数学物理学报(英文版) ›› 2010, Vol. 30 ›› Issue (3): 713-725.doi: 10.1016/S0252-9602(10)60072-0

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EXISTENCE AND NONEXISTENCE OF GLOBAL POSITIVE SOLUTIONS FOR DEGENERATE PARABOLIC#br# EQUATIONS IN EXTERIOR DOMAINS

曾宪忠1|2, 刘振海2   

  1. 1.Department |of Mathematics and Computational Science,  Hunan University |of Science and Technology, |Xiangtan |411201, China;
    2.Department |of Mathematics, Central South University, Changsha  410083, China
  • 收稿日期:2007-06-05 出版日期:2010-05-20 发布日期:2010-05-20
  • 基金资助:

    This work  was supported by the National Natural Science Foundations of China (10971061) and Hunan Provincial Natural Science Foundation of China (09JJ6013)

EXISTENCE AND NONEXISTENCE OF GLOBAL POSITIVE SOLUTIONS FOR DEGENERATE PARABOLIC#br# EQUATIONS IN EXTERIOR DOMAINS

 ZENG Xian-Zhong1|2, LIU Zhen-Hai2   

  1. 1.Department |of Mathematics and Computational Science,  Hunan University of Science and Technology, |Xiangtan |411201, China;
    2.Department |of Mathematics, Central South University, Changsha  410083, China
  • Received:2007-06-05 Online:2010-05-20 Published:2010-05-20
  • Supported by:

    This work  was supported by the National Natural Science Foundations of China (10971061) and Hunan Provincial Natural Science Foundation of China (09JJ6013)

摘要:

This article deals with the degenerate parabolic equations in exterior domains and with  inhomogeneous Dirichlet boundary conditions. We obtain  that pc= (σ+m)n(n-σ-2) is its critical exponent provided {-1, [(1-m)n-2](n+1)} < σ < n-2. This critical exponent is not the same as that for the corresponding equations with the boundary value 0, but is more closely tied to the critical exponent of the  elliptic type degenerate equations. Furthermore, we demonstrate  that if max{1, σ + m} < p ≤ pc,  then every positive solution of the equations blows  up in finite time; whereas for p < pc, the equations admit global positive solutions for some boundary values  and initial data. Meantime, we also  demonstrate that its positive solutions blow  up in finite time provided n ≤ σ + 2.

关键词: Degenerate parabolic equations, exterior domains, inhomogeneous dirichlet boundary conditions, critical exponent, blow-up, global existence

Abstract:

This article deals with the degenerate parabolic equations in exterior domains and with  inhomogeneous Dirichlet boundary conditions. We obtain  that pc= (σ+m)n(n-σ-2) is its critical exponent provided {-1, [(1-m)n-2](n+1)} < σ < n-2. This critical exponent is not the same as that for the corresponding equations with the boundary value 0, but is more closely tied to the critical exponent of the  elliptic type degenerate equations. Furthermore, we demonstrate  that if max{1, σ + m} < p ≤ pc,  then every positive solution of the equations blows  up in finite time; whereas for p < pc, the equations admit global positive solutions for some boundary values  and initial data. Meantime, we also  demonstrate that its positive solutions blow  up in finite time provided n ≤ σ + 2.

Key words: Degenerate parabolic equations, exterior domains, inhomogeneous dirichlet boundary conditions, critical exponent, blow-up, global existence

中图分类号: 

  • 35B33