A degenerate parabolic system with localized sources and nonlocal boundary condition

Abstract

This paper deals with the blow-up properties of the positive solutions to a degenerate parabolic system with localized sources and nonlocal boundary conditions. We investigate the influence of the reaction terms, the weight functions, local terms and localized source on the blow-up properties. We will show that the weight functions play the substantial roles in determining whether the solutions will blow-up or not, and obtain the blow-up conditions and its blow-up rate estimate.

Keywords

Nonlocal boundary condition / localized sources / blow-up rate / Porous medium equation

References

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

[2] Andreucci D, Herrero M A, Velaázquez J L. Liouville theorems and blow up behavior in semilinear reaction- diffusion systems. Ann Inst H Poincaré Anal Non Linéaire, 1997, 14: 1-53

[3] Aronson D G, Crandall M G, Peletier L A. Stabilization of solutions of a degenerate nonlinear diffusion problem. Nonlinear Anal, 1982, 6: 1001-1022

[4] Bebernes J, Bressan A, Lacey A. Total explosions versus single-point blow-up. J Differential Equations, 1988, 73: 30-44

[5] Cantrell R S, Cosner C. Diffusive logistic equation with indefinite weights: Population models in disrupted environments II. SIAM J Math Anal, 1991, 22: 1043-1064

[6] Chadam J M, Peirce A, Yin H M. The blowup property of solutions to some diffusion equations with localized nonlinear reactions. J Math Anal Appl, 1992, 169: 313-328

[7] Chen Y, Xie C. Blow-up for a porous medium equation with a localized source. Appl Math Comput, 2004, 159: 79-93

[8] Chen Y P, Liu Q L, Gao H J. Uniform blow-up rate for diffusion equations with localized nonlinear source. J Math Anal Appl, 2006, 320: 771-778

[9] Chen Y P, Liu Q L, Gao H J. Boundedness of global solutions of a porous medium equation with a localized source. Nonlinear Anal, 2006, 64: 2168-2182

[10] Chen Y P, Liu Q L, Gao H J. Boundedness of global positive solutions of a porous medium equation with a moving localized source. J Math Anal Appl, 2007, 333: 1008-1023

[11] Cui Z J, Yang Z D. Boundedness of global solutions for a nonlinear degenerate parabolic (porous medium) system, with localized sources. Appl Math Comput, 2008, 198: 882-895

[12] Cui Z J, Yang Z D. Roles of weight functions to a nonlinear porous medium equation with nonlocal source and nonlocal boundary condition. J Math Anal Appl, 2008, 342: 559-570

[13] Day W A. Extensions of property of heat equation to linear thermoelasticity and other theories. Quart Appl Math, 1982, 40: 319-330

[14] Day W A. A decreasing property of solutions of parabolic equations with applications to thermoelasticity. Quart Appl Math, 1983, 41: 468-475

[15] Day W A. Heat Conduction within Linear Thermoelasticity. Springer Tracts in Natural Philosophy, Vol 30. New York: Springer, 1985

[16] Deng K. Comparison principle for some nonlocal problems. Quart Appl Math, 1992, 50: 517-522

[17] Deng W B. Global existence and finite time blow up for a degenerate reaction-diffusion system. Nonlinear Anal, 2005, 60: 977-991

[18] Diaz J I. On a nonlinear degenerate parabolic equation in infiltration or evaporation through a porous medium. J Differential Equations, 1987, 69: 368-403

[19] Duan Z W, Deng W B, Xie C H. Uniform blow-up profile for a degenerate parabolic system with nonlocal source. Comput Math Appl, 2004, 47: 977-995

[20] Friedman A, Mcleod B. Blow-up of positive solutions of semilinear heat equations. Indiana Univ Math J, 1985, 34: 425-447

[21] Friedman A. Monotonic decay of solutions of parabolic equations with nonlocal boundary conditions. Quart Appl Math, 1986, 44: 401-407

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

[23] Han Y Z, Gao W J. Global existence and blow-up for a class of degenerate parabolic systems with localized source. Acta Appl Math,

[24] Kong L H, Wang M X. Global existence and blow-up of solutions to a parabolic system with nonlocal sources and boundaries. Science in China, Ser A, 2007, 50: 1251-1266

