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

  • JUN ZHOU (a1) and CHUNLAI MU (a1)
Abstract
Abstract

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.

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Corresponding author
For correspondence; e-mail: zhoujun_math@hotmail.com
References
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[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) 10691087.
[2]Bidanut-Véon M. F. and Garcıá-Huidobro M., “Regular and singular solutions of a quasilinear equation with weights”, Asymptot. Anal. 28 (2001) 115150.
[3]Deng W. B., “Global existence and finite time blow up for a degenerate reaction-diffusion system”, Nonlinear Anal. 60 (2005) 977991.
[4]Deng K. and Levine H. A., “The role of critical exponents in blow-up theorems: the sequel”, J. Math. Anal. Appl. 243 (2000) 85126.
[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) 199217.
[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) 6998.
[7]Díaz J. I., “Nonlinear partial differential equations and free boundaries”, in Elliptic equations, Volume 1 (Pitman, London, 1985).
[8]Dibenedetto E., Degenerate parabolic equations (Springer, Berlin, 1993).
[9]Du L. L., “Blow-up for a degenerate reaction-diffusion system with nonlinear nonlocal sources”, J. Comput. Appl. Math. 202 (2007) 237247.
[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) 977995.
[11]Galaktionov V. A., Kurdyumov S. P. and Samarskii A. A., “A parabolic system of quasi-linear equations I”, Differ. Equ. 19 (1983) 15581571.
[12]Galaktionov V. A., Kurdyumov S. P. and Samarskii A. A., “A parabolic system of quasi-linear equations II”, Differ. Equ. 21 (1985) 10491062.
[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) 125146.
[14]Galaktionov V. A. and Vázquez J. L., “The problem of blow-up in nonlinear parabolic equations”, Dist. Cont. Dyn. Systems 8 (2002) 399433.
[15]Ishii H., “Asymptotic stability and blowing up of solutions of some nonlinear equations”, J. Differential Equations 26 (1997) 291319.
[16]Kalashnikov A. S., “Some problems of the qualitative theory of nonlinear degenerate parabolic equations of second order”, Russian Math. Surveys 42 (1987) 169222.
[17]Levine H. A., “The role of critical exponents in blow-up theorems”, SIAM Rev. 32 (1990) 262288.
[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) 319334.
[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) 15251533.
[20]Li F. C. and Xie C. H., “Global existence and blow-up for a nonlinear porous medium equation”, Appl. Math. Lett. 16 (2003) 185192.
[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) 629654.
[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).
[23]Sun W. J. and Wang S., “Nonlinear degenerate parabolic equation with nonlinear boundary condition”, Acta Math. Sin. (Engl. Ser.) 21 (2005) 847854.
[24]Tsutsumi M., “Existence and nonexistence of global solutions for nonlinear parabolic equations”, Publ. Res. Inst. Math. Sci. 8 (1972) 221229.
[25]Vázquez J. L., The porous medium equations: mathematical theory (Clarendon Press, Oxford, 2007).
[26]Wang S., “Doubly nonlinear degenerate parabolic systems with coupled nonlinear boundary conditions”, J. Differential Equations 182 (2002) 431469.
[27]Wu Z. Q., Zhao J. N., Yin J. X. and Li H. L., Nonlinear diffusion equations (Word Scientific, River Edge, NJ, 2001).
[28]Zhao J., “Existence and nonexistence of solutions for inline-graphic
$u_t-\nabla \cdot (|\nabla u|^{p-2}\nabla u)=f(\nabla u, u, x,t)$
”, J. Math. Anal. Appl. 173 (1993) 130146.
[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) 308324.
[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) 785805.
[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) 111.
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