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GLOBAL EXISTENCE AND BLOW-UP FOR NON-NEWTON POLYTROPIC FILTRATION SYSTEM COUPLED WITH LOCAL SOURCE

Published online by Cambridge University Press:  01 January 2009

JUN ZHOU  and
CHUNLAI MU
JUN ZHOU*
Affiliation:
School of Mathematics and Statistics, Southwest University, Chongqing 400715, P. R. China e-mail: zhoujun_math@hotmail.com.
CHUNLAI MU
Affiliation:
School of Mathematics and Physics, Chongqing University, Chongqing 400044, P. R. China
*
*Corresponding author.
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Abstract

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This paper deals with the global existence and blow-up properties of the following non-Newton polytropic filtration system coupled with local source: ut − Δm,pu = avα, vt − Δn,qv = buβ. Under appropriate hypotheses, we prove that the solution either exists globally or blows up in finite time depending on the initial data and the relations between αβ and mn(p − 1)(q − 1).

Keywords

35K5035K5535K6535B33
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 51 , Issue 1 , January 2009 , pp. 39 - 47
Copyright
Copyright © Glasgow Mathematical Journal Trust 2008

References

REFERENCES

1.Anderson, J. R. and Deng, K., Global existence for degenerate parabolic equations with a non-local forcing, Math. Anal. Methods Appl. Sci. 20 (1997), 1069–1087.Google Scholar
2.Bidanut-Véon, M. F. and García-Huidobro, M., Regular and singular solutions of a quasilinear equation with weights, Asymptotic Anal. 28 (2001), 115–150.Google Scholar
3.Deng, K. and Levine, H. A., The role of critical exponents in blow-up theorems: The sequel, J. Math. Anal. Appl. 243 (2000), 85–126.Google Scholar
4.Dibenedetto, E., Degenerate parabolic equations (Springer-Verlag, Berlin, New York, 1993).Google Scholar
5.Díaz, J. I., Nonlinear partial differential equations and free boundaries, In: Elliptic equations, vol. 1. (Pitman, London, 1985).Google Scholar
6.Deng, W. B., Li, Y. X. and Xie, C. H., Blow-up and global existence for a nonlocal degenerate parabolic system, J. Math. Anal. Appl. 277 (2003), 199–217.CrossRefGoogle Scholar
7.Deng, W. B., Global existence and finite time blow up for a degenerate reaction-diffusion system, Nonlinear Anal. 60 (2005), 977–991.CrossRefGoogle Scholar
8.Du, L. L., Blow-up for a degenerate reaction-diffusion system with nonlinear nonlocal sources, J. Comput. Appl. Math. 202 (2007), 237–247.CrossRefGoogle Scholar
9.Duan, Z. W., Deng, W. B. and Xie, C. H., Uniform blow-up profile for a degenerate parabolic system with nonlocal source, Comput. Math. Appl. 47 (2004), 977–995.Google Scholar
10.Galaktionov, V. A., Kurdyumov, S. P. and Samarskii, A. A., A parabolic system of quasi-linear equations I, Diff. Eq. 19 (1983), 1558–1571.Google Scholar
11.Galaktionov, V. A., Kurdyumov, S. P. and Samarskii, A. A., A parabolic system of quasi-linear equations II, Diff. Eq. 21 (1985), 1049–1062.Google Scholar
12.Galaktionov, V. A. and Levine, H. A., On critical Fujita exponents for heat equations with nonlinear flux conditions on the boundary, Israel J. Math. 94 (1996), 125–146.CrossRefGoogle Scholar
13.Galaktionov, V. A. and Vázquez, J. L., The problem of blow-up in nonlinear parabolic equations, Dist. Cont. Dyn. Syst. 8 (2002), 399–433.Google Scholar
14.Ishii, H., Asymptotic stability and blowing up of solutions of some nonlinear equations, J. Diff. Eq. 26 (1997), 291–319.CrossRefGoogle Scholar
