On critical exponents for the heat equation with a nonlinear boundary condition
Annales de l'I.H.P. Analyse non linéaire, Tome 13 (1996) no. 6, p. 707-732
@article{AIHPC_1996__13_6_707_0,
     author = {Hu, Bei and Yin, Hong-Ming},
     title = {On critical exponents for the heat equation with a nonlinear boundary condition},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Gauthier-Villars},
     volume = {13},
     number = {6},
     year = {1996},
     pages = {707-732},
     zbl = {0908.35066},
     mrnumber = {1420495},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_1996__13_6_707_0}
}
Hu, Bei; Yin, Hong-Ming. On critical exponents for the heat equation with a nonlinear boundary condition. Annales de l'I.H.P. Analyse non linéaire, Tome 13 (1996) no. 6, pp. 707-732. http://www.numdam.org/item/AIHPC_1996__13_6_707_0/

[1] C. Bandle and H.A. Levine, On the existence and nonexistence of global solutions of reaction-diffusion equations in sectorial domains, Trans. Amer. Math. Soc., Vol. 316, 1989, pp. 595-624. | MR 937878 | Zbl 0693.35081

[2] K. Deng, M. Fila and H.A. Levine, On critical exponents for a system of heat equations coupled in the boundary conditions, Acta Math. Univ. Comenianae, Vol. 63, 1994, pp. 169-192. | MR 1319438 | Zbl 0824.35048

[3] M. Fila, Boundedness of global solutions for the heat equation with nonlinear boundary conditions, Comment. Math. Univ. Carolinate, Vol. 30, 1989, pp. 479-484. | MR 1031865 | Zbl 0702.35141

[4] M. Fila and J. Filo, Blow-up on the boundary: a survey, Proceedings of the Banach Center, to appear. | MR 1449147 | Zbl 0858.35065

[5] H. Fujita, On the blowing up of solutions of the Cauchy problem for ut = Δu + u1+α, J. Fac. Sci. Univ. Tokyo Sect. A. Math., Vol. 16, 1966, pp. 105-113.

[6] V.A. Galaktionov and H.A. Levine, On critical Fujita exponents for heat equations with a nonlinear flux boundary condition on the boundary, Israel J. Math., to appear. | Zbl 0851.35067

[7] M.G. Garroni and J.L. Menaldi, Green functions for second order parabolic integrodifferential problems, Longman Scientific and Technical, New York, 1992. | MR 1202037 | Zbl 0806.45007

[8] J.L. Gomez, V. Marquez and N. Wolanski, Blow up and localization of blow up points for the heat equation with a nonlinear boundary condition, J. Diff. Eq., Vol. 2, 1992, pp. 384-401. | MR 1120912 | Zbl 0735.35016

[9] B. Hu, Nonexistence of a positive solution of the Laplace equation with a nonlinear boundary condition, Differential and Integral Equations, Vol. 7, 1994, pp. 301-313. | MR 1255890 | Zbl 0820.35062

[10] B. Hu and H.-M. Yin, The profile near blow up time for solutions of the heat equation with a nonlinear boundary condition, Trans. Amer. Math. Soc., Vol. 346, 1994, pp. 117-135. | MR 1270664 | Zbl 0823.35020

[11] O.A. Ladyzenskaja, V.A. Solonnikov and N.N. Ural'Ceva, Linear and Quasilinear Equations of Parabolic type, AMS Monograph translation, Vol. 23, Providence, RI, 1968. | MR 241822 | Zbl 0174.15403

[12] H.A. Levine, The role of critical exponents in blow up theorems, SIAM Review, Vol. 32, 1990, pp. 262-288. | MR 1056055 | Zbl 0706.35008

[13] H.A. Levine and P. Meier, The value of the critical exponent for reaction-diffusion equations in cones, Arch. Rational Mech. Anal., Vol. 109, 1990, pp. 73-80. | MR 1019170 | Zbl 0702.35131

[14] G. Lieberman, Study of global solutions of parabolic equations via a priori estimates, Part I: Equations with principal elliptic part equal to the Laplacian, Math. Meth. in the Appl. Sci., Vol. 16, 1993, pp. 457-474. | MR 1230123 | Zbl 0797.35093

[15] V.G. Maz'Ja, Sobolev Spaces, Springer-Verlag, New York, 1985.

[16] S. Ohta and A. Kaneko, Critical exponent of blow up for semilinear heat equation on a product domain, J. Fac. Sci. Univ. Tokyo, Sect. IA, Math., Vol. 40, 1993, pp. 635-650. | MR 1269031 | Zbl 0799.35126

[17] C.V. Pao, Nonlinear Parabolic Equation and Elliptic Equations, Plenum Press, New York, 1992. | Zbl 0777.35001

[18] W. Walter, On existence and nonexistence in the large of solutions of parabolic differential equations with a nonlinear boundary condition, SIAM J. Math. Anal., Vol. 6, 1975, pp. 85-90. | MR 364868 | Zbl 0268.35052

[19] F.B. Weissler, Existence and nonexistence of global solutions for a semilinear heat equation, Israel J. Math., Vol. 38, 1981, pp. 29-40. | MR 599472 | Zbl 0476.35043