APM  Vol.3 No.1 , January 2013
Existence and Nonexistence of Global Solutions of a Fully Nonlinear Parabolic Equation
Author(s) Zhihao Ge*
In the paper, we study the global existence of weak solution of the fully nonlinear parabolic problem (1.1)-(1.3) with nonlinear boundary conditions for the situation without strong absorption terms. Also, we consider the blow up of global solution of the problem (1.1)-(1.3) by using the convexity method.

Cite this paper
Z. Ge, "Existence and Nonexistence of Global Solutions of a Fully Nonlinear Parabolic Equation," Advances in Pure Mathematics, Vol. 3 No. 1, 2013, pp. 20-23. doi: 10.4236/apm.2013.31004.
[1]   H. Amann, “Periodic Solutions of Semi-Linear Parabolic Equations,” Academic Press, New York, San Francisco London, 1978.

[2]   A. Friedman and B. Mcleod, “Blow Up of Solutions of Semilinear Heat Equations,” Indiana University Mathematics Journal, Vol. 34, No. 1, 1985, pp. 425-447. doi:10.1512/iumj.1985.34.34025

[3]   V. Galaktionov and J. Vazquez, “The Problem of Blow Up in Nonlinear Parabolic Equations,” Discrete and Continuous Dynamical Systems, Vol. 8, No. 2, 2002, pp. 399- 433. doi:10.3934/dcds.2002.8.399

[4]   X. Song and S. Zheng, “Multinonlinear Interactions in Quasi-Linear Reaction-Diffusion Equations with Nonlinear Boundary Flux,” Mathematical and Computer Modelling, Vol. 39, No. 2-3, 2004, pp. 133-144. doi:10.1016/S0895-7177(04)90002-7

[5]   D. G. Aronson, “The Porous Medium Equation,” Springer Verlag, 1986.

[6]   M. Gurtin and R. Cammy, “On the Diffusion of Biological Population,” Mathematical and Theoretical Biology, Vol. 33, No. 1-2, 1977, pp. 35-49.

[7]   C. V. Pao, “Nonlinear Parabolic and Elliptic Equations,” Plenum Press, New York, 1992.

[8]   O. Ladyzenskaja, V. Solonnikov and N. Uralceva, “Linear and Quasilinear Equations of Parabolic Type,” American Mathematical Society, Providence, 1968.

[9]   R. Temam, “Infinite Dimensional Systems in Mechanics and Physics,” Springer, New York, 1997.

[10]   P. Quittner and P. Souplet, “Bounds of Global Solutions of Parabolic Problems with Nonlinear Boundary Conditions,” Indiana University Mathematics Journal, Vol. 52, No. 4, 2003, pp. 875-900. doi:10.1512/iumj.2003.52.2353

[11]   P. Quittner and P. Souplet, “A Priori Estimates of Global Solutions of Superlinear Parabolic Problems without Variational Structure,” Discrete and Continuous Dynamical Systems, Vol. 9, No. 5, 2003, pp. 1277-1292. doi:10.3934/dcds.2003.9.1277

[12]   A. Bernal and A. Tajdine, “Nonlinear Balance for Reaction-Diffusion Equations under Nonliear Boundary Conditions: Dissipativity and Blow Up,” Journal of Differential Equation, Vol. 169, No. 2, 2001, pp. 332-372. doi:10.1006/jdeq.2000.3903

[13]   J. D. Rossi, “The Blow-Up Rate for a Semilinear Parabolic Equation with a Nonlinear Boundary Condition,” Acta Mathematica Universitatis Comenianae, Vol. 67, No. 2, 1998, pp. 343-350.

[14]   J. D. Rossi and N. Wolanski, “Global Existence and Non-existence for a Parabolic System with Nonlinear Boundary Conditions,” Differential and Integral Equations, Vol. 11, No. 1, 1998, pp. 179-190.

[15]   A. Samarskii, V. Galaktionov and S. Mikhailov, "Blow-Up in Quasilinear Parabolic Equations,” Walter de Gruyter, Berlin/New York, 1995. doi:10.1515/9783110889864

[16]   K. Taira, “Semi-Linear Parabolic Problems in Combustion Theory,” Journal of Mathematical Sciences—The University of Tokyo, Vol. 10, No. 3, 2003, pp. 455-494.

[17]   J. Filo and P. D. Mottoni, “Global Existence and Decay of Solutions of the Porous Medium Equation with Nonlinear Boundary Conditions,” Communications in Partial Differential Equations, Vol. 17, No. 5-6, 1992, pp. 737-765. doi:10.1080/03605309208820862

[18]   H. A. Levine, “Instability and Nonexistence of Global Solutions of Nonlinear Wave Equations of the Form Pun=Au+F(u),” Transactions of the American Mathematical Society, Vol. 192, 1974, pp. 1-21.

[19]   H. A. Levine, “Some Additional Remarks on the Nonexistence of Global Solutions to Nonlinear Wave Equations,” SIAM Journal on Mathematical Analysis, Vol. 5, No. 1, 1974, pp. 138-146. doi:10.1137/0505015

[20]   Z. H. Ge and Y. N. He, “Global Existence of Solutions of a Nonlinear Degenerate Parabolic Problem,” Journal of Physics: Conference Series, Vol. 96, No. 1, 2008, Article ID: 012024.