数学物理学报(英文版) ›› 2017, Vol. 37 ›› Issue (2): 301-315.doi: 10.1016/S0252-9602(17)30003-6

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UNIFORM QUASI-DIFFERENTIABILITY OF SEMIGROUP TO NONLINEAR BEACTION-DIFFUSION EQUATIONS WITH SUPERCRITICAL EXPONENT

钟延生1, 孙春友2   

  1. 1. Department of Mathematics, Fujian Normal University, Fuzhou 350117, China;
    2. School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China
  • 收稿日期:2015-01-22 修回日期:2016-09-02 出版日期:2017-04-25 发布日期:2017-04-25
  • 通讯作者: Yansheng ZHONG,E-mail:zhyansheng08@163.com E-mail:zhyansheng08@163.com
  • 作者简介:Yansheng ZHONG,E-mail:zhyansheng08@163.com;Chunyou SUN,E-mail:sunchy@lzu.edu.cn
  • 基金资助:

    Supported by NSFC Grant (11401100, 10601021), the foundation of Fujian Education Department (JB14021), and the innovation foundation of Fujian Normal University (IRTL1206).

UNIFORM QUASI-DIFFERENTIABILITY OF SEMIGROUP TO NONLINEAR BEACTION-DIFFUSION EQUATIONS WITH SUPERCRITICAL EXPONENT

Yansheng ZHONG1, Chunyou SUN2   

  1. 1. Department of Mathematics, Fujian Normal University, Fuzhou 350117, China;
    2. School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China
  • Received:2015-01-22 Revised:2016-09-02 Online:2017-04-25 Published:2017-04-25
  • Contact: Yansheng ZHONG,E-mail:zhyansheng08@163.com E-mail:zhyansheng08@163.com
  • Supported by:

    Supported by NSFC Grant (11401100, 10601021), the foundation of Fujian Education Department (JB14021), and the innovation foundation of Fujian Normal University (IRTL1206).

摘要:

A new approach is established to show that the semigroup {S (t)}t ≥ 0 generated by a reaction-diffusion equation with supercritical exponent is uniformly quasi-differentiable in Lq(Ω) (2 ≤ q < ∞) with respect to the initial value. As an application, this proves the upper-bound of fractal dimension for its global attractor in the corresponding space.

关键词: Uniform quasi-differentiability, semigroup, reaction-diffusion equation

Abstract:

A new approach is established to show that the semigroup {S (t)}t ≥ 0 generated by a reaction-diffusion equation with supercritical exponent is uniformly quasi-differentiable in Lq(Ω) (2 ≤ q < ∞) with respect to the initial value. As an application, this proves the upper-bound of fractal dimension for its global attractor in the corresponding space.

Key words: Uniform quasi-differentiability, semigroup, reaction-diffusion equation