ZHANG Zheng-ce, WANG Biao. Blow-up Rate Estimate for Degenerate Parabolic Equation With Nonlinear Gradient Term[J]. Applied Mathematics and Mechanics, 2010, 31(6): 756-764. doi: 10.3879/j.issn.1000-0887.2010.06.013
Citation: ZHANG Zheng-ce, WANG Biao. Blow-up Rate Estimate for Degenerate Parabolic Equation With Nonlinear Gradient Term[J]. Applied Mathematics and Mechanics, 2010, 31(6): 756-764. doi: 10.3879/j.issn.1000-0887.2010.06.013

Blow-up Rate Estimate for Degenerate Parabolic Equation With Nonlinear Gradient Term

doi: 10.3879/j.issn.1000-0887.2010.06.013
  • Received Date: 1900-01-01
  • Rev Recd Date: 2010-04-19
  • Publish Date: 2010-06-15
  • Blow-up rate was obtained for a porous medium equation with non linear gradient term and a non linear boundary flux. By using the scaling method and the regularity estmiates of parabolic equations, the blow-up rate which was deter mined by the interaction between the diffusion and the boundary flux was gotten. Interestingly, compared with the previous results, the gradientterm which exponent does not exceed 2 will not affect the blow-up rate for solutions.
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  • [1]
    Song X F, Zheng S N. Multi-nonlinear interaction in quasilinear reaction-diffusion equation with nonlinear boundary flux[J]. Math Comput Modelling, 2004, 39(2/3): 133-144. doi: 10.1016/S0895-7177(04)90002-7
    [2]
    Galaktionov V A, Levine H A. On critical Fujita exponents for heat equtions with nonlinear flux conditions on the boundary[J]. Israel J Math, 1996, 94(1): 125-146. doi: 10.1007/BF02762700
    [3]
    Quirs F, Rossi J D. Blow-up sets and Fujita type curves for a degenerate parabolic system with nonlinear boundary condition[J]. Indiana Univ Math J, 2001, 50(1):629-654. doi: 10.1512/iumj.2001.50.1828
    [4]
    Deng K, Levine H A. The role of critical exponents in blow-up theorems: the sequel[J]. Math Anal Appl, 2000,243(1): 85-106. doi: 10.1006/jmaa.1999.6663
    [5]
    Galaktionov V A, Vzquez J L. The problem of blow-up in nonlinear parabolic equations[J].Discrete Contin Dyn Sys, 2002, 8(2): 399-433. doi: 10.3934/dcds.2002.8.399
    [6]
    Levine H A. The role of critical exponents in blow-up theorems[J]. SIAM Rev, 1990, 32(2): 262-288. doi: 10.1137/1032046
    [7]
    Pao C V. Nonlinear Parabolic and Elliptic Equations[M]. New York, London: Plenum Press, 1992.
    [8]
    Samarskii A A, Galaktionov V A, Kurdyumov S P, Mikhailov A P. Blow-Up in Quasilinear Parabolic Equations[M].Berlin: Walter de Gruyter, 1995.
    [9]
    Vzquez J L. The Porous Medium Equations: Mathmatical Theory[M]. New York: Oxford Univ Press Inc, 2007.
    [10]
    Wu Z Q, Zhao J N, Yin J X, Li H L. Nonlinear Diffusion Equations[M]. River Edge, NJ: World Scientific Publishing Co, 2001.
    [11]
    Souplet P H. Recent results and open problems on parabolic equations with gradient nonlinearities[J]. Electronic J Differ Equations, 2001, 2001(20): 1-19.
    [12]
    Andreu F, Mazn J M, Simondon F, Toledo J. Global existence for a degenerate nonlinear diffusion problem with nonlinear gradient term and source[J]. Math Annalen, 1999, 314(4): 703-728. doi: 10.1007/s002080050313
    [13]
    Zheng S N, Liu B C. A nonlinear diffusion system with convection[J]. Nonlinear Anal, 2005, 63(1): 123-135. doi: 10.1016/j.na.2005.04.039
    [14]
    Zhou J, Mu C L. Blow-up rate for a porous medium equation with convection[J]. Global J Pure Appl Math,2007, 3(1): 13-18.
