CRITICAL EXTINCTION EXPONENTS FOR POLYTROPIC FILTRATION EQUATIONS WITH NONLOCAL SOURCE AND ABSORPTION

doi:10.1016/S0252-9602(15)60008-X

Abstract

Abstract:

In this article, by applying the super-solution and sub-solution methods, instead of energy estimate methods, the authors investigate the critical extinction exponents for a polytropic filtration equation with a nonlocal source and an absorption term, and give a classification of the exponents and coefficients for the solutions to vanish in finite time or not, which improve one of our results (Applicable Analysis, 92(2013), 636-650) and the results of Zheng et al (Math. Meth. Appl. Sci., 36(2013), 730-743).

Key words: Polytropic filtration equation, critical exponent, extinction, super-solution and sub-solution

CLC Number:

35K55

Haixia LI, Yuzhu HAN, Wenjie GAO. CRITICAL EXTINCTION EXPONENTS FOR POLYTROPIC FILTRATION EQUATIONS WITH NONLOCAL SOURCE AND ABSORPTION[J].Acta mathematica scientia,Series B, 2015, 35(2): 366-374.

Trendmd

References

[1] Dibenedetto E. Degenerate Parabolic Equations. Berlin: Springer-Verlag, 1993

[2] Vázquez J L. The porousmedium equation, Mathematical Theory. Oxford Mathematical Monographs. Ox- ford: Oxford University Press, 2007

[3] Calsina A, Perello C, Saldana J. Non-local reaction-diffusion equations modelling predator-prey coevolution. Publ Math, 1994, 32: 315-325

[4] Furter J, Grinfeld M. Local vs nonlocal interactions in population dynamics. J Math Biol, 1989, 27: 65-80

[5] Galaktionov V A, Peletier L A, Vazquez J L. Asymptotics of fast-diffusion equation with critical exponent. SIAM J Math Anal, 2000, 31: 1157-1174

[6] Galaktionov V A, Vazquez J L. Asymptotic behavior of nonlinear parabolic equations with critical expo- nents. A dynamical system approach. J Funct Anal, 1991, 100: 435-462

[7] Galaktionov V A, Vazquez J L. Extinction for a quasilinear heat equation with absorption I. Technique of intersection comparison. Commun Partial Differential Equations, 1994, 19: 1075-1106

[8] Galaktionov V A, Vazquez J L. Extinction for a quasilinear heat equation with absorption II. A dynamical system approach. Commun Partial Differential Equations, 1994, 19: 1107-1137

[9] Han Y Z, Gao W J. Extinction for a fast diffusion equation with a nonlinear nonlocal source. Arch Math, 2011, 97: 353-363

[10] Han Y Z, Gao W J. Extinction and non-extinction for a polytropic filtration equation with a nonlocal source. Appl Anal, 2013, 92: 636-650

[11] Leoni G. A very singular solution for the porous media equation u_t = Δu^m - u^p when 0
[12] Liu W J, Wang M X, Wu B. Extinction and Decay Estimates of Solutions for a Class of Porous Medium Equations. J Inequalities and Applications, 2007, 2007: 1-8

[13] Wang Y F, Yin J X. Critical extinction exponents for a polytropic filtration equation with absorption and source. Math Method Appl Sci, 2013, 36: 1591-1597

[14] Yin J X, Li J, Jin C H. Non-extinction and critical exponent for a polytropic filtration equation. Nonl Anal, 2009, 71: 347-357

[15] Yin J X, Jin C H. Critical extinction and blow-up exponents for fast diffusive polytropic filtration equation with sources. Proceedings of the Edinburgh Mathematical Society, 2009, 52: 419-444

[16] Zheng P, Mu C L. Extinction and decay estimates of solutions for a polytropic filtration equation with the nonlocal source and interior absorption. Math Method Appl Sci, 2013, 36: 730-743

[17] Anderson J R. Local existence and uniqueness of solutions of degenerate parabolic equations. Commun Partial Differential Equations, 1991, 16: 105-143