EXISTENCE AND ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO A SINGULAR PARABOLIC EQUATION

doi:10.1016/S0252-9602(12)60146-5

数学物理学报(英文版) ›› 2012, Vol. 32 ›› Issue (5): 1875-1882.doi: 10.1016/S0252-9602(12)60146-5

EXISTENCE AND ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO A SINGULAR PARABOLIC EQUATION

夏莉¹|李敬娜²|姚正安³

1. Department of Mathematics, Shenzhen University, Shenzhen 518060, China；
2. Department of Mathematics, Jinan University, Guangzhou 510632, China；
3. Department of Mathematics, Sun Yat-Sen University, Guangzhou 510275, China

收稿日期:2011-03-01 修回日期:2011-07-13 出版日期:2012-09-20 发布日期:2012-09-20
基金资助:
Supported by Natural Science Foundation of Youth and Tianyuan (11001177, 11026156, 10926141) and Startup Program of Shenzhen University.

EXISTENCE AND ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO A SINGULAR PARABOLIC EQUATION

XIA Li¹, LI Jing-Na², YAO Zheng-An³

1. Department of Mathematics, Shenzhen University, Shenzhen 518060, China；
2. Department of Mathematics, Jinan University, Guangzhou 510632, China；
3. Department of Mathematics, Sun Yat-Sen University, Guangzhou 510275, China

Received:2011-03-01 Revised:2011-07-13 Online:2012-09-20 Published:2012-09-20
Supported by:
Supported by Natural Science Foundation of Youth and Tianyuan (11001177, 11026156, 10926141) and Startup Program of Shenzhen University.

摘要/Abstract

摘要：

In this paper, we study the initial-boundary value problem for a class of sin-gular parabolic equations. Under some conditions, we obtain the existence and asymptotic behavior of solutions to the problem by parabolic regularization method and the sub-super solutions method. As a byproduct, we prove the existence of solutions to some problems with gradient terms, which blow up on the boundary.

关键词: existence, singular parabolic equation, asymptotic behavior

Abstract:

Key words: existence, singular parabolic equation, asymptotic behavior

中图分类号:

35K15

夏莉, 李敬娜, 姚正安. EXISTENCE AND ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO A SINGULAR PARABOLIC EQUATION[J]. 数学物理学报(英文版), 2012, 32(5): 1875-1882.

XIA Li, LI Jing-Na, YAO Zheng-An. EXISTENCE AND ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO A SINGULAR PARABOLIC EQUATION[J]. Acta mathematica scientia,Series B, 2012, 32(5): 1875-1882.

参考文献

[1] Bandle C, Brunner H. Blowup in diffusion equations: A survey. J Comput Appl Math, 1998, 97: 3–22

[2] Chen S H. Global existence and blowup of solutions for a parabolic equation with a gradient term. Proc Amer Math Soc, 2001, 129: 975–981

[3] Salin T. Quenching and blowup problems for reaction diffusion equations [D]. Espoo: Helsinki Univ Techn, 2004

[4] Abdellaoui B, Aglio A D, Peral I. Regularity and nonuniqueness results for parabolic problems arising in some physical models, having natural growth in the gradient. J Math Pures Appl, 2008, 90: 242–269

[5] Salin T. On quenching with logarithmic singularity. Nonlinear Anal, 2003, 52: 261–289

[6] D´avila J, Montenegro M. Existence and asymptotic behavior for a singular parabolic equation. Trans Amer Math Soc, 2005, 357: 1801–1828

[7] Winkler M. Nonuniqueness in the quenching problem. Math Ann, 2007, 339: 559–597

[8] Xia L, Yao Z A. Existence, uniqueness and asymptotic behavior of solutions for a singular parabolic equation. J Math Anal Appl, 2009, 358: 182–188

[9] Zhou W S, Lei P D. A one-dimensional nonlinear heat equation with a singular term. J Math Anal Appl, 2010, 368: 711–726

[10] BenArtzi M, Goodman J, Levy A. Remarks on a nonlinear parabolic equation. Trans Amer Math Soc, 2000, 352: 731–751

[11] Gilding B H. The Cauchy problem for ut = u + |∇u|q, large-time behaviour. J Math Pures Appl, 2005, 84: 753–785

[12] Li Y X, Souplet P. Single-point gradient blow-up on the boundary for diffusive Hamilton-Jacobi equations in planar domains. Comm Math Phys, 2010, 293: 499–517

[13] Yin J X, Jin C H. Critical exponents and traveling wavefronts of a degenerate-singular parabolic equation in non-divergence form. Discrete Contin Dyn Syst Ser B, 2010, 13: 213–227

[14] Xu Y J, Liu Z H. Controllability for a parabolic equation with a nonlinear term involving the state and the gradient. Acta Math Sci, 2010, 30B(5): 1593–1604

[15] Zhang Z J. Nonlinear elliptic equations with singular boundary conditions. J Math Anal Appl, 1997, 216: 390–397

[16] Lieberman G M. Second Order Parabolic Differential Equations. Beijing: World Scientific Publishing Co Pte Ltd, 2003

[17] Ladyzenskaja O A, Solonnikov V A, Ural´ceva N N. Linear and Quasi-Linear Equations of Parabolic Type. Providence RI: Amer Math Soc, 1968

EXISTENCE AND ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO A SINGULAR PARABOLIC EQUATION

EXISTENCE AND ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO A SINGULAR PARABOLIC EQUATION

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