15.Kalashnikov, A. S., Some problems of the qualitative theory of nonlinear degenerate parabolic equations of second order, Russi. Math. Surv. 42 (1987), 169–222.CrossRefGoogle Scholar
16.Levine, H. A. and Payne, L. E., Nonexistence theorems for the heat equation with nonlinear boundary conditions for the porous medium equation backward in time, J. Diff. Eq. 16 (1974), 319–334.Google Scholar
17.Levine, H. A., The role of critical exponents in blow up theorems, SIAM Rev. 32 (1990), 262–288.CrossRefGoogle Scholar
18.Li, F. C. and Xie, C. H., Global and blow-up solutions to a p-Laplacian equation with nonlocal source, Comput. Math. Appl. 46 (2003), 1525–1533.Google Scholar
19.Li, F. C. and Xie, C. H., Global existence and blow-up for a nonlinear porous medium equation, Appl. Math. Lett. 16 (2003), 185–192.CrossRefGoogle Scholar
20.Li, Y. X. and Xie, C. H., Blow-up for p-Laplacian parabolic equations, J. Diff. Eq. 20 (2003), 1–12.Google Scholar
21.Lindqvist, P., On the equation , Proc. Am. Math. Soc. 109 (1990), 157–164.Google Scholar
22.Lindqvist, P., On the equation , Proc. Am. Math. Soc. 116 (1992), 583–584.Google Scholar
23.de Pablo, A., Quiros, F. and Rossi, J. D., Asymptotic simplification for a reaction-diffusion problem with a nonlinear boundary condition, IMA J. Appl. Math. 67 (2002), 69–98.Google Scholar
24.Quiros, 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.Google Scholar
25.Samarskii, A. A., Galaktionov, V. A., Kurdyumov, S. P. and Mikhailov, A. P., Blow-up in quasilinear parabolic equations (Walter de Gruyter, Berlin, 1985).Google Scholar
26.Sun, W. J. and Wang, S., Nonlinear degenerate parabolic equation with nonlinear boundary condition, Acta Math. Sinica, English Ser. 21 (2005), 847–854.CrossRefGoogle Scholar
27.Tsutsumi, M., Existence and nonexistence of global solutions for nonlinear parabolic equations, Publ. Res. Inst. Math. Sci. 8 (1972), 221–229.CrossRefGoogle Scholar
28.Tsutsumi, M., On solutions of some doubly nonlinear degenerate parabolic equations with absorption, J. Math. Anal. Appl. 132 (1988) 187–212.CrossRefGoogle Scholar
29.Vázquez, J. L., The porous medium equations: Mathematical theory Clarendon Press, Oxford, 2007.Google Scholar
30.Wang, S., Doubly nonlinear degenerate parabolic systems with coupled nonlinear boundary conditions, J. Diff Eq. 182 (2002), 431–469.CrossRefGoogle Scholar
31.Wu, Z. Q., Zhao, J. N., Yin, J. X. and Li, H. L., Nonlinear diffusion equations (Word Scientific Publishing Co.; River Edge, NJ, 2001).Google Scholar
32.Yuan, H. J., Extinction and positivity of the evolution p-Laplacian equation, J. Math. Anal. Appl. 196 (1995), 754–763.CrossRefGoogle Scholar
33.Zheng, S. N., Song, X. F. and Jiang, Z. X., Fujita exponents for degenerate parabolic equations coupled via nonlinear boundary flux, J. Math. Anal. Appl. 298 (2004), 308–324.Google Scholar
34.Zhao, J., Existence and nonexistence of solutions for = f(▽u, u, x, t), J. Math. Anal. Appl. 173 (1993), 130–146.CrossRefGoogle Scholar
35.Zhou, J. and Mu, C. L., On critical Fujita exponent for degenerate parabolic system coupled via nonlinear boundary flux, Proc. Edinb. Math. Soc. 51 (2008), 785–805.Google Scholar
36.Zhou, J. and Mu, C. L., The critical curve for a non-Newtonian polytropic filtration system coupled via nonlinear boundary flux, Nonlinear Anal. 68 (2008), 1–11.CrossRefGoogle Scholar