Existence and Nonexistence of Global Solutions of a Fully Nonlinear Parabolic Equation

Affiliation(s)

Institute of Applied Mathematics, School of Mathematics and Information Sciences, Henan University, Kaifeng, China.

Institute of Applied Mathematics, School of Mathematics and Information Sciences, Henan University, Kaifeng, China.

ABSTRACT

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.

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.

Z. Ge, "Existence and Nonexistence of Global Solutions of a Fully Nonlinear Parabolic Equation,"

References

[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.

[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.