Browse > Article
http://dx.doi.org/10.4134/CKMS.2008.23.4.529

BLOW-UP FOR A NON-NEWTON POLYTROPIC FILTRATION SYSTEM WITH NONLINEAR NONLOCAL SOURCE  

Zhou, Jun (SCHOOL OF MATHEMATICS AND STATISTICS SOUTHWEST UNIVERSITY)
Mu, Chunlai (COLLEGE OF MATHEMATICS AND PHYSICS CHONGQING UNIVERSITY)
Publication Information
Communications of the Korean Mathematical Society / v.23, no.4, 2008 , pp. 529-540 More about this Journal
Abstract
This paper deals the global existence and blow-up properties of the following non-Newton polytropic filtration system, $${u_t}-{\triangle}_{m,p}u=u^{{\alpha}_1}\;{\int}_{\Omega}\;{\upsilon}^{{\beta}_1}\;(x,\;t)dx,\;{\upsilon}_t-{\triangle}_{n,p}{\upsilon}={\upsilon}^{{\alpha}_2}\;{\int}_{\Omega}\;u^{{\beta}_2}\;(x,{\;}t)dx,$$ with homogeneous Dirichlet boundary condition. Under appropriate hypotheses, we prove that the solution either exists globally or blows up in finite time depends on the initial data and the relations of the parameters in the system.
Keywords
non-Newtonian polytropic system; nonlocal source; global existence; blow-up;
Citations & Related Records
연도 인용수 순위
  • Reference
1 V. A. Galaktionov, A parabolic system of quasi-linear equations II, Differential Equations 21 (1985), 1049-1062
2 V. A. Galaktionov and H. A. Levine, On critical Fujita exponents for heat equations with nonlinear flux conditions on the boundary, Israel J. Math. 94 (1996), 125-146   DOI
3 V. A. Galaktionov and J. L. V'azquez, The problem of blow-up in nonlinear parabolic equations, Dist. Cont. Dyn. Systems 8 (2002), 399-433   DOI
4 F. C. Li, Global existence and blow-up for a nonlinear porous medium equation, Appl. Math. Lett. 16 (2003), 185-192   DOI   ScienceOn
5 Y. X. Li and C. H. Xie, Blow-up for p-Laplacian parabolic equations, J. Differential Equations 20 (2003), 1-12   DOI
6 P. Lindqvist, On the equation ${\nabla}{\cdot}$$({\left|}{\nabla}{u}{\right|}^{p-2}\nabla{u})$ = 0, Pro. Amer. Math. Soc. 109 (1990), 157-164   DOI
7 P. Lindqvist, On the equation ${\nabla}{\cdot}$$({\left|}{\nabla}{u}{\right|}^{p-2}\nabla{u})$= 0, Pro. Amer. Math. Soc. 116 (1992), 583-584   DOI
8 A. de Pablo, F. Quiros, and J. D. Rossi, Asymptotic simplification for a reactiondiffusion problem with a nonlinear boundary condition, IMA J. Appl. Math. 67 (2002), 69-98   DOI   ScienceOn
9 A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov, and A. P. Mikhailov, Blow-up in Quasilinear Parabolic Equations, Walter de Gruyter, Berlin, 1985
10 K. Deng and H. A. Levine, The role of critical exponents in blow-up theorems: The sequel, J. Math. Anal. Appl. 243 (2000), 85-126   DOI   ScienceOn
11 J. R. Anderson and K. Deng, Global existence for degenerate parabolic equations with a non-local forcing, Math. Anal. Methods Appl. Sci. 20 (1997), 1069-1087   DOI   ScienceOn
12 M. F. Bidanut-Veon and M. Garcıa-Huidobro, Regular and singular solutions of a quasilinear equation with weights, Asymptotic Anal. 28 (2001), 115-150
13 J. I. Diaz, Nonlinear Partial Differential Equations and Free Boundaries, in Elliptic Equations, Vol 1, Pitman, London, 1985
14 W. B. Deng, Global existence and finite time blow up for a degenerate reaction-diffusion system, Nonlinear Anal. 60 (2005), 977-991   DOI   ScienceOn
