The profile near blowup time for solution of the heat equation with a nonlinear boundary condition

Hu, Bei; Yin, Hong-Ming

doi:10.1090/S0002-9947-1994-1270664-3

The profile near blowup time for solution of the heat equation with a nonlinear boundary condition
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by Bei Hu and Hong-Ming Yin

Trans. Amer. Math. Soc. 346 (1994), 117-135

DOI: https://doi.org/10.1090/S0002-9947-1994-1270664-3

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Abstract:

This paper studies the blowup profile near the blowup time for the heat equation ${u_t} = \Delta u$ with the nonlinear boundary condition ${u_n} = {u^p}$ on $\partial \Omega \times [0,T)$. 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.

References

Herbert Amann, Parabolic evolution equations and nonlinear boundary conditions, J. Differential Equations 72 (1988), no. 2, 201–269. MR 932367, DOI 10.1016/0022-0396(88)90156-8
Jerrold Bebernes and David Eberly, Mathematical problems from combustion theory, Applied Mathematical Sciences, vol. 83, Springer-Verlag, New York, 1989. MR 1012946, DOI 10.1007/978-1-4612-4546-9
Luis A. Caffarelli, Basilis Gidas, and Joel Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math. 42 (1989), no. 3, 271–297. MR 982351, DOI 10.1002/cpa.3160420304
Luis A. Caffarelli and Avner Friedman, Blowup of solutions of nonlinear heat equations, J. Math. Anal. Appl. 129 (1988), no. 2, 409–419. MR 924300, DOI 10.1016/0022-247X(88)90261-2
John M. Chadam and Hong-Ming Yin, A diffusion equation with localized chemical reactions, Proc. Edinburgh Math. Soc. (2) 37 (1994), no. 1, 101–118. MR 1258034, DOI 10.1017/S0013091500018721
M. Chipot, M. Fila, and P. Quittner, Stationary solutions, blow up and convergence to stationary solutions for semilinear parabolic equations with nonlinear boundary conditions, Acta Math. Univ. Comenian. (N.S.) 60 (1991), no. 1, 35–103. MR 1120596
B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math. 34 (1981), no. 4, 525–598. MR 615628, DOI 10.1002/cpa.3160340406
B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations 6 (1981), no. 8, 883–901. MR 619749, DOI 10.1080/03605308108820196
Marek Fila, Boundedness of global solutions for the heat equation with nonlinear boundary conditions, Comment. Math. Univ. Carolin. 30 (1989), no. 3, 479–484. MR 1031865
Marek Fila and Pavol Quittner, The blow-up rate for the heat equation with a nonlinear boundary condition, Math. Methods Appl. Sci. 14 (1991), no. 3, 197–205. MR 1099325, DOI 10.1002/mma.1670140304
Avner Friedman and Bryce McLeod, Blow-up of positive solutions of semilinear heat equations, Indiana Univ. Math. J. 34 (1985), no. 2, 425–447. MR 783924, DOI 10.1512/iumj.1985.34.34025
Avner Friedman, Partial differential equations of parabolic type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR 0181836
Yoshikazu Giga and Robert V. Kohn, Asymptotically self-similar blow-up of semilinear heat equations, Comm. Pure Appl. Math. 38 (1985), no. 3, 297–319. MR 784476, DOI 10.1002/cpa.3160380304
Yoshikazu Giga and Robert V. Kohn, Characterizing blowup using similarity variables, Indiana Univ. Math. J. 36 (1987), no. 1, 1–40. MR 876989, DOI 10.1512/iumj.1987.36.36001
Yoshikazu Giga and Robert V. Kohn, Nondegeneracy of blowup for semilinear heat equations, Comm. Pure Appl. Math. 42 (1989), no. 6, 845–884. MR 1003437, DOI 10.1002/cpa.3160420607
David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190, DOI 10.1007/978-3-642-61798-0

Linear and quasi-linear equations of parabolic type

23

Howard A. Levine and Lawrence 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), 319–334. MR 470481, DOI 10.1016/0022-0396(74)90018-7
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), no. 2, 127–136. MR 897262, DOI 10.1002/mma.1670090111
Wen Xiong Liu, The blow-up rate of solutions of semilinear heat equations, J. Differential Equations 77 (1989), no. 1, 104–122. MR 980545, DOI 10.1016/0022-0396(89)90159-9

Blowup behavior for semilinear heat equations: multi-dimensional case

Pavol Quittner, On global existence and stationary solutions for two classes of semilinear parabolic problems, Comment. Math. Univ. Carolin. 34 (1993), no. 1, 105–124. MR 1240209

On new methods of studying the asymptotic properties of parabolic equations

158

Wolfgang 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), 85–90. MR 364868, DOI 10.1137/0506008
Fred B. Weissler, Existence and nonexistence of global solutions for a semilinear heat equation, Israel J. Math. 38 (1981), no. 1-2, 29–40. MR 599472, DOI 10.1007/BF02761845
Fred B. Weissler, An $L^\infty$ blow-up estimate for a nonlinear heat equation, Comm. Pure Appl. Math. 38 (1985), no. 3, 291–295. MR 784475, DOI 10.1002/cpa.3160380303

Blowup versus global solvability for a class of nonlinear parabolic equations