The profile near blowup time for solution of the heat equation with a nonlinear boundary condition
Authors:
Bei Hu and HongMing Yin
Journal:
Trans. Amer. Math. Soc. 346 (1994), 117135
MSC:
Primary 35B40; Secondary 35B05, 35K60
MathSciNet review:
1270664
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Abstract: This paper studies the blowup profile near the blowup time for the heat equation with the nonlinear boundary condition on . Under certain assumptions, the exact rate of the blowup is established. It is also proved that the blowup will not occur in the interior of the domain. The asymptotic behavior near the blowup point is also studied.
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 [1]
 H. Amann, Parabolic evolution equations and nonlinear boundary conditions, J. Differential Equations 72 (1988), 201269. MR 932367 (89e:35066)
 [2]
 J. Bebernes and D. Eberly, Mathematical problems from combustion theory, SpringerVerlag, New York, 1989. MR 1012946 (91d:35165)
 [3]
 L. A. Caffarelli, B. Gidas, and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Solev growth, Comm. Pure Appl. Math. 42 (1989), 271297. MR 982351 (90c:35075)
 [4]
 L. A. Caffarelli and A. Friedman, Blowup of solutions of nonlinear heat equations, J. Math. Anal. Appl. 12 (1988), 409419. MR 924300 (89c:35077)
 [5]
 J. M. Chadam and H. M. Yin, A diffusion equation with localized chemical reactions, Proc. Edinburgh Math. Soc. 37 (1993), 101118. MR 1258034 (94m:35156)
 [6]
 M. Chipot, M. Fila, and P. Quittner, Stationary solutions, blowup and convergence to stationary solutions for semilinear parabolic equations with nonlinear boundary conditions, Acta Math. Univ. Comenian. 60 (1991), 35103. MR 1120596 (92h:35110)
 [7]
 B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math. 34 (1981), 525598. MR 615628 (83f:35045)
 [8]
 , A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations 8 (1981), 883901. MR 619749 (82h:35033)
 [9]
 M. Fila, Boundedness of global solutions for the heat equation with nonlinear boundary conditions, Comment. Math. Univ. Carolinae 30 (1989), 479484. MR 1031865 (91b:35017)
 [10]
 M. Fila and P. Quittner, The blowup rate for the heat equation with a nonlinear boundary condition, Math. Methods Appl. Sci. 14 (1991), 197205. MR 1099325 (92a:35023)
 [11]
 A. Friedman and B. McLeod, Blowup of positive solutions of semilinear heat equations, Indiana Univ. Math. J. 34 (1985), 425477. MR 783924 (86j:35089)
 [12]
 A. Friedman, Partial differential equations of parabolic type, PrenticeHall, Englewood Cliffs, NJ, 1964. MR 0181836 (31:6062)
 [13]
 Y. Giga and R. V. Kohn, Asymptotic selfsimilar blowup of semilinear heat equations, Comm. Pure Appl. Math. 38 (1985), 297319. MR 784476 (86k:35065)
 [14]
 , Characterizing blowup using similarity variables, Indiana Univ. Math. J. 36 (1987), 425447. MR 876989 (88c:35021)
 [15]
 , Nondegeneracy of blowup for semilinear heat equations, Comm. Pure Appl. Math. 42 (1989), 845884. MR 1003437 (90k:35034)
 [16]
 D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, SpringerVerlag, New York, 1983. MR 737190 (86c:35035)
 [17]
 O. A. Ladyzhenskaja, V. A. Solonnikov, and N. N. Ural'ceva, Linear and quasilinear equations of parabolic type, Amer. Math. Soc. Transl. (2) 23 (1968).
 [18]
 H. A. Levine and L. E. Payne, Nonexistence theorems for the heat equation with nonlinear boundary conditions and for the porous medium equation backward in time, J. Differential Equations 16 (1974), 319334. MR 0470481 (57:10235)
 [19]
 H. A. Levine and R. A. Smith, A potential well theory for the heat equation with a nonlinear boundary condition, Math. Methods Appl. Sci. 9 (1987), 127136. MR 897262 (89f:35114)
 [20]
 W. Liu, The blowup rate of solutions of semilinear heat equations, J. Differential Equations 77 (1989), 104122. MR 980545 (90e:35022)
 [21]
 , Blowup behavior for semilinear heat equations: multidimensional case, IMA preprint series no. 711, 1990.
 [22]
 P. Quittner, On global existence and stationary solutions for two classes of semilinear parabolic problems, Centre de recherche de mathematiques de la decision, Universite Paris IX, preprint 9208, 1992. MR 1240209 (94g:35127)
 [23]
 A. A. Samarskii, On new methods of studying the asymptotic properties of parabolic equations, Proc. Steklov Inst. Math. 158 (1983), 165176.
 [24]
 W. Walter, On existence and nonexistence in the large of solutions of parabolic differential equations with a nonlinear boundary condition, SIAM J. Math. Anal. 6 (1975), 8590. MR 0364868 (51:1122)
 [25]
 F. B. Weissler, Existence and nonexistence of global solutions for a semilinear heat equation, Israel J. Math. 38 (1981), 2940. MR 599472 (82g:35059)
 [26]
 , An blowup estimate for a nonlinear heat equation, Comm. Pure Appl. Math. 38 (1985), 291295. MR 784475 (86k:35064)
 [27]
 H. M. Yin, Blowup versus global solvability for a class of nonlinear parabolic equations, Nonlinear Anal. TMA (to appear).
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947199412706643
PII:
S 00029947(1994)12706643
Keywords:
Blowup rate,
asymptotic behavior,
elliptic estimates,
parabolic estimates
Article copyright:
© Copyright 1994
American Mathematical Society
