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Abstract:

In this paper we study the large time behavior of positive solutions of the heat equation under the nonlinear boundary condition ∂u ∂ν = f(u), where η is the outward normal and f is nondecreasing with f(u) > 0 for u > 0. We show that if Ω = BR and 1/f is integrable at infinity there is finite time blow up for any initial datum. In the two dimensional case we show that this is true for any smooth simply connected domain. In the radially symmetric case if f ε{lunate} C2 is convex and satisfies the properties above we show that blow up occurs only at the boundary. © 1991.

Registro:

Documento: Artículo
Título:Blow up results and localization of blow up points for the heat equation with a nonlinear boundary condition
Autor:Gómez, J.L.; Márquez, V.; Wolanski, N.
Filiación:Departamento de Matemática Aplicada, Universidad Complutense de Madrid, 28040 Madrid, Spain
Departamento de Matemática, Facultad de Ciencias Exactas, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina
Año:1991
Volumen:92
Número:2
Página de inicio:384
Página de fin:401
DOI: http://dx.doi.org/10.1016/0022-0396(91)90056-F
Título revista:Journal of Differential Equations
Título revista abreviado:J. Differ. Equ.
ISSN:00220396
CODEN:JDEQA
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00220396_v92_n2_p384_Gomez

Referencias:

  • Amann, Quasilinear parabolic systems under nonlinear boundary conditions (1986) Arch. Rational Mech. Anal., 92 (2), pp. 153-192
  • Friedman, (1964) Partial Differential Equations of Parabolic Type, , Prentice-Hall, Englewood Cliffs, NJ
  • Ladyženskaja, Solonnikov, Ural'ceva, Linear and quasilinear equations of parabolic type (1968) Transl. Math. Monographs, 23
  • Levine, Stability and instability for solutions of Burguers' equation with a semilinear boundary condition (1988) SIAM Journal on Mathematical Analysis, 19 (2), pp. 312-336
  • Levine, Payne, Nonexistence theorems for the hear equations with nonlinear boundary conditions and for the porous medium equation backward in time (1974) J. Differential Equations, 16 (2), pp. 319-334
  • Walter, On existence and nonexistence in the large of solutions of parabolic differential equations with a nonlinear boundary condition (1975) SIAM Journal on Mathematical Analysis, 6 (1), pp. 85-90
  • Weissler, Single point blow up for a semilinear initial value problem (1984) J. Differential Equations, 55, pp. 204-224

Citas:

---------- APA ----------
Gómez, J.L., Márquez, V. & Wolanski, N. (1991) . Blow up results and localization of blow up points for the heat equation with a nonlinear boundary condition. Journal of Differential Equations, 92(2), 384-401.
http://dx.doi.org/10.1016/0022-0396(91)90056-F
---------- CHICAGO ----------
Gómez, J.L., Márquez, V., Wolanski, N. "Blow up results and localization of blow up points for the heat equation with a nonlinear boundary condition" . Journal of Differential Equations 92, no. 2 (1991) : 384-401.
http://dx.doi.org/10.1016/0022-0396(91)90056-F
---------- MLA ----------
Gómez, J.L., Márquez, V., Wolanski, N. "Blow up results and localization of blow up points for the heat equation with a nonlinear boundary condition" . Journal of Differential Equations, vol. 92, no. 2, 1991, pp. 384-401.
http://dx.doi.org/10.1016/0022-0396(91)90056-F
---------- VANCOUVER ----------
Gómez, J.L., Márquez, V., Wolanski, N. Blow up results and localization of blow up points for the heat equation with a nonlinear boundary condition. J. Differ. Equ. 1991;92(2):384-401.
http://dx.doi.org/10.1016/0022-0396(91)90056-F