Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-25T06:28:16.832Z Has data issue: false hasContentIssue false
  • English
  • Français

GLOBAL EXISTENCE AND BLOW-UP FOR A NON-NEWTON POLYTROPIC FILTRATION SYSTEM WITH NONLOCAL SOURCE

Part of: Parabolic equations and systems Partial differential equations

Published online by Cambridge University Press:  01 July 2008

JUN ZHOU  and
CHUNLAI MU
JUN ZHOU*
Affiliation:
School of Mathematics and Physics, Chongqing University, Chongqing, 400044, People’s Republic of China (email: [email protected])
CHUNLAI MU
Affiliation:
School of Mathematics and Physics, Chongqing University, Chongqing, 400044, People’s Republic of China (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper deals the global existence and blow-up properties of the following non-Newton polytropic filtration system with nonlocal source, 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). In the special case, α=n(q−1), β=m(p−1), we also give a criteria for the solution to exist globally or blow up in finite time, which depends on a,b and ζ(x),ϑ(x) as defined in our main results.

Keywords

non-Newtonian polytropic systemnonlocal sourceglobal existenceblow-up

MSC classification

Secondary: 35K65: Degenerate parabolic equations 35B33: Critical exponents
Type
Research Article
Information
The ANZIAM Journal , Volume 50 , Issue 1 , July 2008 , pp. 13 - 29
Copyright
Copyright © Australian Mathematical Society 2009

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 Garcıá-Huidobro, M., “Regular and singular solutions of a quasilinear equation with weights”, Asymptot. Anal. 28 (2001) 115–150.Google Scholar
[3]Deng, W. B., “Global existence and finite time blow up for a degenerate reaction-diffusion system”, Nonlinear Anal. 60 (2005) 977–991.CrossRefGoogle Scholar
[4]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
[5]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
[6]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
[7]Díaz, J. I., “Nonlinear partial differential equations and free boundaries”, in Elliptic equations, Volume 1 (Pitman, London, 1985).Google Scholar
[8]Dibenedetto, E., Degenerate parabolic equations (Springer, Berlin, 1993).CrossRefGoogle Scholar
[9]Du, L. L., “Blow-up for a degenerate reaction-diffusion system with nonlinear nonlocal sources”, J. Comput. Appl. Math. 202 (2007) 237–247.Google Scholar
[10]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
[11]Galaktionov, V. A., Kurdyumov, S. P. and Samarskii, A. A., “A parabolic system of quasi-linear equations I”, Differ. Equ. 19 (1983) 1558–1571.Google Scholar
[12]Galaktionov, V. A., Kurdyumov, S. P. and Samarskii, A. A., “A parabolic system of quasi-linear equations II”, Differ. Equ. 21 (1985) 1049–1062.Google Scholar
[13]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.Google Scholar
[14]Galaktionov, V. A. and Vázquez, J. L., “The problem of blow-up in nonlinear parabolic equations”, Dist. Cont. Dyn. Systems 8 (2002) 399–433.Google Scholar
[15]Ishii, H., “Asymptotic stability and blowing up of solutions of some nonlinear equations”, J. Differential Equations 26 (1997) 291–319.Google Scholar
[16]Kalashnikov, A. S., “Some problems of the qualitative theory of nonlinear degenerate parabolic equations of second order”, Russian Math. Surveys 42 (1987) 169–222.Google Scholar
[17]Levine, H. A., “The role of critical exponents in blow-up theorems”, SIAM Rev. 32 (1990) 262–288.Google Scholar
[18]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. Differential Equations 16 (1974) 319–334.CrossRefGoogle Scholar
[19]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
[20]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.Google Scholar
[21]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.CrossRefGoogle Scholar
[22]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
[23]Sun, W. J. and Wang, S., “Nonlinear degenerate parabolic equation with nonlinear boundary condition”, Acta Math. Sin. (Engl. Ser.) 21 (2005) 847–854.Google Scholar
[24]Tsutsumi, M., “Existence and nonexistence of global solutions for nonlinear parabolic equations”, Publ. Res. Inst. Math. Sci. 8 (1972) 221–229.Google Scholar
[25]Vázquez, J. L., The porous medium equations: mathematical theory (Clarendon Press, Oxford, 2007).Google Scholar
[26]Wang, S., “Doubly nonlinear degenerate parabolic systems with coupled nonlinear boundary conditions”, J. Differential Equations 182 (2002) 431–469.Google Scholar
[27]Wu, Z. Q., Zhao, J. N., Yin, J. X. and Li, H. L., Nonlinear diffusion equations (Word Scientific, River Edge, NJ, 2001).Google Scholar
[28]Zhao, J., “Existence and nonexistence of solutions for ”, J. Math. Anal. Appl. 173 (1993) 130–146.CrossRefGoogle Scholar
[29]Zheng, S. N., Song, X. F. and Jiang, Z. X., “Critical Fujita exponents for degenerate parabolic equations coupled via nonlinear boundary flux”, J. Math. Anal. Appl. 298 (2004) 308–324.Google Scholar
[30]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
[31]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.Google Scholar