    [15]
    Hu B, Yin H M. The profile near blow-up time for solution for the heat equation with a nonlinear boundary conditions[J]. Trans Amer Math Soc, 1994, 346(1): 117-135. doi: 10.1090/S0002-9947-1994-1270664-3
    [16]
    Guo J S, Hu B. Blow-up rate estimates for the heat equation with a nonlinear gradient source term[J]. Discrete Contin Dyn Sys, 2008, 20(4): 927-937. doi: 10.3934/dcds.2008.20.927
    [17]
    Fila M, Lieberman G. Derivative blow-up and beyond for quasilinear parabolic equations[J]. Differential Integral Equations, 1994, 7(3/4): 811-821.
    [18]
    Li Y X, Souplet P H. Single-point gradient blow-up on the boundary for diffusive Hamilton-Jacobi equations in planar domains[J]. Comm Math Phys, 2010, 293(2): 499-517. doi: 10.1007/s00220-009-0936-8
    [19]
    Zhang Z C, Hu B. Rate estimates of gradient blow-up for a heat equation with exponential nonlinearity[J]. Nonlinear Analysis, 2010, 72(12): 4594-4601. doi: 10.1016/j.na.2010.02.036
    [20]
    Zhang Z C, Hu B. Gradient blow-up rate for a semilinear parabolic equation[J]. Discrete and Continuous Dynamical Systems, 2010, 26(2): 767-779.
    [21]
    白占兵. 一类四阶p-Laplace方程正解的存在性及多解性[J]. 应用数学和力学, 2001, 22(12):1324-1328.
    [22]
    Zheng S N, Wang W. Blow-up rate for a nonlinear diffusion equation[J]. Appl Math Lett, 2006, 19(12):1385-1389. doi: 10.1016/j.aml.2006.02.008
    [23]
    杨作东,陆启韶. 一类非牛顿渗流系统爆破界的估计[J]. 应用数学和力学, 2001, 22(3): 287-294.
    [24]
    张正策,李开泰. 奇异扰动的p-Laplace方程非负非平凡解和正解的结构[J]. 应用数学和力学, 2004, 25(8): 847-854.
    [25]
    Jiang Z X, Zheng S N. Blow-up rate for a nonlinear diffusion equation with absorption and nonlinear boundary flux[J]. Adv Math (China), 2004, 33(5): 615-620.
    [26]
    Jiang Z X, Zheng S N, Song X F. Blow-up analysis for a nonlinear diffusion equation with nonlinear boundary conditions[J]. Appl Math Letters, 2004, 17(2): 193-199. doi: 10.1016/S0893-9659(04)90032-8
    [27]
    Chipot M, Fila M, Quitter P. Stationary solutions, blow up and convergence to stationary solutions for semilinear parabolic equations with nonlinear boundary conditions[J]. Acta Math Univ Comenian, 1991, 60(1): 35-103.
    [28]
    Giadas B, Spruck J. A prior bounds positive solutions of nonlinear elliptic equations[J]. Comm Partial Differential Equations, 1981, 6(8): 883-901. doi: 10.1080/03605308108820196
    [29]
    Giga Y, Kohn R V. Asymptotically self-similar blow-up of semilinear heat equations[J]. Comm Pure Appl Math, 1985, 38(3): 297-319. doi: 10.1002/cpa.3160380304
    [30]
    Ladyzenskaya O A, Solonikiv V A, Ural’ceva N N.Linear and Quasilinear Equations of Parabolic Type[M]. Translations of Mathematical Monographs. Rhode Island: Amer Math Soc,1967.
    [31]
    Ziemer W P. Interior and boundary continuity of weak solutions of degenerate parabolic equations[J].Trans Amer Math Soc, 1982, 271(2): 733-748. doi: 10.1090/S0002-9947-1982-0654859-7
    [32]
    Lieberman G M. Second Order Parabolic Differential Equations[M]. River Edge: World Scientific Co, 1996.
    [33]
    Smoller J. Shock Waves and Reaction-Diffusion Equations[M]. New York: Springer-Verlag, 1983.
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