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CRITICAL CURVES FOR A COUPLED SYSTEM OF FAST DIFFUSIVE NEWTONIAN FILTRATION EQUATIONS

Part of: Parabolic equations and systems

Published online by Cambridge University Press:  07 June 2012

RUNMEI DU  and
ZEJIA WANG
RUNMEI DU
Affiliation:
Institute of Mathematics, Jilin University, Changchun 130012, PR China
ZEJIA WANG*
Affiliation:
College of Mathematics, Jilin University, Changchun 130012, PR China (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
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Abstract

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This paper deals with the large-time behaviour of solutions to the fast diffusive Newtonian filtration equations coupled via the nonlinear boundary sources. A result of Fujita type is obtained by constructing various kinds of upper and lower solutions. In particular, it is shown that the critical global existence curve and the critical Fujita curve concide for the multi-dimensional system. This is quite different from the known results obtained in Wang, Zhou and Lou [‘Critical exponents for porous medium systems coupled via nonlinear boundary flux’, Nonlinear Anal.7(1) (2009), 2134–2140] for the corresponding one-dimensional problem.

Keywords

global existenceblow-upNewtonian filtration equationcritical curve

MSC classification

Secondary: 35K65: Degenerate parabolic equations
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 86 , Issue 3 , December 2012 , pp. 440 - 447
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2012

Footnotes

Supported by the NNSF.

References

[1]Deng, K., Fila, M. and Levine, H. A., ‘On critical exponent for a system of heat equations coupled in the boundary conditions’, Acta Math. Univ. Comenian. 63(2) (1994), 169–192.Google Scholar
[2]Deng, K. and Levine, H. A., ‘The role of critical exponents in blow-up theorems: the sequel’, J. Math. Anal. Appl. 243(1) (2000), 85–126.CrossRefGoogle Scholar
[3]Ferreira, R., Pablo, A., de Quirós, F. and Rossi, J. D., ‘The blow-up profile for a fast diffusion equation with a nonlinear boundary condition’, Rocky Mountain J. Math. 33 (2003), 123–146.CrossRefGoogle Scholar
[4]Fujita, H., ‘On the blowing up of solutions of the Cauchy problem for u t=Δu+u 1+α’, J. Fac. Sci. Univ. Tokyo Sect. I 13 (1966), 109–124.Google Scholar
[5]Galaktionov, V. A. and Levine, H. A., ‘A general approach to critical Fujita exponents in nonlinear parabolic problems’, Nonlinear Anal. 34(7) (1998), 1005–1027.CrossRefGoogle Scholar
[6]Kalashnikov, A. S., ‘Some problems of the qualitative theory of nonlinear degenerate second-order parabolic equations’, Russian Math. Surveys 42(2) (1987), 169–222.CrossRefGoogle Scholar
[7]Pang, P. Y. H., Wang, Z. J. and Yin, J. X., ‘Critical exponents for nonlinear diffusion equations with nonlinear boundary sources’, J. Math. Anal. Appl. 343(2) (2008), 654–662.CrossRefGoogle Scholar
[8]de Quirós, F. and Rossi, J. D., ‘Blow-up sets and Fujita type curves for a degenerate parabolic system with nonlinear boundary conditions’, Indiana Univ. Math. J. 50 (2001), 629–654.CrossRefGoogle Scholar
[9]Wang, Z. J., Zhou, Q. and Lou, W. Q., ‘Critical exponents for porous medium systems coupled via nonlinear boundary flux’, Nonlinear Anal. 7(1) (2009), 2134–2140.CrossRefGoogle Scholar
[10]Wu, Z. Q., Zhao, J. N., Yin, J. X. and Li, H. L., Nonlinear Diffusion Equations (World Scientific, River Edge, NJ, 2001).CrossRefGoogle Scholar