[25] Li F J, Liu B C. Non-simultaneous blow-up in parabolic equations coupled via localized sources. Appl Math Lett (to appear)

[26] Li H L, Wang M X. Uniform blow-up profiles and boundary layer for a parabolic system with localized nonlinear reaction terms. Sci China, Ser A Math, 2005, 48: 185-197

[27] Li H L, Wang M X. Properties of blow-up solutions to a parabolic system with nonlinear localized terms. Discrete Contin Dyn Syst, 2005, 13: 683-700

[28] Li J, Cui Z J, Mu C J. Global existence and blow-up for degenerate and singular parabolic system with localized sources. Appl Math Comput, 2008, 199: 292-300

[29] Li Y X, Gao W J, Han Y Z. Boundedness of global solutions for a porous medium system with moving localized sources. Nonlinear Anal, 2010, 72: 3080-3090

[30] Lin Z G, Liu Y R. Uniform blow-up profiles for diffusion equations with nonlocal source and nonlocal boundary. Acta Math Sci, Ser B, 2004, 24: 443-450

[31] Mi Y S, Mu C L. Blowup properties for nonlinear degenerate diffusion equations with localized sources. Preprint

[32] Ortoleva P, Ross J. Local structures in chemical reaction with heterogeneous catalysis. J Chem Phys, 1972, 56: 43-97

[33] Pao C V. Nonlinear Parabolic and Elliptic Equations. New York: Plenum Press, 1992

[34] Pao C V. Blowing-up of solution of a nonlocal reaction-diffusion problem in combustion theory. J Math Anal Appl, 1992, 166: 591-600

[35] Pao C V. Asymptotic behavior of solutions of reaction-diffusion equations with nonlocal boundary conditions. Comput Math Appl, 1998, 88: 225-238

[36] Pedersen M, Lin Z. The profile near blowup time for solutions of diffusion systems coupled with localized nonlinear reactions. Nonlinear Anal, 2002, 50: 1013-1024

[37] Seo S. Blowup of solutions to heat equations with nonlocal boundary conditions. Kobe Journal of Mathematics, 1996, 13: 123-132

[38] Souplet P. Blow-up in nonlocal reaction-diffusion equations. SIAM J Math Anal, 1998, 29: 1301-1334

[39] Souplet P. Uniform blow-up profiles and boundary behavior for diffusion equations with nonlocal nonlinear source. J Differential Equations, 1999, 153: 374-406

[40] Wang L, Chen Q. The asymptotic behavior of blow-up solution of localized nonlinear equation. J Math Anal Appl, 1996, 200: 315-321

[41] Wang M X, Wen Y F, Blow-up properties for a degenerate parabolic system with nonlinear localized sources. J Math Anal Appl, 2008, 343: 621-635

[42] Wang Y L, Mu C L, Xiang Z Y. Blowup of solutions to a porous medium equation with nonlocal boundary condition. Appl Math Comput, 2007, 192: 579-585

[43] Wang Y L, Mu C L, Xiang Z Y. Properties of positive solution for nonlocal reactiondiffusion equation with nonlocal boundary. Boundary Value Problems, 2007, Article ID 64579, 12 pages

[44] Wang Y L, Xiang Z Y. Blowup analysis for a semilinear parabolic system with nonlocal boundary condition. Boundary Value Problems, 2009, Article ID 516390, 14 pages,

[45] Yin H M. On a class of parabolic equations with nonlocal boundary conditions. J Math Anal Appl, 2004, 294: 712-728

[46] Yin Y F. On nonlinear parabolic equations with nonlocal boundary condition. J Math Anal Appl, 1994, 185: 161-174

[47] Zheng S N, Kong L. Roles of weight functions in a nonlinear nonlocal parabolic system. Nonlinear Anal, 2008, 68: 2406-2416

[48] Zhao L Z, Zheng S N. Critical exponents and asymptotic estimates of solutions to parabolic systems with localized nonlinear sources. J Math Anal Appl, 2004, 292: 621-635

[49] Zhou J, Mu C L. Uniform blow-up profiles and boundary layer for a parabolic system with localized sources. Nonlinear Anal, 2008, 69: 24-34

RIGHTS & PERMISSIONS

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg