A nonlinear integral equation occurring in a singular free boundary problem
Authors:
Klaus Höllig and John A. Nohel
Journal:
Trans. Amer. Math. Soc. 283 (1984), 145155
MSC:
Primary 35R35; Secondary 35K55, 45G10
MathSciNet review:
735412
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Abstract: We study the Cauchy problem with the piecewise linear constitutive function and with smooth initial data which satisfy , , and . We prove that free boundary , given by , is of the form where the constant is the (numerical) solution of a particular nonlinear equation. Moreover, we show that for any , The proof involves the analysis of a nonlinear singular integral equation.
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 [1]
 P. Benilan, M. G. Crandall and A. Pazy, accretive operators (in preparation).
 [2]
 L. C. Evans, Application of nonlinear semigroup theory to certain partial differential equations, Nonlinear Evolution Equations (M. G. Crandall, ed.), Academic Press, New York, 1978. MR 513818 (81b:47078)
 [3]
 A. Fasano and M. Primicerio, General free boundary problems for the heat equation. I, J. Math. Anal. Appl. 57 (1977), 694723. MR 0487016 (58:6695a)
 [4]
 , General free boundary problems for the heat equation. II, J. Math. Anal. Appl. 58 (1977), 202231. MR 0487017 (58:6695b)
 [5]
 K. Höllig, Existence of infinitely many solutions for a forward backward heat equation, Trans. Amer. Math. Soc. 278 (1983), 299316. MR 697076 (84m:35062)
 [6]
 K. Höllig and J. A. Nohel, A diffusion equation with a nonmonotone constitutive function, Systems of Partial Differential Equations (J. M. Ball, ed.), Reidel, Dordrecht, 1983, pp. 409422.
 [7]
 D. Kinderlehrer and L. Nirenberg, Regularity in free boundary problems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 4 (1977), 373391. MR 0440187 (55:13066)
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 D. Schaeffer, A new proof of the infinite differentiability of the free boundary in the Stefan problem, J. Differential Equations 20 (1976), 266269. MR 0390499 (52:11325)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198407354125
PII:
S 00029947(1984)07354125
Keywords:
Cauchy problem,
parabolic,
nonlinear,
free boundary regularity,
nonlinear singular integral equation
Article copyright:
© Copyright 1984
American Mathematical Society
