-dimensional Euler equations: a statistical mechanics description. Commun. Math. Phys. 143, 501–525 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Kavallaris N.I., Lacey A.A., Tzanetis D.E.: Global existence and divergence of critical solutions of a non-local parabolic problem in Ohmic heating process. Nonlinear Anal. TMA 58, 787–812 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Krzywicki A., Nadzieja T.: Some results concerning the Poisson–Boltzmann equation. Zastosowania Mat. (Appl. Math. (Warsaw)) 21, 265–272 (1991)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Lacey A.A.: Thermal runaway in a non-local problem modelling Ohmic heating. Part I: Model derivation and some specail cases. Eur. J. Appl. Math. 6, 127–144 (1995)zbMATHMathSciNetGoogle Scholar
  9. 9.
    Lacey A.A.: Thermal runaway in a non–local problem modelling Ohmic heating. Part II: General proof of blow-up and asymptotics of runaway. Eur. J. Appl. Math. 6, 201–224 (1995)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Latos, E.A., Tzanetis, D.E.: Existence and blow-up of solutions for a non-local filtration and porous medium problem. In: Proceedings of the Edinburgh Mathmatical Society (2009, in press)Google Scholar
  11. 11.
    Liu Q., Liang F., Li Y.: Asymptotic behavior of a nonlocal parabolic problem in Ohmic heating process. Eur. J. Appl. Math. 20(3), 247–267 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Ockendon J., Howison S., Lacey A., Movchan A.: Applied Partial Differential Equations. Oxford Univercity Press, Oxford (1999)zbMATHGoogle Scholar
  13. 13.
    Sattinger D.H.: Monotone methods in nonlinear elliptic and parabolic boundary value problems. Indiana Univ. Math. J. 21, 979–1000 (1972)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Tzanetis D.E.: Blow-up of radially symmetric solutions of a non-local problem modelling Ohmic heating. Electron. J. Differ. Equ. 11, 1–26 (2002)MathSciNetGoogle Scholar
  15. 15.
    Tzanetis D.E., Vlamos P.M.: A nonlocal problem modelling Ohmic heating with variable thermal conductivity. Nonlinear Anal. RWA 2, 443–454 (2001)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  1. 1.Department of Mathematics, School of Applied Mathematical and Physical SciencesNational Technical University of AthensAthensGreece

Personalised recommendations