# Advances in Quenching

Chapter

## Abstract

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.

## Keywords

Stationary Solution Finite Time Parabolic Problem Infinite Time Nonlinear Boundary Condition
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## References

- 1.A. Ackor, B. Kawohl, “Remarks on Quenching,” Nonlinear Analysis, TMA
**13**(1989), 53–61.CrossRefGoogle Scholar - 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.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.C. Y. Chan, C. S. Chen, “Critical lengths for global existence of solutions for coupled semilinear singular parabolic problems.” (preprint)Google Scholar
- 5.C. Y. Chan, H. G. Kaper, “Quenching for semilinear singular parabolic problems,” SIAM J. Math. Anal.
**20**(1989), 558–566.MathSciNetMATHCrossRefGoogle Scholar - 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.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.C. Y. Chan, M. K. Kwong, “Quenching phenomena for singular nonlinear parabolic equations,” Nonlinear Analysis
**12**(1988), 1377–1383.MathSciNetMATHCrossRefGoogle Scholar - 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.M. Fila, J. Hulshof, “A note on the quenching rate.” (preprint).Google Scholar
- 11.M. Fila, B. Kawohl, “Is quenching in infinite time possible?” Quail. Appl. Math., (in press).Google Scholar
- 12.M. S. Floater, “Blow up at the boundary for degenerate semilinear parabolic equations,” (in press).Google Scholar
- 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.J. S. Guo, “On the semilinear elliptic equation Δ
*w*−1/2*y*. Δ*w*+ λ*w*-*w*^{-β}= 0 in*R*^{N}.” (manuscript).Google Scholar - 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.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.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.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.S. R. Park, “The phenomenon of quenching in the presence of convection,” Ph.D. Thesis, Iowa State University, (1989).Google Scholar
- 20.D. Phillips, “Existence of solutions to a quenching problem,” Applicable Analysis 24 (1987), 253–264.MathSciNetMATHCrossRefGoogle Scholar

## Hyperbolic Problems

- 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
- 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
- 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
- 24.R. A. Smith, “On a Hyperbolic Quenching problem in several dimensions,” SIAM J. Math Anal.
**20**(1989), 1081–1094.MathSciNetMATHCrossRefGoogle Scholar - 25.N. Sternberg, “Blowup near higher modes of nonlinear wave equations,” TAMS,
**296**(1986), 315–325.MathSciNetMATHCrossRefGoogle Scholar

## Other References

- 26.Dziuk and Kawohl, “On radially symmetric mean curvature flow,” (to appear).Google Scholar
- 27.Y. Giga, R. V. Kohn, “Nondegeneracy of blowup for semilinear heat equations,” Hokkaido U. Preprint Series #46, October 1988.Google Scholar
- 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 - 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 - 30.Jong-Sheng Guo, “On the quenching rate estimate, ” Quarterly of Applied Mathematics (submitted).Google Scholar

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