Abstract
This paper deals with blow-up of positive solution of the nonlinear heat equation with absorption subject to a nonlinear boundary condition. The conditions under which the solutions may exist globally or blow-up are obtained by the comparison principles.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Acosta G, Rossi J D. Blow-up vs global existence for quasilinear prabolic systems with a nonlinear boundary condition [J]. Journal of Applied Mathematics and Physics, 1997, 48(5): 711–724.
Ding J, Li S. Blow-up solutions and global solutions for a class of quasilinear parabolic equations with Robin boundary conditions [J]. Computers and Mathematics with Applications, 2005, 49(5): 689–701.
Friedman A, McLeod J B. Blow-up of positive solutions of semi-linear heat equations [J]. Indiana University Mathematics Journal, 1985, 34(2): 425–447.
Fujita H. On the blowing up of solutions of the Cauchy problem for ut = Δu+u1+α [J]. Journal of the Faculty of Science, 1966, 16(1): 105–113.
Pinsky R. Existence and noexistence of global solutions for ut = Δu+a(x)up in Rd [J]. Journal of Differential Equations, 1997, 133(1): 152–177.
Yu Wei, Wang Yuan-di. Properties of parabolic partial differential equation of the first kind boundary condition problems [J]. Journal of Shanghai University (Natural Science Edition), 2005, 11(4): 391–401 (in Chinese).
Zhang H, Liu Z, Zhang W. Growth estimates and blow-up in quasilinear parabolic problems [J]. Applicable Analysis, 2007, 86(2): 261–268.
Hu B, Yin H M. The profile near blowup time for solution of the heat equation with a nonlinear boundary condition [J]. Transactions of the American Mathematical Society, 1994, 346(1): 117–135.
Levine H, Payne L. Nonexistce theorems for the heat equation with nonlinear boundary conditions and for the porous medium equation backward in time [J]. Journal of Differential Equations, 1974, 16(2): 319–334.
Wang M, Wu Y. Global existence and blow-up problems for quasilinear parabolic equations with nonlinear boundary conditions [J]. SIAM Journal on Mathematical Analysis, 1993, 24(6): 1515–1521.
Ying H M. Blow-up versus global solvability for a class of nonlinear parabolic equations [J]. Nonlinear Analysis, 1994, 23(7): 911–924.
Amann A. Quasilinear parabolic systems under nonlinear boundary conditions [J]. Archive for Rational Mechanics and Analysis, 1986, 92(2): 153–192.
Boni T K. Sur i’explosion et le comportement asymptotique de la solution d’une equation parabolique semilinaire du second ordre [J]. Comptes Rendus de UA-cadémie des Scuebces, Série I, 1998, 326(3): 317–322.
Galaktionov V A, Vazquez J L. Asymptic behaviour of nonlinear parabolic equations with critical exponents [J]. Journal of Functional Ananlysis, 1991, 100(2): 435–462.
Quittner P. On global existence and stationary solutions for two classes of semilinear parabolic problems [J]. Commentationes Mathematicae Universitatis, 1993, 34(1): 105–124.
Author information
Authors and Affiliations
Corresponding author
Additional information
Project supported by the Education Department of Zhejiang Province (Grant No.Y200805137), and the Zhejiang Ocean University (Grant No.X08Z04)
About this article
Cite this article
Zhang, Hl., Guo, Xl. Blow-up and global existence for the heat equation with nonlinear absorption-diffusion. J. Shanghai Univ.(Engl. Ed.) 14, 170–173 (2010). https://doi.org/10.1007/s11741-010-0624-2
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11741-010-0624-2