Advances in Quenching

  • Howard A. Levine
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 7)


In this paper we shall survey the literature on the so called quenching problem since 1985, when the last survey on the subject appeared [18]. We shall also present some open problems which have arisen in consequence of the results of the recent literature.


Stationary Solution Finite Time Parabolic Problem Infinite Time Nonlinear Boundary Condition 
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  1. 1.
    A. Ackor, B. Kawohl, “Remarks on Quenching,” Nonlinear Analysis, TMA 13 (1989), 53–61.CrossRefGoogle Scholar
  2. 2.
    C. Bandle, C. M. Brauner, “Singular Perturbation method in a parabolic problem with free boundary,” Proc BAIL IVth Conf. Novosibirsk, eds, S. K. Godunov, J. J. H. Miller, V. A. Novikov, Boole Press, Dublin (1987), 7–14.Google Scholar
  3. 3.
    P. Baras, L. Cohen, “Complete blow up for the solution of a semilinear heat equation,” J. Funct. Anal. 71 (1987), 142–174.MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    C. Y. Chan, C. S. Chen, “Critical lengths for global existence of solutions for coupled semilinear singular parabolic problems.” (preprint)Google Scholar
  5. 5.
    C. Y. Chan, H. G. Kaper, “Quenching for semilinear singular parabolic problems,” SIAM J. Math. Anal. 20 (1989), 558–566.MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    C. Y. Chan, C. S. Chen, “A numerical method for semilinear singular parabolic quenching problems,” Quart. Appl. Math. 47 (1989), 45–57.MathSciNetMATHGoogle Scholar
  7. 7.
    C. Y. Chan, M. K. Kwong, “Existence results of steady states of semilinear reaction-diffusion equations and their applications,” JDE 77 (1989), 304–321.MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    C. Y. Chan, M. K. Kwong, “Quenching phenomena for singular nonlinear parabolic equations,” Nonlinear Analysis 12 (1988), 1377–1383.MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    K. Deng, H. A. Levine, “On the blow up of u t at quenching,” Proc. A.M.S. 106 (1989) 1049–1056.MathSciNetMATHGoogle Scholar
  10. 10.
    M. Fila, J. Hulshof, “A note on the quenching rate.” (preprint).Google Scholar
  11. 11.
    M. Fila, B. Kawohl, “Is quenching in infinite time possible?” Quail. Appl. Math., (in press).Google Scholar
  12. 12.
    M. S. Floater, “Blow up at the boundary for degenerate semilinear parabolic equations,” (in press).Google Scholar
  13. 13.
    A. Friedman, B. McLeod, “Blow-up of positive solutions of semilinear heat equations,” Indiana University Math. J. 34 (1985), 425–447.MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    J. S. Guo, “On the semilinear elliptic equation Δw−1/2y. Δw + λw-w = 0 in R N.” (manuscript).Google Scholar
  15. 15.
    J. S. Guo, “On the quenching behavior of the solution of a semilinear parabolic equation,” J. Math. Anal. Appl., (in press).Google Scholar
  16. 16.
    H. Kawarada, “On solutions of the initial-boundary value problem for u t = u xx + 1/(1-u)”, RIMS Kyoto U. 10 (1975), 729–736.MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    H. A. Levine, “Quenching, nonquenching, and beyond quenching for solutions of some parabolic equations,” Annali di Mat. Pura et Applic. (in press).Google Scholar
  18. 18.
    H. A. Levine, “The phenomenon of quenching: A survey,” in Proc. Vlth Int. Conf. on Trends in the Theory and Practice of Nonlinear Analysis, North Holland, N.Y., (1985).Google Scholar
  19. 19.
    S. R. Park, “The phenomenon of quenching in the presence of convection,” Ph.D. Thesis, Iowa State University, (1989).Google Scholar
  20. 20.
    D. Phillips, “Existence of solutions to a quenching problem,” Applicable Analysis 24 (1987), 253–264.MathSciNetMATHCrossRefGoogle Scholar

Hyperbolic Problems

  1. 21.
    J. Axtell, “A numerical study of the derivatives of solutions of the wave equation with a singular forcing term at quenching,” Num. Meth. for PDE 1 (1989), 53–76.MathSciNetCrossRefGoogle Scholar
  2. 22.
    P. H. Chang, H.A. Levine, “The quenching of solutions of semilinear hyperbolic equations,” SIAM J. Math Anal. 12 (1982), 893–903.MathSciNetCrossRefGoogle Scholar
  3. 23.
    M. A. Rammaha, “On the quenching of solutions of the wave equation with a nonlinear boundary condition,” J. Reine Ang. Mat. (in print).Google Scholar
  4. 24.
    R. A. Smith, “On a Hyperbolic Quenching problem in several dimensions,” SIAM J. Math Anal. 20 (1989), 1081–1094.MathSciNetMATHCrossRefGoogle Scholar
  5. 25.
    N. Sternberg, “Blowup near higher modes of nonlinear wave equations,” TAMS, 296 (1986), 315–325.MathSciNetMATHCrossRefGoogle Scholar

Other References

  1. 26.
    Dziuk and Kawohl, “On radially symmetric mean curvature flow,” (to appear).Google Scholar
  2. 27.
    Y. Giga, R. V. Kohn, “Nondegeneracy of blowup for semilinear heat equations,” Hokkaido U. Preprint Series #46, October 1988.Google Scholar
  3. 28.
    H. A. Levine, “The quenching of solutions of linear parabolic and hyperbolic equations with nonlinear boundary conditions, ” SIAM J. Math. Anal. 14 (1983) 1139–1153.MathSciNetMATHCrossRefGoogle Scholar
  4. 29.
    H. A. Levine, “Some nonexistence and instability theorems for solutions of formally parabolic equations of the form Pu t =-Au + F(u), Arch. Rat. Mech. Anal. 51 (1973) 371–386.MATHCrossRefGoogle Scholar
  5. 30.
    Jong-Sheng Guo, “On the quenching rate estimate, ” Quarterly of Applied Mathematics (submitted).Google Scholar

Copyright information

© Springer Science+Business Media New York 1992

Authors and Affiliations

  • Howard A. Levine
    • 1
  1. 1.Department of MathematicsIowa State UniversityAmes

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