Blow-Up Phenomena for a Class of Parabolic Systems with Time Dependent Coefficients

Lawrence E. Payne, Gérard A. Philippin

Département de Mathématiques et de Statistique, Université Laval, Québec City, Canada.
Department of Mathematics, Cornell University, Ithaca, USA.

DOI: 10.4236/am.2012.34049                

Abstract

Blow-up phenomena for solutions of some nonlinear parabolic systems with time dependent coefficients are investigated. Both lower and upper bounds for the blow-up time are derived when blow-up occurs.

Keywords

Parabolic Systems; Blow-Up; Sobolev Type Inequality

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	L. E. Payne, P. W. Schaefer, “Blow-Up Phenomena for Some Nonlinear Parabolic Systems,” International Journal of Pure and Applied Mathematics, Vol. 48, No. 2, 2008, pp. 193-202.
[2]	L. E. Payne and G. A. Philippin, “On Blow-Up Phenomena for Solutions of a Class of Nonlinear Parabolic Problems with Time Dependent Coefficients under Dirichlet Boundary Conditions,” Proceedings of the American Mathematical Sociery, accepted.
[3]	V. A. Galaktionov and J. L. Vazquez, “The Problem of Blow-Up in Nonlinear Parabolic Equations,” Journal of Dynamical and Control System, Vol. 8, No. 3, 2002, pp. 399-433. doi:10.1023/A:1016334621818
[4]	A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov and A. P. Mikhailov, “Blow-Up in Quasilinear Parabolic Equations,” Walter de Gruyter & Co., Berlin, 1995. doi:10.1515/9783110889864
[5]	B. Straughan, “Explosive Instabilities in Mechanics,” Springer, Berlin, 1998. doi:10.1007/978-3-642-58807-5
[6]	C. Bandle and H. Brunner, “Blow-Up in Diffusion Equations: A Survey,” Journal of Computational and Applied Mathematics, Vol. 97, No. 1-2, 1998, pp. 3-22. doi:10.1016/S0377-0427(98)00100-9
[7]	L. E. Payne, G. A. Philippin and P. W. Schaefer, “Bounds for Blow-Up Time in Nonlinear Parabolic Problems,” Journal of Mathematical Analysis and Applications, Vol. 338, No. 1, 2008, pp. 438-447. doi:10.1016/j.jmaa.2007.05.022
[8]	L. E. Payne, G. A. Philippin and P. W. Schaefer, “BlowUp Phenomena for Some Nonlinear Parabolic Problems,” Nonlinear Analysis: Theory, Methods & Applications, Vol. 69, No. 10, 2008, pp. 3495-3502. doi:10.1016/j.na.2007.09.035
[9]	L. E. Payne, G. A. Philippin and S. Vernier-Piro, “BlowUp Phenomena for a Semilinear Heat Equation with Nonlinear Boundary Condition, I,” Zeitschrift für Angewandte Mathematik und Physik, Vol. 61, No. 6, 2010, pp. 9991007. doi:10.1007/s00033-010-0071-6
[10]	L. E. Payne, G. A. Philippin and S. Vernier-Piro, “BlowUp Phenomena for a semilinear Heat Equation with Nonlinear Boundary Condition, II,” Nonlinear Analysis: Theory, Methods & Applications, Vol. 73, No. 4, 2010, pp. 971-978. doi:10.1016/j.na.2010.04.023
[11]	L. E. Payne and P. W. Schaefer, “Lower Bound for BlowUp Time in Parabolic Problems under Neumann Conditions,” Applicable Analysis, Vol. 85, No. 10, 2006, pp. 1301-1311. doi:10.1080/00036810600915730
[12]	L. E. Payne and P. W. Schaefer, “Lower Bound for Blow -Up Time in Parabolic Problems under Dirichlet Conditions,” Journal of Mathematical Analysis and Applications, Vol. 328, No. 2, 2007, pp. 1196-1205. doi:10.1016/j.jmaa.2006.06.015
[13]	L. E. Payne and P. W. Schaefer, “Bounds for the BlowUp Time for the Heat Equation under Nonlinear Boundary Conditions,” Proceedings of the Royal Society of Edinburgh, Vol. 139, No. 6, 2009, pp. 1289-1296.
[14]	L. E. Payne and J. C. Song, “Lower Bounds for the BlowUp Time in a Temperature Dependent Navier-Stokes Flow,” Journal of Mathematical Analysis and Applications, Vol. 335, No. 1, 2007, pp. 371-376. doi:10.1016/j.jmaa.2007.01.083
[15]	P. Quittner, “On Global Existence and Stationary Solutions of Two Classes of Semilinear Parabolic Equations,” Commemtationes Mathematicae Universitatis Carolinae, Vol. 34, No. 1, 1993, pp. 105-124.
[16]	P. Quittner and P. Souplet, “Superlinear Parabolic Problems. Blow-Up, Global Existence and Steady States,” Birkh?user, Basel, 2007.
[17]	J. L. Vazquez, “The Problem of Blow-Up for Nonlinear Heat Equations. Complete Blow-Up and Avalanche Formation,” Rendiconti Lincei Matematica e Applicazioni, Vol. 15, No. 34, 2004, pp. 281-300.
[18]	F. B. Weissler, “Local Existence and Nonexistence for Semilinear Parabolic Equations in LP,” Indiana University Mathematics Journal, Vol. 29, No. 1, 1980, pp. 79-102. doi:10.1512/iumj.1980.29.29007
[19]	F. B. Weissler, “Existence and Nonexistence of Global Solutions for a Heat Equation,” Isra?l Journal of Mathematics, Vol. 38, No. 1-2, 1981, pp. 29-40.
[20]	G. Talenti, “Best Constant in Sobolev Inequality,” Annali di Matematica Pura ed Applicata, Vol. 110, No. 1, 1976, pp. 353-372.
[21]	L. E. Payne, “Uniqueness Criteria for Steady State Solutions of the Navier-Stokes Equations,” In: Atti del Simposio Internazionale Sulle Applicazioni Dell’Analisi Alla Fisica Matematica, Cagliari-Sassari, 1964, pp. 130-153.