|
|
|
A degenerate parabolic system with localized sources and nonlocal boundary condition |
Yongsheng MI1,2( ), Chunlai MU1 |
| 1. College of Mathematics and Statistics, Chongqing University, Chongqing 400044, China; 2. College of Mathematics and Computer Sciences, Yangtze Normal University, Fuling 408100, China |
|
|
|
|
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
|
|
Corresponding Author(s):
MI Yongsheng,Email:miyongshen@163.com
|
|
Issue Date: 01 February 2012
|
|
| 1 |
Anderson J R. Local existence and uniqueness of solutions of degenerate parabolic equations. Comm Partial Differential Equations , 1991, 16: 105-143
doi: 10.1080/03605309108820753
|
| 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
doi: 10.1016/0362-546X(82)90072-4
|
| 4 |
Bebernes J, Bressan A, Lacey A. Total explosions versus single-point blow-up. J Differential Equations , 1988, 73: 30-44
doi: 10.1016/0022-0396(88)90116-7
|
| 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
doi: 10.1137/0522068
|
| 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
doi: 10.1016/0022-247X(92)90081-N
|
| 7 |
Chen Y, Xie C. Blow-up for a porous medium equation with a localized source. Appl Math Comput , 2004, 159: 79-93
doi: 10.1016/j.amc.2003.10.032
|
| 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
doi: 10.1016/j.jmaa.2005.07.058
|
| 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
doi: 10.1016/j.na.2005.08.004
|
| 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
doi: 10.1016/j.jmaa.2006.11.048
|
| 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
doi: 10.1016/j.amc.2007.09.037
|
| 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
doi: 10.1016/j.jmaa.2007.11.055
|
| 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
doi: 10.1007/978-1-4613-9555-3
|
| 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
doi: 10.1016/j.na.2004.10.016
|
| 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
doi: 10.1016/0022-0396(87)90125-2
|
| 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
doi: 10.1016/S0898-1221(04)90081-8
|
| 20 |
Friedman A, Mcleod B. Blow-up of positive solutions of semilinear heat equations. Indiana Univ Math J , 1985, 34: 425-447
doi: 10.1512/iumj.1985.34.34025
|
| 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
doi: 10.1007/BF00276081
|
| 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 , pmid:10.1007/s10440-010-9563-9" target="blank">
doi: 10.1007/s10440-010-9563-9 pmid:10.1007/s10440-010-9563-9" target="blank">
doi: 10.1007/s10440-010-9563-9
|
| 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
doi: 10.1007/s11425-007-0105-5
|
| 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
doi: 10.3934/dcds.2005.13.683
|
| 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
doi: 10.1016/j.amc.2007.09.054
|
| 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
doi: 10.1016/j.na.2009.11.047
|
| 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
doi: 10.1063/1.1677879
|
| 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
doi: 10.1016/0022-247X(92)90318-8
|
| 35 |
Pao C V. Asymptotic behavior of solutions of reaction-diffusion equations with nonlocal boundary conditions. Comput Math Appl , 1998, 88: 225-238
doi: 10.1016/S0377-0427(97)00215-X
|
| 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
doi: 10.1016/S0362-546X(01)00798-2
|
| 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
doi: 10.1137/S0036141097318900
|
| 39 |
Souplet P. Uniform blow-up profiles and boundary behavior for diffusion equations with nonlocal nonlinear source. J Differential Equations , 1999, 153: 374-406
doi: 10.1006/jdeq.1998.3535
|
| 40 |
Wang L, Chen Q. The asymptotic behavior of blow-up solution of localized nonlinear equation. J Math Anal Appl , 1996, 200: 315-321
doi: 10.1006/jmaa.1996.0207
|
| 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
doi: 10.1016/j.jmaa.2008.01.073
|
| 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
doi: 10.1016/j.amc.2007.03.036
|
| 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,
doi: 10.1155/2009/516390
|
| 45 |
Yin H M. On a class of parabolic equations with nonlocal boundary conditions. J Math Anal Appl , 2004, 294: 712-728
doi: 10.1016/j.jmaa.2004.03.021
|
| 46 |
Yin Y F. On nonlinear parabolic equations with nonlocal boundary condition. J Math Anal Appl , 1994, 185: 161-174
doi: 10.1006/jmaa.1994.1239
|
| 47 |
Zheng S N, Kong L. Roles of weight functions in a nonlinear nonlocal parabolic system. Nonlinear Anal , 2008, 68: 2406-2416
doi: 10.1016/j.na.2007.01.067
|
| 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
doi: 10.1016/j.jmaa.2003.12.011
|
| 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
doi: 10.1016/j.na.2007.04.038
|
|
Viewed |
|
|
|
Full text
|
|
|
|
|
Abstract
|
|
|
|
|
Cited |
|
|
|
|
| |
Shared |
|
|
|
|
| |
Discussed |
|
|
|
|