15 W. B. Deng, Y. X. Li, and C. H. Xie, Blow-up and global existence for a nonlocal degenerate parabolic system, J. Math. Anal. Appl. 277 (2003), 199-217   DOI   ScienceOn
16 J. Zhao, Existence and nonexistence of solutions for ut −${\nabla}{\cdot}$$({\left|}{\nabla}{u}{\right|}^{p-2}\nabla{u})$ = f(${\nabla}$u, u, x, t), J. Math. Anal. Appl. 173 (1993), 130-146   DOI   ScienceOn
17 S. N. Zheng, X. F. Song, and Z. X. Jiang, Critical Fujita exponents for degenerate parabolic equations coupled via nonlinear boundary flux, J. Math. Anal. Appl. 298 (2004), 308-324   DOI   ScienceOn
18 J. Zhou and C. L. Mu, On critical Fujita exponent for degenerate parabolic system coupled via nonlinear boundary flux, Proc. Edinb. Math. Soc. (in press)   DOI   ScienceOn
19 J. Zhou, The critical curve for a non-Newtonian polytropic filtration system coupled via nonlinear boundary flux, Nonlinear Anal. 68 (2008), 1-11   DOI   ScienceOn
20 H. Ishii, Asymptotic stability and blowing up of solutions of some nonlinear equations, J. Differential Equations 26 (1997), 291-319   DOI
21 H. A. Levine, The role of critical exponents in blow up theorems, SIAM Rev. 32 (1990), 262-288   DOI   ScienceOn
22 J. Zhou, Global existence and blow-up for non-Newton polytropic filtration system with nonlocal source, ANZIAM J. (in press)   DOI
23 J. L. Vazquez, The porous Medium Equations: Mathematical Theory, Oxford Univ. Press to appear
24 S. Wang, Doubly Nonlinear Degenerate parabolic Systems with coupled nonlinear boundary conditions, J. Differential Equations 182 (2002), 431-469   DOI   ScienceOn
25 Z. Q. Wu, J. N. Zhao, J. X. Yin, and H. L. Li, Nonlinear Diffusion Equations, Word Scientific Publishing Co., Inc., River Edge, NJ, 2001
26 F. C. Li and C. H. Xie, Global and blow-up solutions to a p-Laplacian equation with nonlocal source, Comput. Math. Appl. 46 (2003), 1525-1533   DOI   ScienceOn
27 L. L. Du, Blow-up for a degenerate reaction-diffusion system with nonlinear nonlocal sources, J. Comput. Appl. Math. 202 (2007), 237-247   DOI   ScienceOn
28 Z. W. Duan, W. B. Deng, and C. H. Xie, Uniform blow-up profile for a degenerate parabolic system with nonlocal source, Comput. Math. Appl. 47 (2004), 977-995   DOI   ScienceOn
29 V. A. Galaktionov, S. P. Kurdyumov, and A. A. Samarskii, A parabolic system of quasilinear equations I, Differential Equations 19 (1983), 1558-1571
30 H. A. Levine and L. E. Payne, 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   DOI
31 W. J. Sun and S. Wang, Nonlinear degenerate parabolic equation with nonlinear boundary condition, Acta Mathematica Sinica, English Series 21 (2005), 847-854   DOI
32 M. Tsutsumi, Existence and nonexistence of global solutions for nonlinear parabolic equations, Publ. Res. Inst. Math. Sci. 8 (1972), 221-229   DOI
33 J. Zhou, Global existence and blow-up for non-Newton polytropic filtration system coupled with local source, Glasgow Math. J. (in press)   DOI   ScienceOn
34 F. Quiros and J. D. Rossi, Blow-up sets and Fujita type curves for a degenerate parabolic system with nonlinear boundary conditions, Indiana Univ. Math. J. 50 (2001), 629-654   DOI
35 E. Dibenedetto, Degenerate Parabolic Equations, Springer-Verlag, Berlin, New York, 1993
36 A. S. Kalashnikov, Some Problems of the qualitative theory of nonlinear degenerate parabolic equations of second order, Russian Math. Surveys 42 (1987), 169-222   DOI   ScienceOn