On the blow up of at quenching
Authors:
Keng Deng and Howard A. Levine
Journal:
Proc. Amer. Math. Soc. 106 (1989), 10491056
MSC:
Primary 35B40; Secondary 35K55
MathSciNet review:
969520
Fulltext PDF Free Access
Abstract 
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Abstract: Let be a bounded convex domain in with smooth boundary. We consider the problems in , while on and . Here satisfies , and , while satisfies . We show that if quenches (reaches in finite time), then the quenching points are in a compact subset of and blows up. We also extend the result to the third boundary value problem.
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 [1]
 H. Kawarada, On the solutions of initialboundary value problems for , Publ. Res. Inst. Math. Sci. 10 (1975), 729736. MR 0385328 (52:6192)
 [2]
 A. Acker and W. Walter, The quenching problem for nonlinear parabolic equations, Lecture Notes in Math., vol. 564, SpringerVerlag, New York, 1976. MR 0604032 (58:29265)
 [3]
 , On the global existence of solutions of parabolic differential equations with a singular nonlinear term, Nonlinear Anal. 2 (1978), 449505. MR 512487 (80c:35054)
 [4]
 H. A. Levine and J. T. Montgomery, The quenching of solutions of some nonlinear parabolic equations, SIAM J. Math. Anal. 11 (1980), 842847. MR 586912 (83b:35093)
 [5]
 H. A. Levine, The quenching of solutions of linear parabolic and hyperbolic equations with nonlinear boundary conditions, SIAM J. Math. Anal. 14 (1983), 11391153. MR 718814 (85i:35085a)
 [6]
 , The phenomenon of quenching: A survey, Trends in the Theory and Practice of Nonlinear Analysis (V. Lakshmikantham, ed.) Elsevier Science Publ., North Holland, 1985, pp. 275286. MR 817500 (87b:35082)
 [7]
 , Quenching, nonquenching, and beyond quenching for solutions of some parabolic equations, Ann. Mat. Pura Appl. (in press).
 [8]
 A. Acker and B. Kawohl, Remarks on quenching, Nonlinear Anal. TMA 13 (1989), 5361. MR 973368 (90c:35007)
 [9]
 A. Friedman and B. McLeod, Blowup of positive solutions of semilinear heat equations, Indiana Univ. Math. J. 34 (1985), 425447. MR 783924 (86j:35089)
 [10]
 B. Gidas, W.M. Ni, and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), 209243. MR 544879 (80h:35043)
 [11]
 M. Chipot and F. B. Weissler, Some blow up results for a nonlinear parabolic equation with a gradient term, IMA Preprint Series no. 298.
 [12]
 C. Y. Chan and M. K. Kwong, Quenching phenomena for singular nonlinear parabolic equations, Nonlinear Analysis, TMA 12 (1988), 13771383. MR 972406 (90a:35110)
 [13]
 J. Guo, On the quenching behavior of the solution of a semilinear parabolic equation, J. Math. Anal. Appl. (in print).
 [14]
 , On the semilinear elliptic equation, in , (in print).
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939198909695200
PII:
S 00029939(1989)09695200
Article copyright:
© Copyright 1989
American Mathematical Society
